F5.qti

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  <objectbank ident="F5">
    <qtimetadata>
      <qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- F5</fieldentry></qtimetadatafield>
    </qtimetadata>
  <item ident="F5-0946" title="F5 | Substitution strategies | ver. 0946"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t - y + {y'} = -\frac{2}{t + y + 2} + 3" alt="-t - y + {y'} = -\frac{2}{t + y + 2} + 3" title="-t - y + {y'} = -\frac{2}{t + y + 2} + 3" data-latex="-t - y + {y'} = -\frac{2}{t + y + 2} + 3"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} - 4 \, t^{2} + y^{2} = 0" alt="t y {y'} - 4 \, t^{2} + y^{2} = 0" title="t y {y'} - 4 \, t^{2} + y^{2} = 0" data-latex="t y {y'} - 4 \, t^{2} + y^{2} = 0"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20-%20y%20+%20%7By'%7D%20=%20-%5Cfrac%7B2%7D%7Bt%20+%20y%20+%202%7D%20+%203" alt="-t - y + {y'} = -\frac{2}{t + y + 2} + 3" title="-t - y + {y'} = -\frac{2}{t + y + 2} + 3" data-latex="-t - y + {y'} = -\frac{2}{t + y + 2} + 3"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20-%204%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D%20=%200" alt="t y {y'} - 4 \, t^{2} + y^{2} = 0" title="t y {y'} - 4 \, t^{2} + y^{2} = 0" data-latex="t y {y'} - 4 \, t^{2} + y^{2} = 0"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = 1" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-7987" title="F5 | Substitution strategies | ver. 7987"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + \frac{3}{3 \, t - y + 2} + 5 = -3 \, t + y" alt="{y'} + \frac{3}{3 \, t - y + 2} + 5 = -3 \, t + y" title="{y'} + \frac{3}{3 \, t - y + 2} + 5 = -3 \, t + y" data-latex="{y'} + \frac{3}{3 \, t - y + 2} + 5 = -3 \, t + y"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, t^{2} = -t y {y'} - y^{2}" alt="-2 \, t^{2} = -t y {y'} - y^{2}" title="-2 \, t^{2} = -t y {y'} - y^{2}" data-latex="-2 \, t^{2} = -t y {y'} - y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%20%5Cfrac%7B3%7D%7B3%20%5C,%20t%20-%20y%20+%202%7D%20+%205%20=%20-3%20%5C,%20t%20+%20y" alt="{y'} + \frac{3}{3 \, t - y + 2} + 5 = -3 \, t + y" title="{y'} + \frac{3}{3 \, t - y + 2} + 5 = -3 \, t + y" data-latex="{y'} + \frac{3}{3 \, t - y + 2} + 5 = -3 \, t + y"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20t%5E%7B2%7D%20=%20-t%20y%20%7By'%7D%20-%20y%5E%7B2%7D" alt="-2 \, t^{2} = -t y {y'} - y^{2}" title="-2 \, t^{2} = -t y {y'} - y^{2}" data-latex="-2 \, t^{2} = -t y {y'} - y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = 1" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-9524" title="F5 | Substitution strategies | ver. 9524"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, t - {y'} = -y - \frac{3}{3 \, t - y - 3}" alt="-3 \, t - {y'} = -y - \frac{3}{3 \, t - y - 3}" title="-3 \, t - {y'} = -y - \frac{3}{3 \, t - y - 3}" data-latex="-3 \, t - {y'} = -y - \frac{3}{3 \, t - y - 3}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} = -t y {y'} - y^{2}" alt="t^{2} = -t y {y'} - y^{2}" title="t^{2} = -t y {y'} - y^{2}" data-latex="t^{2} = -t y {y'} - y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20t%20-%20%7By'%7D%20=%20-y%20-%20%5Cfrac%7B3%7D%7B3%20%5C,%20t%20-%20y%20-%203%7D" alt="-3 \, t - {y'} = -y - \frac{3}{3 \, t - y - 3}" title="-3 \, t - {y'} = -y - \frac{3}{3 \, t - y - 3}" data-latex="-3 \, t - {y'} = -y - \frac{3}{3 \, t - y - 3}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%5E%7B2%7D%20=%20-t%20y%20%7By'%7D%20-%20y%5E%7B2%7D" alt="t^{2} = -t y {y'} - y^{2}" title="t^{2} = -t y {y'} - y^{2}" data-latex="t^{2} = -t y {y'} - y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = 1" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-6626" title="F5 | Substitution strategies | ver. 6626"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - 3 \, t^{2} - y^{2} = 0" alt="-t y {y'} - 3 \, t^{2} - y^{2} = 0" title="-t y {y'} - 3 \, t^{2} - y^{2} = 0" data-latex="-t y {y'} - 3 \, t^{2} - y^{2} = 0"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, t - \frac{2}{3 \, t + y - 2} - 1 = y - {y'}" alt="-3 \, t - \frac{2}{3 \, t + y - 2} - 1 = y - {y'}" title="-3 \, t - \frac{2}{3 \, t + y - 2} - 1 = y - {y'}" data-latex="-3 \, t - \frac{2}{3 \, t + y - 2} - 1 = y - {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%203%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D%20=%200" alt="-t y {y'} - 3 \, t^{2} - y^{2} = 0" title="-t y {y'} - 3 \, t^{2} - y^{2} = 0" data-latex="-t y {y'} - 3 \, t^{2} - y^{2} = 0"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20t%20-%20%5Cfrac%7B2%7D%7B3%20%5C,%20t%20+%20y%20-%202%7D%20-%201%20=%20y%20-%20%7By'%7D" alt="-3 \, t - \frac{2}{3 \, t + y - 2} - 1 = y - {y'}" title="-3 \, t - \frac{2}{3 \, t + y - 2} - 1 = y - {y'}" data-latex="-3 \, t - \frac{2}{3 \, t + y - 2} - 1 = y - {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = 1" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-9025" title="F5 | Substitution strategies | ver. 9025"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - 5 \, t^{2} = y^{2}" alt="-t y {y'} - 5 \, t^{2} = y^{2}" title="-t y {y'} - 5 \, t^{2} = y^{2}" data-latex="-t y {y'} - 5 \, t^{2} = y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y = -4 \, t + {y'} + \frac{3}{4 \, t + y + 3} - 7" alt="y = -4 \, t + {y'} + \frac{3}{4 \, t + y + 3} - 7" title="y = -4 \, t + {y'} + \frac{3}{4 \, t + y + 3} - 7" data-latex="y = -4 \, t + {y'} + \frac{3}{4 \, t + y + 3} - 7"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%205%20%5C,%20t%5E%7B2%7D%20=%20y%5E%7B2%7D" alt="-t y {y'} - 5 \, t^{2} = y^{2}" title="-t y {y'} - 5 \, t^{2} = y^{2}" data-latex="-t y {y'} - 5 \, t^{2} = y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y%20=%20-4%20%5C,%20t%20+%20%7By'%7D%20+%20%5Cfrac%7B3%7D%7B4%20%5C,%20t%20+%20y%20+%203%7D%20-%207" alt="y = -4 \, t + {y'} + \frac{3}{4 \, t + y + 3} - 7" title="y = -4 \, t + {y'} + \frac{3}{4 \, t + y + 3} - 7" data-latex="y = -4 \, t + {y'} + \frac{3}{4 \, t + y + 3} - 7"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = 1" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-4033" title="F5 | Substitution strategies | ver. 4033"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} + 2 \, t^{2} = y^{2}" alt="-t y {y'} + 2 \, t^{2} = y^{2}" title="-t y {y'} + 2 \, t^{2} = y^{2}" data-latex="-t y {y'} + 2 \, t^{2} = y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y - {y'} - 4 = 2 \, t - \frac{4}{2 \, t - y + 2}" alt="y - {y'} - 4 = 2 \, t - \frac{4}{2 \, t - y + 2}" title="y - {y'} - 4 = 2 \, t - \frac{4}{2 \, t - y + 2}" data-latex="y - {y'} - 4 = 2 \, t - \frac{4}{2 \, t - y + 2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20+%202%20%5C,%20t%5E%7B2%7D%20=%20y%5E%7B2%7D" alt="-t y {y'} + 2 \, t^{2} = y^{2}" title="-t y {y'} + 2 \, t^{2} = y^{2}" data-latex="-t y {y'} + 2 \, t^{2} = y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y%20-%20%7By'%7D%20-%204%20=%202%20%5C,%20t%20-%20%5Cfrac%7B4%7D%7B2%20%5C,%20t%20-%20y%20+%202%7D" alt="y - {y'} - 4 = 2 \, t - \frac{4}{2 \, t - y + 2}" title="y - {y'} - 4 = 2 \, t - \frac{4}{2 \, t - y + 2}" data-latex="y - {y'} - 4 = 2 \, t - \frac{4}{2 \, t - y + 2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = 1" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-7548" title="F5 | Substitution strategies | ver. 7548"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, t + {y'} = y - \frac{2}{3 \, t + y - 5} - 2" alt="-3 \, t + {y'} = y - \frac{2}{3 \, t + y - 5} - 2" title="-3 \, t + {y'} = y - \frac{2}{3 \, t + y - 5} - 2" data-latex="-3 \, t + {y'} = y - \frac{2}{3 \, t + y - 5} - 2"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - 2 \, t^{2} - y^{2} = 0" alt="-t y {y'} - 2 \, t^{2} - y^{2} = 0" title="-t y {y'} - 2 \, t^{2} - y^{2} = 0" data-latex="-t y {y'} - 2 \, t^{2} - y^{2} = 0"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20t%20+%20%7By'%7D%20=%20y%20-%20%5Cfrac%7B2%7D%7B3%20%5C,%20t%20+%20y%20-%205%7D%20-%202" alt="-3 \, t + {y'} = y - \frac{2}{3 \, t + y - 5} - 2" title="-3 \, t + {y'} = y - \frac{2}{3 \, t + y - 5} - 2" data-latex="-3 \, t + {y'} = y - \frac{2}{3 \, t + y - 5} - 2"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%202%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D%20=%200" alt="-t y {y'} - 2 \, t^{2} - y^{2} = 0" title="-t y {y'} - 2 \, t^{2} - y^{2} = 0" data-latex="-t y {y'} - 2 \, t^{2} - y^{2} = 0"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = 1" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-2212" title="F5 | Substitution strategies | ver. 2212"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t - {y'} - 1 = -y + \frac{1}{4 \, t - y - 3}" alt="-4 \, t - {y'} - 1 = -y + \frac{1}{4 \, t - y - 3}" title="-4 \, t - {y'} - 1 = -y + \frac{1}{4 \, t - y - 3}" data-latex="-4 \, t - {y'} - 1 = -y + \frac{1}{4 \, t - y - 3}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} + 5 \, t^{2} = -y^{2}" alt="t y {y'} + 5 \, t^{2} = -y^{2}" title="t y {y'} + 5 \, t^{2} = -y^{2}" data-latex="t y {y'} + 5 \, t^{2} = -y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%20-%20%7By'%7D%20-%201%20=%20-y%20+%20%5Cfrac%7B1%7D%7B4%20%5C,%20t%20-%20y%20-%203%7D" alt="-4 \, t - {y'} - 1 = -y + \frac{1}{4 \, t - y - 3}" title="-4 \, t - {y'} - 1 = -y + \frac{1}{4 \, t - y - 3}" data-latex="-4 \, t - {y'} - 1 = -y + \frac{1}{4 \, t - y - 3}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20+%205%20%5C,%20t%5E%7B2%7D%20=%20-y%5E%7B2%7D" alt="t y {y'} + 5 \, t^{2} = -y^{2}" title="t y {y'} + 5 \, t^{2} = -y^{2}" data-latex="t y {y'} + 5 \, t^{2} = -y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = 1" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-6868" title="F5 | Substitution strategies | ver. 6868"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, t^{2} = t y {y'} + y^{2}" alt="-5 \, t^{2} = t y {y'} + y^{2}" title="-5 \, t^{2} = t y {y'} + y^{2}" data-latex="-5 \, t^{2} = t y {y'} + y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, t - {y'} = -y - \frac{3}{2 \, t - y - 2}" alt="-2 \, t - {y'} = -y - \frac{3}{2 \, t - y - 2}" title="-2 \, t - {y'} = -y - \frac{3}{2 \, t - y - 2}" data-latex="-2 \, t - {y'} = -y - \frac{3}{2 \, t - y - 2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5%20%5C,%20t%5E%7B2%7D%20=%20t%20y%20%7By'%7D%20+%20y%5E%7B2%7D" alt="-5 \, t^{2} = t y {y'} + y^{2}" title="-5 \, t^{2} = t y {y'} + y^{2}" data-latex="-5 \, t^{2} = t y {y'} + y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20t%20-%20%7By'%7D%20=%20-y%20-%20%5Cfrac%7B3%7D%7B2%20%5C,%20t%20-%20y%20-%202%7D" alt="-2 \, t - {y'} = -y - \frac{3}{2 \, t - y - 2}" title="-2 \, t - {y'} = -y - \frac{3}{2 \, t - y - 2}" data-latex="-2 \, t - {y'} = -y - \frac{3}{2 \, t - y - 2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = 1" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-3207" title="F5 | Substitution strategies | ver. 3207"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = t y {y'} + 5 \, t^{2} + y^{2}" alt="0 = t y {y'} + 5 \, t^{2} + y^{2}" title="0 = t y {y'} + 5 \, t^{2} + y^{2}" data-latex="0 = t y {y'} + 5 \, t^{2} + y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y + {y'} - 7 = 5 \, t - \frac{4}{5 \, t + y + 2}" alt="-y + {y'} - 7 = 5 \, t - \frac{4}{5 \, t + y + 2}" title="-y + {y'} - 7 = 5 \, t - \frac{4}{5 \, t + y + 2}" data-latex="-y + {y'} - 7 = 5 \, t - \frac{4}{5 \, t + y + 2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20t%20y%20%7By'%7D%20+%205%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="0 = t y {y'} + 5 \, t^{2} + y^{2}" title="0 = t y {y'} + 5 \, t^{2} + y^{2}" data-latex="0 = t y {y'} + 5 \, t^{2} + y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20+%20%7By'%7D%20-%207%20=%205%20%5C,%20t%20-%20%5Cfrac%7B4%7D%7B5%20%5C,%20t%20+%20y%20+%202%7D" alt="-y + {y'} - 7 = 5 \, t - \frac{4}{5 \, t + y + 2}" title="-y + {y'} - 7 = 5 \, t - \frac{4}{5 \, t + y + 2}" data-latex="-y + {y'} - 7 = 5 \, t - \frac{4}{5 \, t + y + 2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = 1" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-2778" title="F5 | Substitution strategies | ver. 2778"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} - 4 \, t^{2} + y^{2} = 0" alt="t y {y'} - 4 \, t^{2} + y^{2} = 0" title="t y {y'} - 4 \, t^{2} + y^{2} = 0" data-latex="t y {y'} - 4 \, t^{2} + y^{2} = 0"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y - \frac{4}{4 \, t - y + 5} - 9 = 4 \, t + {y'}" alt="y - \frac{4}{4 \, t - y + 5} - 9 = 4 \, t + {y'}" title="y - \frac{4}{4 \, t - y + 5} - 9 = 4 \, t + {y'}" data-latex="y - \frac{4}{4 \, t - y + 5} - 9 = 4 \, t + {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20-%204%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D%20=%200" alt="t y {y'} - 4 \, t^{2} + y^{2} = 0" title="t y {y'} - 4 \, t^{2} + y^{2} = 0" data-latex="t y {y'} - 4 \, t^{2} + y^{2} = 0"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y%20-%20%5Cfrac%7B4%7D%7B4%20%5C,%20t%20-%20y%20+%205%7D%20-%209%20=%204%20%5C,%20t%20+%20%7By'%7D" alt="y - \frac{4}{4 \, t - y + 5} - 9 = 4 \, t + {y'}" title="y - \frac{4}{4 \, t - y + 5} - 9 = 4 \, t + {y'}" data-latex="y - \frac{4}{4 \, t - y + 5} - 9 = 4 \, t + {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = 1" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-7332" title="F5 | Substitution strategies | ver. 7332"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -t y {y'} - 3 \, t^{2} - y^{2}" alt="0 = -t y {y'} - 3 \, t^{2} - y^{2}" title="0 = -t y {y'} - 3 \, t^{2} - y^{2}" data-latex="0 = -t y {y'} - 3 \, t^{2} - y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + \frac{2}{2 \, t + y + 2} = 2 \, t + y + 4" alt="{y'} + \frac{2}{2 \, t + y + 2} = 2 \, t + y + 4" title="{y'} + \frac{2}{2 \, t + y + 2} = 2 \, t + y + 4" data-latex="{y'} + \frac{2}{2 \, t + y + 2} = 2 \, t + y + 4"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-t%20y%20%7By'%7D%20-%203%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D" alt="0 = -t y {y'} - 3 \, t^{2} - y^{2}" title="0 = -t y {y'} - 3 \, t^{2} - y^{2}" data-latex="0 = -t y {y'} - 3 \, t^{2} - y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%20%5Cfrac%7B2%7D%7B2%20%5C,%20t%20+%20y%20+%202%7D%20=%202%20%5C,%20t%20+%20y%20+%204" alt="{y'} + \frac{2}{2 \, t + y + 2} = 2 \, t + y + 4" title="{y'} + \frac{2}{2 \, t + y + 2} = 2 \, t + y + 4" data-latex="{y'} + \frac{2}{2 \, t + y + 2} = 2 \, t + y + 4"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = 1" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8975" title="F5 | Substitution strategies | ver. 8975"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-\frac{3}{3 \, t - y + 3} = -3 \, t + y - {y'} - 6" alt="-\frac{3}{3 \, t - y + 3} = -3 \, t + y - {y'} - 6" title="-\frac{3}{3 \, t - y + 3} = -3 \, t + y - {y'} - 6" data-latex="-\frac{3}{3 \, t - y + 3} = -3 \, t + y - {y'} - 6"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} = -3 \, t^{2} - y^{2}" alt="t y {y'} = -3 \, t^{2} - y^{2}" title="t y {y'} = -3 \, t^{2} - y^{2}" data-latex="t y {y'} = -3 \, t^{2} - y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%5Cfrac%7B3%7D%7B3%20%5C,%20t%20-%20y%20+%203%7D%20=%20-3%20%5C,%20t%20+%20y%20-%20%7By'%7D%20-%206" alt="-\frac{3}{3 \, t - y + 3} = -3 \, t + y - {y'} - 6" title="-\frac{3}{3 \, t - y + 3} = -3 \, t + y - {y'} - 6" data-latex="-\frac{3}{3 \, t - y + 3} = -3 \, t + y - {y'} - 6"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20=%20-3%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D" alt="t y {y'} = -3 \, t^{2} - y^{2}" title="t y {y'} = -3 \, t^{2} - y^{2}" data-latex="t y {y'} = -3 \, t^{2} - y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = 1" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-6582" title="F5 | Substitution strategies | ver. 6582"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t + \frac{5}{4 \, t + y - 5} = y - {y'} - 1" alt="-4 \, t + \frac{5}{4 \, t + y - 5} = y - {y'} - 1" title="-4 \, t + \frac{5}{4 \, t + y - 5} = y - {y'} - 1" data-latex="-4 \, t + \frac{5}{4 \, t + y - 5} = y - {y'} - 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} + 4 \, t^{2} = y^{2}" alt="-t y {y'} + 4 \, t^{2} = y^{2}" title="-t y {y'} + 4 \, t^{2} = y^{2}" data-latex="-t y {y'} + 4 \, t^{2} = y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%20+%20%5Cfrac%7B5%7D%7B4%20%5C,%20t%20+%20y%20-%205%7D%20=%20y%20-%20%7By'%7D%20-%201" alt="-4 \, t + \frac{5}{4 \, t + y - 5} = y - {y'} - 1" title="-4 \, t + \frac{5}{4 \, t + y - 5} = y - {y'} - 1" data-latex="-4 \, t + \frac{5}{4 \, t + y - 5} = y - {y'} - 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20+%204%20%5C,%20t%5E%7B2%7D%20=%20y%5E%7B2%7D" alt="-t y {y'} + 4 \, t^{2} = y^{2}" title="-t y {y'} + 4 \, t^{2} = y^{2}" data-latex="-t y {y'} + 4 \, t^{2} = y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = 1" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8887" title="F5 | Substitution strategies | ver. 8887"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} = -t y {y'} - y^{2}" alt="-4 \, t^{2} = -t y {y'} - y^{2}" title="-4 \, t^{2} = -t y {y'} - y^{2}" data-latex="-4 \, t^{2} = -t y {y'} - y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t + \frac{2}{4 \, t + y - 3} - 1 = y - {y'}" alt="-4 \, t + \frac{2}{4 \, t + y - 3} - 1 = y - {y'}" title="-4 \, t + \frac{2}{4 \, t + y - 3} - 1 = y - {y'}" data-latex="-4 \, t + \frac{2}{4 \, t + y - 3} - 1 = y - {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%5E%7B2%7D%20=%20-t%20y%20%7By'%7D%20-%20y%5E%7B2%7D" alt="-4 \, t^{2} = -t y {y'} - y^{2}" title="-4 \, t^{2} = -t y {y'} - y^{2}" data-latex="-4 \, t^{2} = -t y {y'} - y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%20+%20%5Cfrac%7B2%7D%7B4%20%5C,%20t%20+%20y%20-%203%7D%20-%201%20=%20y%20-%20%7By'%7D" alt="-4 \, t + \frac{2}{4 \, t + y - 3} - 1 = y - {y'}" title="-4 \, t + \frac{2}{4 \, t + y - 3} - 1 = y - {y'}" data-latex="-4 \, t + \frac{2}{4 \, t + y - 3} - 1 = y - {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = 1" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-3405" title="F5 | Substitution strategies | ver. 3405"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t + y - {y'} - \frac{1}{4 \, t - y - 1} - 3 = 0" alt="-4 \, t + y - {y'} - \frac{1}{4 \, t - y - 1} - 3 = 0" title="-4 \, t + y - {y'} - \frac{1}{4 \, t - y - 1} - 3 = 0" data-latex="-4 \, t + y - {y'} - \frac{1}{4 \, t - y - 1} - 3 = 0"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, t^{2} + y^{2} = -t y {y'}" alt="-2 \, t^{2} + y^{2} = -t y {y'}" title="-2 \, t^{2} + y^{2} = -t y {y'}" data-latex="-2 \, t^{2} + y^{2} = -t y {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%20+%20y%20-%20%7By'%7D%20-%20%5Cfrac%7B1%7D%7B4%20%5C,%20t%20-%20y%20-%201%7D%20-%203%20=%200" alt="-4 \, t + y - {y'} - \frac{1}{4 \, t - y - 1} - 3 = 0" title="-4 \, t + y - {y'} - \frac{1}{4 \, t - y - 1} - 3 = 0" data-latex="-4 \, t + y - {y'} - \frac{1}{4 \, t - y - 1} - 3 = 0"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D%20=%20-t%20y%20%7By'%7D" alt="-2 \, t^{2} + y^{2} = -t y {y'}" title="-2 \, t^{2} + y^{2} = -t y {y'}" data-latex="-2 \, t^{2} + y^{2} = -t y {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = 1" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-9422" title="F5 | Substitution strategies | ver. 9422"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, t^{2} + y^{2} = -t y {y'}" alt="5 \, t^{2} + y^{2} = -t y {y'}" title="5 \, t^{2} + y^{2} = -t y {y'}" data-latex="5 \, t^{2} + y^{2} = -t y {y'}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, t - 9 = -y + {y'} - \frac{3}{5 \, t - y + 4}" alt="-5 \, t - 9 = -y + {y'} - \frac{3}{5 \, t - y + 4}" title="-5 \, t - 9 = -y + {y'} - \frac{3}{5 \, t - y + 4}" data-latex="-5 \, t - 9 = -y + {y'} - \frac{3}{5 \, t - y + 4}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D%20=%20-t%20y%20%7By'%7D" alt="5 \, t^{2} + y^{2} = -t y {y'}" title="5 \, t^{2} + y^{2} = -t y {y'}" data-latex="5 \, t^{2} + y^{2} = -t y {y'}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5%20%5C,%20t%20-%209%20=%20-y%20+%20%7By'%7D%20-%20%5Cfrac%7B3%7D%7B5%20%5C,%20t%20-%20y%20+%204%7D" alt="-5 \, t - 9 = -y + {y'} - \frac{3}{5 \, t - y + 4}" title="-5 \, t - 9 = -y + {y'} - \frac{3}{5 \, t - y + 4}" data-latex="-5 \, t - 9 = -y + {y'} - \frac{3}{5 \, t - y + 4}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = 1" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-0011" title="F5 | Substitution strategies | ver. 0011"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t - \frac{2}{4 \, t + y - 5} + 1 = y - {y'}" alt="-4 \, t - \frac{2}{4 \, t + y - 5} + 1 = y - {y'}" title="-4 \, t - \frac{2}{4 \, t + y - 5} + 1 = y - {y'}" data-latex="-4 \, t - \frac{2}{4 \, t + y - 5} + 1 = y - {y'}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} - y^{2} = t y {y'}" alt="4 \, t^{2} - y^{2} = t y {y'}" title="4 \, t^{2} - y^{2} = t y {y'}" data-latex="4 \, t^{2} - y^{2} = t y {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%20-%20%5Cfrac%7B2%7D%7B4%20%5C,%20t%20+%20y%20-%205%7D%20+%201%20=%20y%20-%20%7By'%7D" alt="-4 \, t - \frac{2}{4 \, t + y - 5} + 1 = y - {y'}" title="-4 \, t - \frac{2}{4 \, t + y - 5} + 1 = y - {y'}" data-latex="-4 \, t - \frac{2}{4 \, t + y - 5} + 1 = y - {y'}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D%20=%20t%20y%20%7By'%7D" alt="4 \, t^{2} - y^{2} = t y {y'}" title="4 \, t^{2} - y^{2} = t y {y'}" data-latex="4 \, t^{2} - y^{2} = t y {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = 1" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8123" title="F5 | Substitution strategies | ver. 8123"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t - y - \frac{1}{4 \, t - y - 5} = -{y'} + 1" alt="4 \, t - y - \frac{1}{4 \, t - y - 5} = -{y'} + 1" title="4 \, t - y - \frac{1}{4 \, t - y - 5} = -{y'} + 1" data-latex="4 \, t - y - \frac{1}{4 \, t - y - 5} = -{y'} + 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} + 5 \, t^{2} = y^{2}" alt="-t y {y'} + 5 \, t^{2} = y^{2}" title="-t y {y'} + 5 \, t^{2} = y^{2}" data-latex="-t y {y'} + 5 \, t^{2} = y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20t%20-%20y%20-%20%5Cfrac%7B1%7D%7B4%20%5C,%20t%20-%20y%20-%205%7D%20=%20-%7By'%7D%20+%201" alt="4 \, t - y - \frac{1}{4 \, t - y - 5} = -{y'} + 1" title="4 \, t - y - \frac{1}{4 \, t - y - 5} = -{y'} + 1" data-latex="4 \, t - y - \frac{1}{4 \, t - y - 5} = -{y'} + 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20+%205%20%5C,%20t%5E%7B2%7D%20=%20y%5E%7B2%7D" alt="-t y {y'} + 5 \, t^{2} = y^{2}" title="-t y {y'} + 5 \, t^{2} = y^{2}" data-latex="-t y {y'} + 5 \, t^{2} = y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 1} = 1" alt="\frac{u {u'}}{u^{2} - 1} = 1" title="\frac{u {u'}}{u^{2} - 1} = 1" data-latex="\frac{u {u'}}{u^{2} - 1} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 1} = 1" title="\frac{u {u'}}{u^{2} - 1} = 1" data-latex="\frac{u {u'}}{u^{2} - 1} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-5219" title="F5 | Substitution strategies | ver. 5219"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} = -5 \, t^{2} + y^{2}" alt="-t y {y'} = -5 \, t^{2} + y^{2}" title="-t y {y'} = -5 \, t^{2} + y^{2}" data-latex="-t y {y'} = -5 \, t^{2} + y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y + {y'} - 3 = 2 \, t + \frac{4}{2 \, t + y + 1}" alt="-y + {y'} - 3 = 2 \, t + \frac{4}{2 \, t + y + 1}" title="-y + {y'} - 3 = 2 \, t + \frac{4}{2 \, t + y + 1}" data-latex="-y + {y'} - 3 = 2 \, t + \frac{4}{2 \, t + y + 1}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20=%20-5%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="-t y {y'} = -5 \, t^{2} + y^{2}" title="-t y {y'} = -5 \, t^{2} + y^{2}" data-latex="-t y {y'} = -5 \, t^{2} + y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20+%20%7By'%7D%20-%203%20=%202%20%5C,%20t%20+%20%5Cfrac%7B4%7D%7B2%20%5C,%20t%20+%20y%20+%201%7D" alt="-y + {y'} - 3 = 2 \, t + \frac{4}{2 \, t + y + 1}" title="-y + {y'} - 3 = 2 \, t + \frac{4}{2 \, t + y + 1}" data-latex="-y + {y'} - 3 = 2 \, t + \frac{4}{2 \, t + y + 1}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = 1" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8381" title="F5 | Substitution strategies | ver. 8381"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-\frac{3}{3 \, t - y + 3} - 6 = 3 \, t - y + {y'}" alt="-\frac{3}{3 \, t - y + 3} - 6 = 3 \, t - y + {y'}" title="-\frac{3}{3 \, t - y + 3} - 6 = 3 \, t - y + {y'}" data-latex="-\frac{3}{3 \, t - y + 3} - 6 = 3 \, t - y + {y'}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} + 5 \, t^{2} = -y^{2}" alt="t y {y'} + 5 \, t^{2} = -y^{2}" title="t y {y'} + 5 \, t^{2} = -y^{2}" data-latex="t y {y'} + 5 \, t^{2} = -y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%5Cfrac%7B3%7D%7B3%20%5C,%20t%20-%20y%20+%203%7D%20-%206%20=%203%20%5C,%20t%20-%20y%20+%20%7By'%7D" alt="-\frac{3}{3 \, t - y + 3} - 6 = 3 \, t - y + {y'}" title="-\frac{3}{3 \, t - y + 3} - 6 = 3 \, t - y + {y'}" data-latex="-\frac{3}{3 \, t - y + 3} - 6 = 3 \, t - y + {y'}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20+%205%20%5C,%20t%5E%7B2%7D%20=%20-y%5E%7B2%7D" alt="t y {y'} + 5 \, t^{2} = -y^{2}" title="t y {y'} + 5 \, t^{2} = -y^{2}" data-latex="t y {y'} + 5 \, t^{2} = -y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = 1" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-9077" title="F5 | Substitution strategies | ver. 9077"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, t + y - {y'} - \frac{3}{2 \, t - y + 2} - 4 = 0" alt="-2 \, t + y - {y'} - \frac{3}{2 \, t - y + 2} - 4 = 0" title="-2 \, t + y - {y'} - \frac{3}{2 \, t - y + 2} - 4 = 0" data-latex="-2 \, t + y - {y'} - \frac{3}{2 \, t - y + 2} - 4 = 0"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, t^{2} - y^{2} = t y {y'}" alt="-3 \, t^{2} - y^{2} = t y {y'}" title="-3 \, t^{2} - y^{2} = t y {y'}" data-latex="-3 \, t^{2} - y^{2} = t y {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20t%20+%20y%20-%20%7By'%7D%20-%20%5Cfrac%7B3%7D%7B2%20%5C,%20t%20-%20y%20+%202%7D%20-%204%20=%200" alt="-2 \, t + y - {y'} - \frac{3}{2 \, t - y + 2} - 4 = 0" title="-2 \, t + y - {y'} - \frac{3}{2 \, t - y + 2} - 4 = 0" data-latex="-2 \, t + y - {y'} - \frac{3}{2 \, t - y + 2} - 4 = 0"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D%20=%20t%20y%20%7By'%7D" alt="-3 \, t^{2} - y^{2} = t y {y'}" title="-3 \, t^{2} - y^{2} = t y {y'}" data-latex="-3 \, t^{2} - y^{2} = t y {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = 1" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-7528" title="F5 | Substitution strategies | ver. 7528"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y + \frac{5}{5 \, t + y + 5} = 5 \, t - {y'} + 10" alt="-y + \frac{5}{5 \, t + y + 5} = 5 \, t - {y'} + 10" title="-y + \frac{5}{5 \, t + y + 5} = 5 \, t - {y'} + 10" data-latex="-y + \frac{5}{5 \, t + y + 5} = 5 \, t - {y'} + 10"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} + t^{2} = -y^{2}" alt="t y {y'} + t^{2} = -y^{2}" title="t y {y'} + t^{2} = -y^{2}" data-latex="t y {y'} + t^{2} = -y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20+%20%5Cfrac%7B5%7D%7B5%20%5C,%20t%20+%20y%20+%205%7D%20=%205%20%5C,%20t%20-%20%7By'%7D%20+%2010" alt="-y + \frac{5}{5 \, t + y + 5} = 5 \, t - {y'} + 10" title="-y + \frac{5}{5 \, t + y + 5} = 5 \, t - {y'} + 10" data-latex="-y + \frac{5}{5 \, t + y + 5} = 5 \, t - {y'} + 10"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20+%20t%5E%7B2%7D%20=%20-y%5E%7B2%7D" alt="t y {y'} + t^{2} = -y^{2}" title="t y {y'} + t^{2} = -y^{2}" data-latex="t y {y'} + t^{2} = -y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = 1" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-2676" title="F5 | Substitution strategies | ver. 2676"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} + y^{2} = 2 \, t^{2}" alt="t y {y'} + y^{2} = 2 \, t^{2}" title="t y {y'} + y^{2} = 2 \, t^{2}" data-latex="t y {y'} + y^{2} = 2 \, t^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 = 2 \, t - y + {y'} - \frac{4}{2 \, t - y - 4}" alt="2 = 2 \, t - y + {y'} - \frac{4}{2 \, t - y - 4}" title="2 = 2 \, t - y + {y'} - \frac{4}{2 \, t - y - 4}" data-latex="2 = 2 \, t - y + {y'} - \frac{4}{2 \, t - y - 4}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20+%20y%5E%7B2%7D%20=%202%20%5C,%20t%5E%7B2%7D" alt="t y {y'} + y^{2} = 2 \, t^{2}" title="t y {y'} + y^{2} = 2 \, t^{2}" data-latex="t y {y'} + y^{2} = 2 \, t^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20=%202%20%5C,%20t%20-%20y%20+%20%7By'%7D%20-%20%5Cfrac%7B4%7D%7B2%20%5C,%20t%20-%20y%20-%204%7D" alt="2 = 2 \, t - y + {y'} - \frac{4}{2 \, t - y - 4}" title="2 = 2 \, t - y + {y'} - \frac{4}{2 \, t - y - 4}" data-latex="2 = 2 \, t - y + {y'} - \frac{4}{2 \, t - y - 4}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = 1" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-5911" title="F5 | Substitution strategies | ver. 5911"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - 2 \, t^{2} - y^{2} = 0" alt="-t y {y'} - 2 \, t^{2} - y^{2} = 0" title="-t y {y'} - 2 \, t^{2} - y^{2} = 0" data-latex="-t y {y'} - 2 \, t^{2} - y^{2} = 0"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, t + {y'} - \frac{4}{3 \, t - y + 1} = y - 4" alt="3 \, t + {y'} - \frac{4}{3 \, t - y + 1} = y - 4" title="3 \, t + {y'} - \frac{4}{3 \, t - y + 1} = y - 4" data-latex="3 \, t + {y'} - \frac{4}{3 \, t - y + 1} = y - 4"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%202%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D%20=%200" alt="-t y {y'} - 2 \, t^{2} - y^{2} = 0" title="-t y {y'} - 2 \, t^{2} - y^{2} = 0" data-latex="-t y {y'} - 2 \, t^{2} - y^{2} = 0"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20t%20+%20%7By'%7D%20-%20%5Cfrac%7B4%7D%7B3%20%5C,%20t%20-%20y%20+%201%7D%20=%20y%20-%204" alt="3 \, t + {y'} - \frac{4}{3 \, t - y + 1} = y - 4" title="3 \, t + {y'} - \frac{4}{3 \, t - y + 1} = y - 4" data-latex="3 \, t + {y'} - \frac{4}{3 \, t - y + 1} = y - 4"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = 1" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-6285" title="F5 | Substitution strategies | ver. 6285"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} = t y {y'} + y^{2}" alt="-4 \, t^{2} = t y {y'} + y^{2}" title="-4 \, t^{2} = t y {y'} + y^{2}" data-latex="-4 \, t^{2} = t y {y'} + y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t = -y + {y'} + \frac{5}{4 \, t - y + 3} + 7" alt="-4 \, t = -y + {y'} + \frac{5}{4 \, t - y + 3} + 7" title="-4 \, t = -y + {y'} + \frac{5}{4 \, t - y + 3} + 7" data-latex="-4 \, t = -y + {y'} + \frac{5}{4 \, t - y + 3} + 7"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%5E%7B2%7D%20=%20t%20y%20%7By'%7D%20+%20y%5E%7B2%7D" alt="-4 \, t^{2} = t y {y'} + y^{2}" title="-4 \, t^{2} = t y {y'} + y^{2}" data-latex="-4 \, t^{2} = t y {y'} + y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%20=%20-y%20+%20%7By'%7D%20+%20%5Cfrac%7B5%7D%7B4%20%5C,%20t%20-%20y%20+%203%7D%20+%207" alt="-4 \, t = -y + {y'} + \frac{5}{4 \, t - y + 3} + 7" title="-4 \, t = -y + {y'} + \frac{5}{4 \, t - y + 3} + 7" data-latex="-4 \, t = -y + {y'} + \frac{5}{4 \, t - y + 3} + 7"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = 1" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-9239" title="F5 | Substitution strategies | ver. 9239"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t - {y'} + 2 = -y - \frac{4}{t - y - 3}" alt="-t - {y'} + 2 = -y - \frac{4}{t - y - 3}" title="-t - {y'} + 2 = -y - \frac{4}{t - y - 3}" data-latex="-t - {y'} + 2 = -y - \frac{4}{t - y - 3}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} = -t^{2} + y^{2}" alt="-t y {y'} = -t^{2} + y^{2}" title="-t y {y'} = -t^{2} + y^{2}" data-latex="-t y {y'} = -t^{2} + y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20-%20%7By'%7D%20+%202%20=%20-y%20-%20%5Cfrac%7B4%7D%7Bt%20-%20y%20-%203%7D" alt="-t - {y'} + 2 = -y - \frac{4}{t - y - 3}" title="-t - {y'} + 2 = -y - \frac{4}{t - y - 3}" data-latex="-t - {y'} + 2 = -y - \frac{4}{t - y - 3}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20=%20-t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="-t y {y'} = -t^{2} + y^{2}" title="-t y {y'} = -t^{2} + y^{2}" data-latex="-t y {y'} = -t^{2} + y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = 1" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-1518" title="F5 | Substitution strategies | ver. 1518"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, t + {y'} - 4 = y + \frac{1}{2 \, t + y + 2}" alt="-2 \, t + {y'} - 4 = y + \frac{1}{2 \, t + y + 2}" title="-2 \, t + {y'} - 4 = y + \frac{1}{2 \, t + y + 2}" data-latex="-2 \, t + {y'} - 4 = y + \frac{1}{2 \, t + y + 2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -t y {y'} + 5 \, t^{2} - y^{2}" alt="0 = -t y {y'} + 5 \, t^{2} - y^{2}" title="0 = -t y {y'} + 5 \, t^{2} - y^{2}" data-latex="0 = -t y {y'} + 5 \, t^{2} - y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20t%20+%20%7By'%7D%20-%204%20=%20y%20+%20%5Cfrac%7B1%7D%7B2%20%5C,%20t%20+%20y%20+%202%7D" alt="-2 \, t + {y'} - 4 = y + \frac{1}{2 \, t + y + 2}" title="-2 \, t + {y'} - 4 = y + \frac{1}{2 \, t + y + 2}" data-latex="-2 \, t + {y'} - 4 = y + \frac{1}{2 \, t + y + 2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-t%20y%20%7By'%7D%20+%205%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D" alt="0 = -t y {y'} + 5 \, t^{2} - y^{2}" title="0 = -t y {y'} + 5 \, t^{2} - y^{2}" data-latex="0 = -t y {y'} + 5 \, t^{2} - y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = 1" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-0746" title="F5 | Substitution strategies | ver. 0746"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, t = y - {y'} - \frac{4}{5 \, t - y - 2} - 3" alt="5 \, t = y - {y'} - \frac{4}{5 \, t - y - 2} - 3" title="5 \, t = y - {y'} - \frac{4}{5 \, t - y - 2} - 3" data-latex="5 \, t = y - {y'} - \frac{4}{5 \, t - y - 2} - 3"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} - 2 \, t^{2} = -y^{2}" alt="t y {y'} - 2 \, t^{2} = -y^{2}" title="t y {y'} - 2 \, t^{2} = -y^{2}" data-latex="t y {y'} - 2 \, t^{2} = -y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20t%20=%20y%20-%20%7By'%7D%20-%20%5Cfrac%7B4%7D%7B5%20%5C,%20t%20-%20y%20-%202%7D%20-%203" alt="5 \, t = y - {y'} - \frac{4}{5 \, t - y - 2} - 3" title="5 \, t = y - {y'} - \frac{4}{5 \, t - y - 2} - 3" data-latex="5 \, t = y - {y'} - \frac{4}{5 \, t - y - 2} - 3"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20-%202%20%5C,%20t%5E%7B2%7D%20=%20-y%5E%7B2%7D" alt="t y {y'} - 2 \, t^{2} = -y^{2}" title="t y {y'} - 2 \, t^{2} = -y^{2}" data-latex="t y {y'} - 2 \, t^{2} = -y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = 1" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-6426" title="F5 | Substitution strategies | ver. 6426"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y + 6 = -4 \, t - {y'} + \frac{1}{4 \, t - y + 2}" alt="-y + 6 = -4 \, t - {y'} + \frac{1}{4 \, t - y + 2}" title="-y + 6 = -4 \, t - {y'} + \frac{1}{4 \, t - y + 2}" data-latex="-y + 6 = -4 \, t - {y'} + \frac{1}{4 \, t - y + 2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, t^{2} + y^{2} = -t y {y'}" alt="3 \, t^{2} + y^{2} = -t y {y'}" title="3 \, t^{2} + y^{2} = -t y {y'}" data-latex="3 \, t^{2} + y^{2} = -t y {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20+%206%20=%20-4%20%5C,%20t%20-%20%7By'%7D%20+%20%5Cfrac%7B1%7D%7B4%20%5C,%20t%20-%20y%20+%202%7D" alt="-y + 6 = -4 \, t - {y'} + \frac{1}{4 \, t - y + 2}" title="-y + 6 = -4 \, t - {y'} + \frac{1}{4 \, t - y + 2}" data-latex="-y + 6 = -4 \, t - {y'} + \frac{1}{4 \, t - y + 2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D%20=%20-t%20y%20%7By'%7D" alt="3 \, t^{2} + y^{2} = -t y {y'}" title="3 \, t^{2} + y^{2} = -t y {y'}" data-latex="3 \, t^{2} + y^{2} = -t y {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 1} = 1" alt="\frac{u {u'}}{u^{2} - 1} = 1" title="\frac{u {u'}}{u^{2} - 1} = 1" data-latex="\frac{u {u'}}{u^{2} - 1} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 1} = 1" title="\frac{u {u'}}{u^{2} - 1} = 1" data-latex="\frac{u {u'}}{u^{2} - 1} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8635" title="F5 | Substitution strategies | ver. 8635"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{2} = t y {y'} + y^{2}" alt="-t^{2} = t y {y'} + y^{2}" title="-t^{2} = t y {y'} + y^{2}" data-latex="-t^{2} = t y {y'} + y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y - {y'} - 3 = 2 \, t + \frac{4}{2 \, t - y + 1}" alt="y - {y'} - 3 = 2 \, t + \frac{4}{2 \, t - y + 1}" title="y - {y'} - 3 = 2 \, t + \frac{4}{2 \, t - y + 1}" data-latex="y - {y'} - 3 = 2 \, t + \frac{4}{2 \, t - y + 1}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B2%7D%20=%20t%20y%20%7By'%7D%20+%20y%5E%7B2%7D" alt="-t^{2} = t y {y'} + y^{2}" title="-t^{2} = t y {y'} + y^{2}" data-latex="-t^{2} = t y {y'} + y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y%20-%20%7By'%7D%20-%203%20=%202%20%5C,%20t%20+%20%5Cfrac%7B4%7D%7B2%20%5C,%20t%20-%20y%20+%201%7D" alt="y - {y'} - 3 = 2 \, t + \frac{4}{2 \, t - y + 1}" title="y - {y'} - 3 = 2 \, t + \frac{4}{2 \, t - y + 1}" data-latex="y - {y'} - 3 = 2 \, t + \frac{4}{2 \, t - y + 1}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = 1" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-6950" title="F5 | Substitution strategies | ver. 6950"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, t^{2} = -t y {y'} - y^{2}" alt="-2 \, t^{2} = -t y {y'} - y^{2}" title="-2 \, t^{2} = -t y {y'} - y^{2}" data-latex="-2 \, t^{2} = -t y {y'} - y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, t + y - {y'} - 6 = \frac{2}{5 \, t - y + 1}" alt="-5 \, t + y - {y'} - 6 = \frac{2}{5 \, t - y + 1}" title="-5 \, t + y - {y'} - 6 = \frac{2}{5 \, t - y + 1}" data-latex="-5 \, t + y - {y'} - 6 = \frac{2}{5 \, t - y + 1}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20t%5E%7B2%7D%20=%20-t%20y%20%7By'%7D%20-%20y%5E%7B2%7D" alt="-2 \, t^{2} = -t y {y'} - y^{2}" title="-2 \, t^{2} = -t y {y'} - y^{2}" data-latex="-2 \, t^{2} = -t y {y'} - y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5%20%5C,%20t%20+%20y%20-%20%7By'%7D%20-%206%20=%20%5Cfrac%7B2%7D%7B5%20%5C,%20t%20-%20y%20+%201%7D" alt="-5 \, t + y - {y'} - 6 = \frac{2}{5 \, t - y + 1}" title="-5 \, t + y - {y'} - 6 = \frac{2}{5 \, t - y + 1}" data-latex="-5 \, t + y - {y'} - 6 = \frac{2}{5 \, t - y + 1}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = 1" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-0246" title="F5 | Substitution strategies | ver. 0246"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, t + \frac{5}{3 \, t + y + 1} + 4 = -y + {y'}" alt="3 \, t + \frac{5}{3 \, t + y + 1} + 4 = -y + {y'}" title="3 \, t + \frac{5}{3 \, t + y + 1} + 4 = -y + {y'}" data-latex="3 \, t + \frac{5}{3 \, t + y + 1} + 4 = -y + {y'}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} + y^{2} = 5 \, t^{2}" alt="t y {y'} + y^{2} = 5 \, t^{2}" title="t y {y'} + y^{2} = 5 \, t^{2}" data-latex="t y {y'} + y^{2} = 5 \, t^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20t%20+%20%5Cfrac%7B5%7D%7B3%20%5C,%20t%20+%20y%20+%201%7D%20+%204%20=%20-y%20+%20%7By'%7D" alt="3 \, t + \frac{5}{3 \, t + y + 1} + 4 = -y + {y'}" title="3 \, t + \frac{5}{3 \, t + y + 1} + 4 = -y + {y'}" data-latex="3 \, t + \frac{5}{3 \, t + y + 1} + 4 = -y + {y'}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20+%20y%5E%7B2%7D%20=%205%20%5C,%20t%5E%7B2%7D" alt="t y {y'} + y^{2} = 5 \, t^{2}" title="t y {y'} + y^{2} = 5 \, t^{2}" data-latex="t y {y'} + y^{2} = 5 \, t^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = 1" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8540" title="F5 | Substitution strategies | ver. 8540"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-\frac{5}{4 \, t - y + 1} - 5 = 4 \, t - y + {y'}" alt="-\frac{5}{4 \, t - y + 1} - 5 = 4 \, t - y + {y'}" title="-\frac{5}{4 \, t - y + 1} - 5 = 4 \, t - y + {y'}" data-latex="-\frac{5}{4 \, t - y + 1} - 5 = 4 \, t - y + {y'}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} = t y {y'} + y^{2}" alt="-4 \, t^{2} = t y {y'} + y^{2}" title="-4 \, t^{2} = t y {y'} + y^{2}" data-latex="-4 \, t^{2} = t y {y'} + y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%5Cfrac%7B5%7D%7B4%20%5C,%20t%20-%20y%20+%201%7D%20-%205%20=%204%20%5C,%20t%20-%20y%20+%20%7By'%7D" alt="-\frac{5}{4 \, t - y + 1} - 5 = 4 \, t - y + {y'}" title="-\frac{5}{4 \, t - y + 1} - 5 = 4 \, t - y + {y'}" data-latex="-\frac{5}{4 \, t - y + 1} - 5 = 4 \, t - y + {y'}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%5E%7B2%7D%20=%20t%20y%20%7By'%7D%20+%20y%5E%7B2%7D" alt="-4 \, t^{2} = t y {y'} + y^{2}" title="-4 \, t^{2} = t y {y'} + y^{2}" data-latex="-4 \, t^{2} = t y {y'} + y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = 1" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-9208" title="F5 | Substitution strategies | ver. 9208"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} - \frac{1}{4 \, t - y - 2} = -4 \, t + y - 2" alt="{y'} - \frac{1}{4 \, t - y - 2} = -4 \, t + y - 2" title="{y'} - \frac{1}{4 \, t - y - 2} = -4 \, t + y - 2" data-latex="{y'} - \frac{1}{4 \, t - y - 2} = -4 \, t + y - 2"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = t y {y'} + 2 \, t^{2} + y^{2}" alt="0 = t y {y'} + 2 \, t^{2} + y^{2}" title="0 = t y {y'} + 2 \, t^{2} + y^{2}" data-latex="0 = t y {y'} + 2 \, t^{2} + y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20-%20%5Cfrac%7B1%7D%7B4%20%5C,%20t%20-%20y%20-%202%7D%20=%20-4%20%5C,%20t%20+%20y%20-%202" alt="{y'} - \frac{1}{4 \, t - y - 2} = -4 \, t + y - 2" title="{y'} - \frac{1}{4 \, t - y - 2} = -4 \, t + y - 2" data-latex="{y'} - \frac{1}{4 \, t - y - 2} = -4 \, t + y - 2"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20t%20y%20%7By'%7D%20+%202%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="0 = t y {y'} + 2 \, t^{2} + y^{2}" title="0 = t y {y'} + 2 \, t^{2} + y^{2}" data-latex="0 = t y {y'} + 2 \, t^{2} + y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 1} = 1" alt="\frac{u {u'}}{u^{2} - 1} = 1" title="\frac{u {u'}}{u^{2} - 1} = 1" data-latex="\frac{u {u'}}{u^{2} - 1} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 1} = 1" title="\frac{u {u'}}{u^{2} - 1} = 1" data-latex="\frac{u {u'}}{u^{2} - 1} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-0670" title="F5 | Substitution strategies | ver. 0670"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} = -t^{2} - y^{2}" alt="t y {y'} = -t^{2} - y^{2}" title="t y {y'} = -t^{2} - y^{2}" data-latex="t y {y'} = -t^{2} - y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, t + {y'} + \frac{1}{2 \, t - y + 1} = y - 3" alt="2 \, t + {y'} + \frac{1}{2 \, t - y + 1} = y - 3" title="2 \, t + {y'} + \frac{1}{2 \, t - y + 1} = y - 3" data-latex="2 \, t + {y'} + \frac{1}{2 \, t - y + 1} = y - 3"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20=%20-t%5E%7B2%7D%20-%20y%5E%7B2%7D" alt="t y {y'} = -t^{2} - y^{2}" title="t y {y'} = -t^{2} - y^{2}" data-latex="t y {y'} = -t^{2} - y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20t%20+%20%7By'%7D%20+%20%5Cfrac%7B1%7D%7B2%20%5C,%20t%20-%20y%20+%201%7D%20=%20y%20-%203" alt="2 \, t + {y'} + \frac{1}{2 \, t - y + 1} = y - 3" title="2 \, t + {y'} + \frac{1}{2 \, t - y + 1} = y - 3" data-latex="2 \, t + {y'} + \frac{1}{2 \, t - y + 1} = y - 3"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = 1" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-7320" title="F5 | Substitution strategies | ver. 7320"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, t - y + {y'} = -\frac{2}{2 \, t - y - 3} + 1" alt="2 \, t - y + {y'} = -\frac{2}{2 \, t - y - 3} + 1" title="2 \, t - y + {y'} = -\frac{2}{2 \, t - y - 3} + 1" data-latex="2 \, t - y + {y'} = -\frac{2}{2 \, t - y - 3} + 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, t^{2} = -t y {y'} - y^{2}" alt="-3 \, t^{2} = -t y {y'} - y^{2}" title="-3 \, t^{2} = -t y {y'} - y^{2}" data-latex="-3 \, t^{2} = -t y {y'} - y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20t%20-%20y%20+%20%7By'%7D%20=%20-%5Cfrac%7B2%7D%7B2%20%5C,%20t%20-%20y%20-%203%7D%20+%201" alt="2 \, t - y + {y'} = -\frac{2}{2 \, t - y - 3} + 1" title="2 \, t - y + {y'} = -\frac{2}{2 \, t - y - 3} + 1" data-latex="2 \, t - y + {y'} = -\frac{2}{2 \, t - y - 3} + 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20t%5E%7B2%7D%20=%20-t%20y%20%7By'%7D%20-%20y%5E%7B2%7D" alt="-3 \, t^{2} = -t y {y'} - y^{2}" title="-3 \, t^{2} = -t y {y'} - y^{2}" data-latex="-3 \, t^{2} = -t y {y'} - y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = 1" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8519" title="F5 | Substitution strategies | ver. 8519"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{t - y - 5} = -t + y - {y'} + 4" alt="\frac{5}{t - y - 5} = -t + y - {y'} + 4" title="\frac{5}{t - y - 5} = -t + y - {y'} + 4" data-latex="\frac{5}{t - y - 5} = -t + y - {y'} + 4"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} - 5 \, t^{2} = -y^{2}" alt="t y {y'} - 5 \, t^{2} = -y^{2}" title="t y {y'} - 5 \, t^{2} = -y^{2}" data-latex="t y {y'} - 5 \, t^{2} = -y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7Bt%20-%20y%20-%205%7D%20=%20-t%20+%20y%20-%20%7By'%7D%20+%204" alt="\frac{5}{t - y - 5} = -t + y - {y'} + 4" title="\frac{5}{t - y - 5} = -t + y - {y'} + 4" data-latex="\frac{5}{t - y - 5} = -t + y - {y'} + 4"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20-%205%20%5C,%20t%5E%7B2%7D%20=%20-y%5E%7B2%7D" alt="t y {y'} - 5 \, t^{2} = -y^{2}" title="t y {y'} - 5 \, t^{2} = -y^{2}" data-latex="t y {y'} - 5 \, t^{2} = -y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = 1" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8788" title="F5 | Substitution strategies | ver. 8788"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, t^{2} = t y {y'} + y^{2}" alt="-5 \, t^{2} = t y {y'} + y^{2}" title="-5 \, t^{2} = t y {y'} + y^{2}" data-latex="-5 \, t^{2} = t y {y'} + y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 2 \, t - y + {y'} + \frac{3}{2 \, t - y - 3} - 1" alt="0 = 2 \, t - y + {y'} + \frac{3}{2 \, t - y - 3} - 1" title="0 = 2 \, t - y + {y'} + \frac{3}{2 \, t - y - 3} - 1" data-latex="0 = 2 \, t - y + {y'} + \frac{3}{2 \, t - y - 3} - 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5%20%5C,%20t%5E%7B2%7D%20=%20t%20y%20%7By'%7D%20+%20y%5E%7B2%7D" alt="-5 \, t^{2} = t y {y'} + y^{2}" title="-5 \, t^{2} = t y {y'} + y^{2}" data-latex="-5 \, t^{2} = t y {y'} + y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%202%20%5C,%20t%20-%20y%20+%20%7By'%7D%20+%20%5Cfrac%7B3%7D%7B2%20%5C,%20t%20-%20y%20-%203%7D%20-%201" alt="0 = 2 \, t - y + {y'} + \frac{3}{2 \, t - y - 3} - 1" title="0 = 2 \, t - y + {y'} + \frac{3}{2 \, t - y - 3} - 1" data-latex="0 = 2 \, t - y + {y'} + \frac{3}{2 \, t - y - 3} - 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = 1" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-0696" title="F5 | Substitution strategies | ver. 0696"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -4 \, t + y - {y'} + \frac{2}{4 \, t - y - 3} - 1" alt="0 = -4 \, t + y - {y'} + \frac{2}{4 \, t - y - 3} - 1" title="0 = -4 \, t + y - {y'} + \frac{2}{4 \, t - y - 3} - 1" data-latex="0 = -4 \, t + y - {y'} + \frac{2}{4 \, t - y - 3} - 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{2} + y^{2} = -t y {y'}" alt="-t^{2} + y^{2} = -t y {y'}" title="-t^{2} + y^{2} = -t y {y'}" data-latex="-t^{2} + y^{2} = -t y {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-4%20%5C,%20t%20+%20y%20-%20%7By'%7D%20+%20%5Cfrac%7B2%7D%7B4%20%5C,%20t%20-%20y%20-%203%7D%20-%201" alt="0 = -4 \, t + y - {y'} + \frac{2}{4 \, t - y - 3} - 1" title="0 = -4 \, t + y - {y'} + \frac{2}{4 \, t - y - 3} - 1" data-latex="0 = -4 \, t + y - {y'} + \frac{2}{4 \, t - y - 3} - 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B2%7D%20+%20y%5E%7B2%7D%20=%20-t%20y%20%7By'%7D" alt="-t^{2} + y^{2} = -t y {y'}" title="-t^{2} + y^{2} = -t y {y'}" data-latex="-t^{2} + y^{2} = -t y {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = 1" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-1603" title="F5 | Substitution strategies | ver. 1603"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, t^{2} = -t y {y'} - y^{2}" alt="-2 \, t^{2} = -t y {y'} - y^{2}" title="-2 \, t^{2} = -t y {y'} - y^{2}" data-latex="-2 \, t^{2} = -t y {y'} - y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t - {y'} = -y - \frac{1}{4 \, t + y + 1} - 5" alt="4 \, t - {y'} = -y - \frac{1}{4 \, t + y + 1} - 5" title="4 \, t - {y'} = -y - \frac{1}{4 \, t + y + 1} - 5" data-latex="4 \, t - {y'} = -y - \frac{1}{4 \, t + y + 1} - 5"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20t%5E%7B2%7D%20=%20-t%20y%20%7By'%7D%20-%20y%5E%7B2%7D" alt="-2 \, t^{2} = -t y {y'} - y^{2}" title="-2 \, t^{2} = -t y {y'} - y^{2}" data-latex="-2 \, t^{2} = -t y {y'} - y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20t%20-%20%7By'%7D%20=%20-y%20-%20%5Cfrac%7B1%7D%7B4%20%5C,%20t%20+%20y%20+%201%7D%20-%205" alt="4 \, t - {y'} = -y - \frac{1}{4 \, t + y + 1} - 5" title="4 \, t - {y'} = -y - \frac{1}{4 \, t + y + 1} - 5" data-latex="4 \, t - {y'} = -y - \frac{1}{4 \, t + y + 1} - 5"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = 1" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-3762" title="F5 | Substitution strategies | ver. 3762"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} = -2 \, t^{2} + y^{2}" alt="-t y {y'} = -2 \, t^{2} + y^{2}" title="-t y {y'} = -2 \, t^{2} + y^{2}" data-latex="-t y {y'} = -2 \, t^{2} + y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} + 6 = -2 \, t - y - \frac{5}{2 \, t + y + 4}" alt="-{y'} + 6 = -2 \, t - y - \frac{5}{2 \, t + y + 4}" title="-{y'} + 6 = -2 \, t - y - \frac{5}{2 \, t + y + 4}" data-latex="-{y'} + 6 = -2 \, t - y - \frac{5}{2 \, t + y + 4}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20=%20-2%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="-t y {y'} = -2 \, t^{2} + y^{2}" title="-t y {y'} = -2 \, t^{2} + y^{2}" data-latex="-t y {y'} = -2 \, t^{2} + y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20+%206%20=%20-2%20%5C,%20t%20-%20y%20-%20%5Cfrac%7B5%7D%7B2%20%5C,%20t%20+%20y%20+%204%7D" alt="-{y'} + 6 = -2 \, t - y - \frac{5}{2 \, t + y + 4}" title="-{y'} + 6 = -2 \, t - y - \frac{5}{2 \, t + y + 4}" data-latex="-{y'} + 6 = -2 \, t - y - \frac{5}{2 \, t + y + 4}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = 1" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-0801" title="F5 | Substitution strategies | ver. 0801"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, t - y + {y'} + \frac{2}{3 \, t + y - 3} = 0" alt="-3 \, t - y + {y'} + \frac{2}{3 \, t + y - 3} = 0" title="-3 \, t - y + {y'} + \frac{2}{3 \, t + y - 3} = 0" data-latex="-3 \, t - y + {y'} + \frac{2}{3 \, t + y - 3} = 0"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - y^{2} = 2 \, t^{2}" alt="-t y {y'} - y^{2} = 2 \, t^{2}" title="-t y {y'} - y^{2} = 2 \, t^{2}" data-latex="-t y {y'} - y^{2} = 2 \, t^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20t%20-%20y%20+%20%7By'%7D%20+%20%5Cfrac%7B2%7D%7B3%20%5C,%20t%20+%20y%20-%203%7D%20=%200" alt="-3 \, t - y + {y'} + \frac{2}{3 \, t + y - 3} = 0" title="-3 \, t - y + {y'} + \frac{2}{3 \, t + y - 3} = 0" data-latex="-3 \, t - y + {y'} + \frac{2}{3 \, t + y - 3} = 0"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%20y%5E%7B2%7D%20=%202%20%5C,%20t%5E%7B2%7D" alt="-t y {y'} - y^{2} = 2 \, t^{2}" title="-t y {y'} - y^{2} = 2 \, t^{2}" data-latex="-t y {y'} - y^{2} = 2 \, t^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = 1" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8801" title="F5 | Substitution strategies | ver. 8801"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} - \frac{4}{3 \, t - y + 2} = 3 \, t - y + 5" alt="-{y'} - \frac{4}{3 \, t - y + 2} = 3 \, t - y + 5" title="-{y'} - \frac{4}{3 \, t - y + 2} = 3 \, t - y + 5" data-latex="-{y'} - \frac{4}{3 \, t - y + 2} = 3 \, t - y + 5"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} = -3 \, t^{2} + y^{2}" alt="-t y {y'} = -3 \, t^{2} + y^{2}" title="-t y {y'} = -3 \, t^{2} + y^{2}" data-latex="-t y {y'} = -3 \, t^{2} + y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20-%20%5Cfrac%7B4%7D%7B3%20%5C,%20t%20-%20y%20+%202%7D%20=%203%20%5C,%20t%20-%20y%20+%205" alt="-{y'} - \frac{4}{3 \, t - y + 2} = 3 \, t - y + 5" title="-{y'} - \frac{4}{3 \, t - y + 2} = 3 \, t - y + 5" data-latex="-{y'} - \frac{4}{3 \, t - y + 2} = 3 \, t - y + 5"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20=%20-3%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="-t y {y'} = -3 \, t^{2} + y^{2}" title="-t y {y'} = -3 \, t^{2} + y^{2}" data-latex="-t y {y'} = -3 \, t^{2} + y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = 1" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-7882" title="F5 | Substitution strategies | ver. 7882"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y^{2} = t y {y'} + 5 \, t^{2}" alt="-y^{2} = t y {y'} + 5 \, t^{2}" title="-y^{2} = t y {y'} + 5 \, t^{2}" data-latex="-y^{2} = t y {y'} + 5 \, t^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} + \frac{1}{t - y + 5} - 6 = t - y" alt="-{y'} + \frac{1}{t - y + 5} - 6 = t - y" title="-{y'} + \frac{1}{t - y + 5} - 6 = t - y" data-latex="-{y'} + \frac{1}{t - y + 5} - 6 = t - y"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%5E%7B2%7D%20=%20t%20y%20%7By'%7D%20+%205%20%5C,%20t%5E%7B2%7D" alt="-y^{2} = t y {y'} + 5 \, t^{2}" title="-y^{2} = t y {y'} + 5 \, t^{2}" data-latex="-y^{2} = t y {y'} + 5 \, t^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20+%20%5Cfrac%7B1%7D%7Bt%20-%20y%20+%205%7D%20-%206%20=%20t%20-%20y" alt="-{y'} + \frac{1}{t - y + 5} - 6 = t - y" title="-{y'} + \frac{1}{t - y + 5} - 6 = t - y" data-latex="-{y'} + \frac{1}{t - y + 5} - 6 = t - y"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 1} = 1" alt="\frac{u {u'}}{u^{2} - 1} = 1" title="\frac{u {u'}}{u^{2} - 1} = 1" data-latex="\frac{u {u'}}{u^{2} - 1} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 1} = 1" title="\frac{u {u'}}{u^{2} - 1} = 1" data-latex="\frac{u {u'}}{u^{2} - 1} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8193" title="F5 | Substitution strategies | ver. 8193"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, t + {y'} + \frac{3}{5 \, t + y - 4} = y + 1" alt="-5 \, t + {y'} + \frac{3}{5 \, t + y - 4} = y + 1" title="-5 \, t + {y'} + \frac{3}{5 \, t + y - 4} = y + 1" data-latex="-5 \, t + {y'} + \frac{3}{5 \, t + y - 4} = y + 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - y^{2} = -4 \, t^{2}" alt="-t y {y'} - y^{2} = -4 \, t^{2}" title="-t y {y'} - y^{2} = -4 \, t^{2}" data-latex="-t y {y'} - y^{2} = -4 \, t^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5%20%5C,%20t%20+%20%7By'%7D%20+%20%5Cfrac%7B3%7D%7B5%20%5C,%20t%20+%20y%20-%204%7D%20=%20y%20+%201" alt="-5 \, t + {y'} + \frac{3}{5 \, t + y - 4} = y + 1" title="-5 \, t + {y'} + \frac{3}{5 \, t + y - 4} = y + 1" data-latex="-5 \, t + {y'} + \frac{3}{5 \, t + y - 4} = y + 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%20y%5E%7B2%7D%20=%20-4%20%5C,%20t%5E%7B2%7D" alt="-t y {y'} - y^{2} = -4 \, t^{2}" title="-t y {y'} - y^{2} = -4 \, t^{2}" data-latex="-t y {y'} - y^{2} = -4 \, t^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = 1" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8664" title="F5 | Substitution strategies | ver. 8664"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y - {y'} - \frac{4}{t - y - 5} = t - 4" alt="y - {y'} - \frac{4}{t - y - 5} = t - 4" title="y - {y'} - \frac{4}{t - y - 5} = t - 4" data-latex="y - {y'} - \frac{4}{t - y - 5} = t - 4"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} = -4 \, t^{2} - y^{2}" alt="t y {y'} = -4 \, t^{2} - y^{2}" title="t y {y'} = -4 \, t^{2} - y^{2}" data-latex="t y {y'} = -4 \, t^{2} - y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y%20-%20%7By'%7D%20-%20%5Cfrac%7B4%7D%7Bt%20-%20y%20-%205%7D%20=%20t%20-%204" alt="y - {y'} - \frac{4}{t - y - 5} = t - 4" title="y - {y'} - \frac{4}{t - y - 5} = t - 4" data-latex="y - {y'} - \frac{4}{t - y - 5} = t - 4"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20=%20-4%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D" alt="t y {y'} = -4 \, t^{2} - y^{2}" title="t y {y'} = -4 \, t^{2} - y^{2}" data-latex="t y {y'} = -4 \, t^{2} - y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = 1" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-2983" title="F5 | Substitution strategies | ver. 2983"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - y^{2} = -3 \, t^{2}" alt="-t y {y'} - y^{2} = -3 \, t^{2}" title="-t y {y'} - y^{2} = -3 \, t^{2}" data-latex="-t y {y'} - y^{2} = -3 \, t^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-\frac{3}{3 \, t - y - 4} = -3 \, t + y - {y'} + 1" alt="-\frac{3}{3 \, t - y - 4} = -3 \, t + y - {y'} + 1" title="-\frac{3}{3 \, t - y - 4} = -3 \, t + y - {y'} + 1" data-latex="-\frac{3}{3 \, t - y - 4} = -3 \, t + y - {y'} + 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%20y%5E%7B2%7D%20=%20-3%20%5C,%20t%5E%7B2%7D" alt="-t y {y'} - y^{2} = -3 \, t^{2}" title="-t y {y'} - y^{2} = -3 \, t^{2}" data-latex="-t y {y'} - y^{2} = -3 \, t^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%5Cfrac%7B3%7D%7B3%20%5C,%20t%20-%20y%20-%204%7D%20=%20-3%20%5C,%20t%20+%20y%20-%20%7By'%7D%20+%201" alt="-\frac{3}{3 \, t - y - 4} = -3 \, t + y - {y'} + 1" title="-\frac{3}{3 \, t - y - 4} = -3 \, t + y - {y'} + 1" data-latex="-\frac{3}{3 \, t - y - 4} = -3 \, t + y - {y'} + 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = 1" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8072" title="F5 | Substitution strategies | ver. 8072"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, t + y - {y'} + \frac{3}{2 \, t + y + 3} = \left(-5\right)" alt="2 \, t + y - {y'} + \frac{3}{2 \, t + y + 3} = \left(-5\right)" title="2 \, t + y - {y'} + \frac{3}{2 \, t + y + 3} = \left(-5\right)" data-latex="2 \, t + y - {y'} + \frac{3}{2 \, t + y + 3} = \left(-5\right)"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -t y {y'} - 2 \, t^{2} - y^{2}" alt="0 = -t y {y'} - 2 \, t^{2} - y^{2}" title="0 = -t y {y'} - 2 \, t^{2} - y^{2}" data-latex="0 = -t y {y'} - 2 \, t^{2} - y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20t%20+%20y%20-%20%7By'%7D%20+%20%5Cfrac%7B3%7D%7B2%20%5C,%20t%20+%20y%20+%203%7D%20=%20%5Cleft(-5%5Cright)" alt="2 \, t + y - {y'} + \frac{3}{2 \, t + y + 3} = \left(-5\right)" title="2 \, t + y - {y'} + \frac{3}{2 \, t + y + 3} = \left(-5\right)" data-latex="2 \, t + y - {y'} + \frac{3}{2 \, t + y + 3} = \left(-5\right)"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-t%20y%20%7By'%7D%20-%202%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D" alt="0 = -t y {y'} - 2 \, t^{2} - y^{2}" title="0 = -t y {y'} - 2 \, t^{2} - y^{2}" data-latex="0 = -t y {y'} - 2 \, t^{2} - y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = 1" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-0668" title="F5 | Substitution strategies | ver. 0668"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t - y + {y'} - \frac{1}{t + y - 1} = 0" alt="-t - y + {y'} - \frac{1}{t + y - 1} = 0" title="-t - y + {y'} - \frac{1}{t + y - 1} = 0" data-latex="-t - y + {y'} - \frac{1}{t + y - 1} = 0"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = t y {y'} - 2 \, t^{2} + y^{2}" alt="0 = t y {y'} - 2 \, t^{2} + y^{2}" title="0 = t y {y'} - 2 \, t^{2} + y^{2}" data-latex="0 = t y {y'} - 2 \, t^{2} + y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20-%20y%20+%20%7By'%7D%20-%20%5Cfrac%7B1%7D%7Bt%20+%20y%20-%201%7D%20=%200" alt="-t - y + {y'} - \frac{1}{t + y - 1} = 0" title="-t - y + {y'} - \frac{1}{t + y - 1} = 0" data-latex="-t - y + {y'} - \frac{1}{t + y - 1} = 0"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20t%20y%20%7By'%7D%20-%202%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="0 = t y {y'} - 2 \, t^{2} + y^{2}" title="0 = t y {y'} - 2 \, t^{2} + y^{2}" data-latex="0 = t y {y'} - 2 \, t^{2} + y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = 1" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-2445" title="F5 | Substitution strategies | ver. 2445"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} + t^{2} = y^{2}" alt="-t y {y'} + t^{2} = y^{2}" title="-t y {y'} + t^{2} = y^{2}" data-latex="-t y {y'} + t^{2} = y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t - y - 5 = -{y'} - \frac{5}{t + y + 4}" alt="-t - y - 5 = -{y'} - \frac{5}{t + y + 4}" title="-t - y - 5 = -{y'} - \frac{5}{t + y + 4}" data-latex="-t - y - 5 = -{y'} - \frac{5}{t + y + 4}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20+%20t%5E%7B2%7D%20=%20y%5E%7B2%7D" alt="-t y {y'} + t^{2} = y^{2}" title="-t y {y'} + t^{2} = y^{2}" data-latex="-t y {y'} + t^{2} = y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20-%20y%20-%205%20=%20-%7By'%7D%20-%20%5Cfrac%7B5%7D%7Bt%20+%20y%20+%204%7D" alt="-t - y - 5 = -{y'} - \frac{5}{t + y + 4}" title="-t - y - 5 = -{y'} - \frac{5}{t + y + 4}" data-latex="-t - y - 5 = -{y'} - \frac{5}{t + y + 4}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = 1" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-9509" title="F5 | Substitution strategies | ver. 9509"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + 5 = -t + y - \frac{1}{t - y + 4}" alt="{y'} + 5 = -t + y - \frac{1}{t - y + 4}" title="{y'} + 5 = -t + y - \frac{1}{t - y + 4}" data-latex="{y'} + 5 = -t + y - \frac{1}{t - y + 4}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, t^{2} = -t y {y'} - y^{2}" alt="3 \, t^{2} = -t y {y'} - y^{2}" title="3 \, t^{2} = -t y {y'} - y^{2}" data-latex="3 \, t^{2} = -t y {y'} - y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%205%20=%20-t%20+%20y%20-%20%5Cfrac%7B1%7D%7Bt%20-%20y%20+%204%7D" alt="{y'} + 5 = -t + y - \frac{1}{t - y + 4}" title="{y'} + 5 = -t + y - \frac{1}{t - y + 4}" data-latex="{y'} + 5 = -t + y - \frac{1}{t - y + 4}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20t%5E%7B2%7D%20=%20-t%20y%20%7By'%7D%20-%20y%5E%7B2%7D" alt="3 \, t^{2} = -t y {y'} - y^{2}" title="3 \, t^{2} = -t y {y'} - y^{2}" data-latex="3 \, t^{2} = -t y {y'} - y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = 1" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-6926" title="F5 | Substitution strategies | ver. 6926"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} - 2 \, t^{2} + y^{2} = 0" alt="t y {y'} - 2 \, t^{2} + y^{2} = 0" title="t y {y'} - 2 \, t^{2} + y^{2} = 0" data-latex="t y {y'} - 2 \, t^{2} + y^{2} = 0"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t = y - {y'} + \frac{2}{4 \, t + y + 3} + 7" alt="-4 \, t = y - {y'} + \frac{2}{4 \, t + y + 3} + 7" title="-4 \, t = y - {y'} + \frac{2}{4 \, t + y + 3} + 7" data-latex="-4 \, t = y - {y'} + \frac{2}{4 \, t + y + 3} + 7"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20-%202%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D%20=%200" alt="t y {y'} - 2 \, t^{2} + y^{2} = 0" title="t y {y'} - 2 \, t^{2} + y^{2} = 0" data-latex="t y {y'} - 2 \, t^{2} + y^{2} = 0"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%20=%20y%20-%20%7By'%7D%20+%20%5Cfrac%7B2%7D%7B4%20%5C,%20t%20+%20y%20+%203%7D%20+%207" alt="-4 \, t = y - {y'} + \frac{2}{4 \, t + y + 3} + 7" title="-4 \, t = y - {y'} + \frac{2}{4 \, t + y + 3} + 7" data-latex="-4 \, t = y - {y'} + \frac{2}{4 \, t + y + 3} + 7"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = 1" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-7284" title="F5 | Substitution strategies | ver. 7284"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3 \, t^{2} + y^{2} = -t y {y'}" alt="-3 \, t^{2} + y^{2} = -t y {y'}" title="-3 \, t^{2} + y^{2} = -t y {y'}" data-latex="-3 \, t^{2} + y^{2} = -t y {y'}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y + \frac{2}{4 \, t + y - 2} - 2 = 4 \, t - {y'}" alt="-y + \frac{2}{4 \, t + y - 2} - 2 = 4 \, t - {y'}" title="-y + \frac{2}{4 \, t + y - 2} - 2 = 4 \, t - {y'}" data-latex="-y + \frac{2}{4 \, t + y - 2} - 2 = 4 \, t - {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-3%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D%20=%20-t%20y%20%7By'%7D" alt="-3 \, t^{2} + y^{2} = -t y {y'}" title="-3 \, t^{2} + y^{2} = -t y {y'}" data-latex="-3 \, t^{2} + y^{2} = -t y {y'}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20+%20%5Cfrac%7B2%7D%7B4%20%5C,%20t%20+%20y%20-%202%7D%20-%202%20=%204%20%5C,%20t%20-%20%7By'%7D" alt="-y + \frac{2}{4 \, t + y - 2} - 2 = 4 \, t - {y'}" title="-y + \frac{2}{4 \, t + y - 2} - 2 = 4 \, t - {y'}" data-latex="-y + \frac{2}{4 \, t + y - 2} - 2 = 4 \, t - {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = 1" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-5155" title="F5 | Substitution strategies | ver. 5155"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, t - y + {y'} + \frac{2}{5 \, t - y - 2} = \left(-3\right)" alt="5 \, t - y + {y'} + \frac{2}{5 \, t - y - 2} = \left(-3\right)" title="5 \, t - y + {y'} + \frac{2}{5 \, t - y - 2} = \left(-3\right)" data-latex="5 \, t - y + {y'} + \frac{2}{5 \, t - y - 2} = \left(-3\right)"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y^{2} = t y {y'} - 2 \, t^{2}" alt="-y^{2} = t y {y'} - 2 \, t^{2}" title="-y^{2} = t y {y'} - 2 \, t^{2}" data-latex="-y^{2} = t y {y'} - 2 \, t^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20t%20-%20y%20+%20%7By'%7D%20+%20%5Cfrac%7B2%7D%7B5%20%5C,%20t%20-%20y%20-%202%7D%20=%20%5Cleft(-3%5Cright)" alt="5 \, t - y + {y'} + \frac{2}{5 \, t - y - 2} = \left(-3\right)" title="5 \, t - y + {y'} + \frac{2}{5 \, t - y - 2} = \left(-3\right)" data-latex="5 \, t - y + {y'} + \frac{2}{5 \, t - y - 2} = \left(-3\right)"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%5E%7B2%7D%20=%20t%20y%20%7By'%7D%20-%202%20%5C,%20t%5E%7B2%7D" alt="-y^{2} = t y {y'} - 2 \, t^{2}" title="-y^{2} = t y {y'} - 2 \, t^{2}" data-latex="-y^{2} = t y {y'} - 2 \, t^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = 1" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-4402" title="F5 | Substitution strategies | ver. 4402"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - 3 \, t^{2} = y^{2}" alt="-t y {y'} - 3 \, t^{2} = y^{2}" title="-t y {y'} - 3 \, t^{2} = y^{2}" data-latex="-t y {y'} - 3 \, t^{2} = y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, t - {y'} - 1 = -y + \frac{4}{3 \, t + y - 4}" alt="3 \, t - {y'} - 1 = -y + \frac{4}{3 \, t + y - 4}" title="3 \, t - {y'} - 1 = -y + \frac{4}{3 \, t + y - 4}" data-latex="3 \, t - {y'} - 1 = -y + \frac{4}{3 \, t + y - 4}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%203%20%5C,%20t%5E%7B2%7D%20=%20y%5E%7B2%7D" alt="-t y {y'} - 3 \, t^{2} = y^{2}" title="-t y {y'} - 3 \, t^{2} = y^{2}" data-latex="-t y {y'} - 3 \, t^{2} = y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20t%20-%20%7By'%7D%20-%201%20=%20-y%20+%20%5Cfrac%7B4%7D%7B3%20%5C,%20t%20+%20y%20-%204%7D" alt="3 \, t - {y'} - 1 = -y + \frac{4}{3 \, t + y - 4}" title="3 \, t - {y'} - 1 = -y + \frac{4}{3 \, t + y - 4}" data-latex="3 \, t - {y'} - 1 = -y + \frac{4}{3 \, t + y - 4}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = 1" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8015" title="F5 | Substitution strategies | ver. 8015"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{2} + y^{2} = -t y {y'}" alt="-t^{2} + y^{2} = -t y {y'}" title="-t^{2} + y^{2} = -t y {y'}" data-latex="-t^{2} + y^{2} = -t y {y'}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t + {y'} + \frac{1}{4 \, t + y + 3} = y + 7" alt="-4 \, t + {y'} + \frac{1}{4 \, t + y + 3} = y + 7" title="-4 \, t + {y'} + \frac{1}{4 \, t + y + 3} = y + 7" data-latex="-4 \, t + {y'} + \frac{1}{4 \, t + y + 3} = y + 7"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B2%7D%20+%20y%5E%7B2%7D%20=%20-t%20y%20%7By'%7D" alt="-t^{2} + y^{2} = -t y {y'}" title="-t^{2} + y^{2} = -t y {y'}" data-latex="-t^{2} + y^{2} = -t y {y'}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%20+%20%7By'%7D%20+%20%5Cfrac%7B1%7D%7B4%20%5C,%20t%20+%20y%20+%203%7D%20=%20y%20+%207" alt="-4 \, t + {y'} + \frac{1}{4 \, t + y + 3} = y + 7" title="-4 \, t + {y'} + \frac{1}{4 \, t + y + 3} = y + 7" data-latex="-4 \, t + {y'} + \frac{1}{4 \, t + y + 3} = y + 7"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 1} = 1" alt="\frac{u {u'}}{u^{2} - 1} = 1" title="\frac{u {u'}}{u^{2} - 1} = 1" data-latex="\frac{u {u'}}{u^{2} - 1} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 1} = 1" title="\frac{u {u'}}{u^{2} - 1} = 1" data-latex="\frac{u {u'}}{u^{2} - 1} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-3666" title="F5 | Substitution strategies | ver. 3666"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - 5 \, t^{2} = y^{2}" alt="-t y {y'} - 5 \, t^{2} = y^{2}" title="-t y {y'} - 5 \, t^{2} = y^{2}" data-latex="-t y {y'} - 5 \, t^{2} = y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y = 4 \, t + {y'} + \frac{4}{4 \, t - y + 5} + 9" alt="y = 4 \, t + {y'} + \frac{4}{4 \, t - y + 5} + 9" title="y = 4 \, t + {y'} + \frac{4}{4 \, t - y + 5} + 9" data-latex="y = 4 \, t + {y'} + \frac{4}{4 \, t - y + 5} + 9"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%205%20%5C,%20t%5E%7B2%7D%20=%20y%5E%7B2%7D" alt="-t y {y'} - 5 \, t^{2} = y^{2}" title="-t y {y'} - 5 \, t^{2} = y^{2}" data-latex="-t y {y'} - 5 \, t^{2} = y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y%20=%204%20%5C,%20t%20+%20%7By'%7D%20+%20%5Cfrac%7B4%7D%7B4%20%5C,%20t%20-%20y%20+%205%7D%20+%209" alt="y = 4 \, t + {y'} + \frac{4}{4 \, t - y + 5} + 9" title="y = 4 \, t + {y'} + \frac{4}{4 \, t - y + 5} + 9" data-latex="y = 4 \, t + {y'} + \frac{4}{4 \, t - y + 5} + 9"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = 1" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8652" title="F5 | Substitution strategies | ver. 8652"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} = 5 \, t^{2} - y^{2}" alt="t y {y'} = 5 \, t^{2} - y^{2}" title="t y {y'} = 5 \, t^{2} - y^{2}" data-latex="t y {y'} = 5 \, t^{2} - y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t - y + {y'} + \frac{5}{t - y + 5} = \left(-6\right)" alt="t - y + {y'} + \frac{5}{t - y + 5} = \left(-6\right)" title="t - y + {y'} + \frac{5}{t - y + 5} = \left(-6\right)" data-latex="t - y + {y'} + \frac{5}{t - y + 5} = \left(-6\right)"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20=%205%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D" alt="t y {y'} = 5 \, t^{2} - y^{2}" title="t y {y'} = 5 \, t^{2} - y^{2}" data-latex="t y {y'} = 5 \, t^{2} - y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20-%20y%20+%20%7By'%7D%20+%20%5Cfrac%7B5%7D%7Bt%20-%20y%20+%205%7D%20=%20%5Cleft(-6%5Cright)" alt="t - y + {y'} + \frac{5}{t - y + 5} = \left(-6\right)" title="t - y + {y'} + \frac{5}{t - y + 5} = \left(-6\right)" data-latex="t - y + {y'} + \frac{5}{t - y + 5} = \left(-6\right)"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = 1" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-3468" title="F5 | Substitution strategies | ver. 3468"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} - y^{2} = t y {y'}" alt="-4 \, t^{2} - y^{2} = t y {y'}" title="-4 \, t^{2} - y^{2} = t y {y'}" data-latex="-4 \, t^{2} - y^{2} = t y {y'}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t + y = {y'} + \frac{1}{4 \, t - y - 4}" alt="-4 \, t + y = {y'} + \frac{1}{4 \, t - y - 4}" title="-4 \, t + y = {y'} + \frac{1}{4 \, t - y - 4}" data-latex="-4 \, t + y = {y'} + \frac{1}{4 \, t - y - 4}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D%20=%20t%20y%20%7By'%7D" alt="-4 \, t^{2} - y^{2} = t y {y'}" title="-4 \, t^{2} - y^{2} = t y {y'}" data-latex="-4 \, t^{2} - y^{2} = t y {y'}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%20+%20y%20=%20%7By'%7D%20+%20%5Cfrac%7B1%7D%7B4%20%5C,%20t%20-%20y%20-%204%7D" alt="-4 \, t + y = {y'} + \frac{1}{4 \, t - y - 4}" title="-4 \, t + y = {y'} + \frac{1}{4 \, t - y - 4}" data-latex="-4 \, t + y = {y'} + \frac{1}{4 \, t - y - 4}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = 1" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-5165" title="F5 | Substitution strategies | ver. 5165"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{2}{4 \, t - y + 2} = 4 \, t - y + {y'} + 6" alt="\frac{2}{4 \, t - y + 2} = 4 \, t - y + {y'} + 6" title="\frac{2}{4 \, t - y + 2} = 4 \, t - y + {y'} + 6" data-latex="\frac{2}{4 \, t - y + 2} = 4 \, t - y + {y'} + 6"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - y^{2} = 4 \, t^{2}" alt="-t y {y'} - y^{2} = 4 \, t^{2}" title="-t y {y'} - y^{2} = 4 \, t^{2}" data-latex="-t y {y'} - y^{2} = 4 \, t^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B2%7D%7B4%20%5C,%20t%20-%20y%20+%202%7D%20=%204%20%5C,%20t%20-%20y%20+%20%7By'%7D%20+%206" alt="\frac{2}{4 \, t - y + 2} = 4 \, t - y + {y'} + 6" title="\frac{2}{4 \, t - y + 2} = 4 \, t - y + {y'} + 6" data-latex="\frac{2}{4 \, t - y + 2} = 4 \, t - y + {y'} + 6"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%20y%5E%7B2%7D%20=%204%20%5C,%20t%5E%7B2%7D" alt="-t y {y'} - y^{2} = 4 \, t^{2}" title="-t y {y'} - y^{2} = 4 \, t^{2}" data-latex="-t y {y'} - y^{2} = 4 \, t^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = 1" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-3819" title="F5 | Substitution strategies | ver. 3819"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} = t^{2} + y^{2}" alt="-t y {y'} = t^{2} + y^{2}" title="-t y {y'} = t^{2} + y^{2}" data-latex="-t y {y'} = t^{2} + y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, t - y + {y'} = \frac{4}{2 \, t + y - 3} - 1" alt="-2 \, t - y + {y'} = \frac{4}{2 \, t + y - 3} - 1" title="-2 \, t - y + {y'} = \frac{4}{2 \, t + y - 3} - 1" data-latex="-2 \, t - y + {y'} = \frac{4}{2 \, t + y - 3} - 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20=%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="-t y {y'} = t^{2} + y^{2}" title="-t y {y'} = t^{2} + y^{2}" data-latex="-t y {y'} = t^{2} + y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20t%20-%20y%20+%20%7By'%7D%20=%20%5Cfrac%7B4%7D%7B2%20%5C,%20t%20+%20y%20-%203%7D%20-%201" alt="-2 \, t - y + {y'} = \frac{4}{2 \, t + y - 3} - 1" title="-2 \, t - y + {y'} = \frac{4}{2 \, t + y - 3} - 1" data-latex="-2 \, t - y + {y'} = \frac{4}{2 \, t + y - 3} - 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = 1" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-2276" title="F5 | Substitution strategies | ver. 2276"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-\frac{1}{4 \, t - y - 4} = 4 \, t - y + {y'}" alt="-\frac{1}{4 \, t - y - 4} = 4 \, t - y + {y'}" title="-\frac{1}{4 \, t - y - 4} = 4 \, t - y + {y'}" data-latex="-\frac{1}{4 \, t - y - 4} = 4 \, t - y + {y'}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, t^{2} - y^{2} = t y {y'}" alt="3 \, t^{2} - y^{2} = t y {y'}" title="3 \, t^{2} - y^{2} = t y {y'}" data-latex="3 \, t^{2} - y^{2} = t y {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%5Cfrac%7B1%7D%7B4%20%5C,%20t%20-%20y%20-%204%7D%20=%204%20%5C,%20t%20-%20y%20+%20%7By'%7D" alt="-\frac{1}{4 \, t - y - 4} = 4 \, t - y + {y'}" title="-\frac{1}{4 \, t - y - 4} = 4 \, t - y + {y'}" data-latex="-\frac{1}{4 \, t - y - 4} = 4 \, t - y + {y'}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D%20=%20t%20y%20%7By'%7D" alt="3 \, t^{2} - y^{2} = t y {y'}" title="3 \, t^{2} - y^{2} = t y {y'}" data-latex="3 \, t^{2} - y^{2} = t y {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = 1" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-5002" title="F5 | Substitution strategies | ver. 5002"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + 2 = -t + y - \frac{2}{t - y + 1}" alt="{y'} + 2 = -t + y - \frac{2}{t - y + 1}" title="{y'} + 2 = -t + y - \frac{2}{t - y + 1}" data-latex="{y'} + 2 = -t + y - \frac{2}{t - y + 1}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{2} + y^{2} = -t y {y'}" alt="-t^{2} + y^{2} = -t y {y'}" title="-t^{2} + y^{2} = -t y {y'}" data-latex="-t^{2} + y^{2} = -t y {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%202%20=%20-t%20+%20y%20-%20%5Cfrac%7B2%7D%7Bt%20-%20y%20+%201%7D" alt="{y'} + 2 = -t + y - \frac{2}{t - y + 1}" title="{y'} + 2 = -t + y - \frac{2}{t - y + 1}" data-latex="{y'} + 2 = -t + y - \frac{2}{t - y + 1}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B2%7D%20+%20y%5E%7B2%7D%20=%20-t%20y%20%7By'%7D" alt="-t^{2} + y^{2} = -t y {y'}" title="-t^{2} + y^{2} = -t y {y'}" data-latex="-t^{2} + y^{2} = -t y {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = 1" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-3876" title="F5 | Substitution strategies | ver. 3876"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = t y {y'} - t^{2} + y^{2}" alt="0 = t y {y'} - t^{2} + y^{2}" title="0 = t y {y'} - t^{2} + y^{2}" data-latex="0 = t y {y'} - t^{2} + y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{4 \, t + y + 2} - 6 = 4 \, t + y - {y'}" alt="\frac{5}{4 \, t + y + 2} - 6 = 4 \, t + y - {y'}" title="\frac{5}{4 \, t + y + 2} - 6 = 4 \, t + y - {y'}" data-latex="\frac{5}{4 \, t + y + 2} - 6 = 4 \, t + y - {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20t%20y%20%7By'%7D%20-%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="0 = t y {y'} - t^{2} + y^{2}" title="0 = t y {y'} - t^{2} + y^{2}" data-latex="0 = t y {y'} - t^{2} + y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7B4%20%5C,%20t%20+%20y%20+%202%7D%20-%206%20=%204%20%5C,%20t%20+%20y%20-%20%7By'%7D" alt="\frac{5}{4 \, t + y + 2} - 6 = 4 \, t + y - {y'}" title="\frac{5}{4 \, t + y + 2} - 6 = 4 \, t + y - {y'}" data-latex="\frac{5}{4 \, t + y + 2} - 6 = 4 \, t + y - {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = 1" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-0152" title="F5 | Substitution strategies | ver. 0152"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y^{2} = t y {y'} + 4 \, t^{2}" alt="-y^{2} = t y {y'} + 4 \, t^{2}" title="-y^{2} = t y {y'} + 4 \, t^{2}" data-latex="-y^{2} = t y {y'} + 4 \, t^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, t - y - \frac{3}{2 \, t + y + 3} = -{y'} + 5" alt="-2 \, t - y - \frac{3}{2 \, t + y + 3} = -{y'} + 5" title="-2 \, t - y - \frac{3}{2 \, t + y + 3} = -{y'} + 5" data-latex="-2 \, t - y - \frac{3}{2 \, t + y + 3} = -{y'} + 5"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%5E%7B2%7D%20=%20t%20y%20%7By'%7D%20+%204%20%5C,%20t%5E%7B2%7D" alt="-y^{2} = t y {y'} + 4 \, t^{2}" title="-y^{2} = t y {y'} + 4 \, t^{2}" data-latex="-y^{2} = t y {y'} + 4 \, t^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20t%20-%20y%20-%20%5Cfrac%7B3%7D%7B2%20%5C,%20t%20+%20y%20+%203%7D%20=%20-%7By'%7D%20+%205" alt="-2 \, t - y - \frac{3}{2 \, t + y + 3} = -{y'} + 5" title="-2 \, t - y - \frac{3}{2 \, t + y + 3} = -{y'} + 5" data-latex="-2 \, t - y - \frac{3}{2 \, t + y + 3} = -{y'} + 5"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = 1" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-1652" title="F5 | Substitution strategies | ver. 1652"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y^{2} = t y {y'} + t^{2}" alt="-y^{2} = t y {y'} + t^{2}" title="-y^{2} = t y {y'} + t^{2}" data-latex="-y^{2} = t y {y'} + t^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y + \frac{4}{2 \, t - y - 3} - 1 = -2 \, t - {y'}" alt="-y + \frac{4}{2 \, t - y - 3} - 1 = -2 \, t - {y'}" title="-y + \frac{4}{2 \, t - y - 3} - 1 = -2 \, t - {y'}" data-latex="-y + \frac{4}{2 \, t - y - 3} - 1 = -2 \, t - {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%5E%7B2%7D%20=%20t%20y%20%7By'%7D%20+%20t%5E%7B2%7D" alt="-y^{2} = t y {y'} + t^{2}" title="-y^{2} = t y {y'} + t^{2}" data-latex="-y^{2} = t y {y'} + t^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20+%20%5Cfrac%7B4%7D%7B2%20%5C,%20t%20-%20y%20-%203%7D%20-%201%20=%20-2%20%5C,%20t%20-%20%7By'%7D" alt="-y + \frac{4}{2 \, t - y - 3} - 1 = -2 \, t - {y'}" title="-y + \frac{4}{2 \, t - y - 3} - 1 = -2 \, t - {y'}" data-latex="-y + \frac{4}{2 \, t - y - 3} - 1 = -2 \, t - {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = 1" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-9577" title="F5 | Substitution strategies | ver. 9577"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y - \frac{1}{2 \, t - y + 1} - 3 = 2 \, t + {y'}" alt="y - \frac{1}{2 \, t - y + 1} - 3 = 2 \, t + {y'}" title="y - \frac{1}{2 \, t - y + 1} - 3 = 2 \, t + {y'}" data-latex="y - \frac{1}{2 \, t - y + 1} - 3 = 2 \, t + {y'}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = t y {y'} - 5 \, t^{2} + y^{2}" alt="0 = t y {y'} - 5 \, t^{2} + y^{2}" title="0 = t y {y'} - 5 \, t^{2} + y^{2}" data-latex="0 = t y {y'} - 5 \, t^{2} + y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y%20-%20%5Cfrac%7B1%7D%7B2%20%5C,%20t%20-%20y%20+%201%7D%20-%203%20=%202%20%5C,%20t%20+%20%7By'%7D" alt="y - \frac{1}{2 \, t - y + 1} - 3 = 2 \, t + {y'}" title="y - \frac{1}{2 \, t - y + 1} - 3 = 2 \, t + {y'}" data-latex="y - \frac{1}{2 \, t - y + 1} - 3 = 2 \, t + {y'}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20t%20y%20%7By'%7D%20-%205%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="0 = t y {y'} - 5 \, t^{2} + y^{2}" title="0 = t y {y'} - 5 \, t^{2} + y^{2}" data-latex="0 = t y {y'} - 5 \, t^{2} + y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = 1" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-6740" title="F5 | Substitution strategies | ver. 6740"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} - \frac{4}{2 \, t - y + 4} - 6 = 2 \, t - y" alt="-{y'} - \frac{4}{2 \, t - y + 4} - 6 = 2 \, t - y" title="-{y'} - \frac{4}{2 \, t - y + 4} - 6 = 2 \, t - y" data-latex="-{y'} - \frac{4}{2 \, t - y + 4} - 6 = 2 \, t - y"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4 \, t^{2} - y^{2} = t y {y'}" alt="4 \, t^{2} - y^{2} = t y {y'}" title="4 \, t^{2} - y^{2} = t y {y'}" data-latex="4 \, t^{2} - y^{2} = t y {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20-%20%5Cfrac%7B4%7D%7B2%20%5C,%20t%20-%20y%20+%204%7D%20-%206%20=%202%20%5C,%20t%20-%20y" alt="-{y'} - \frac{4}{2 \, t - y + 4} - 6 = 2 \, t - y" title="-{y'} - \frac{4}{2 \, t - y + 4} - 6 = 2 \, t - y" data-latex="-{y'} - \frac{4}{2 \, t - y + 4} - 6 = 2 \, t - y"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?4%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D%20=%20t%20y%20%7By'%7D" alt="4 \, t^{2} - y^{2} = t y {y'}" title="4 \, t^{2} - y^{2} = t y {y'}" data-latex="4 \, t^{2} - y^{2} = t y {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = 1" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-9769" title="F5 | Substitution strategies | ver. 9769"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} = -5 \, t^{2} + y^{2}" alt="-t y {y'} = -5 \, t^{2} + y^{2}" title="-t y {y'} = -5 \, t^{2} + y^{2}" data-latex="-t y {y'} = -5 \, t^{2} + y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t + {y'} + \frac{4}{t + y + 5} - 6 = y" alt="-t + {y'} + \frac{4}{t + y + 5} - 6 = y" title="-t + {y'} + \frac{4}{t + y + 5} - 6 = y" data-latex="-t + {y'} + \frac{4}{t + y + 5} - 6 = y"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20=%20-5%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="-t y {y'} = -5 \, t^{2} + y^{2}" title="-t y {y'} = -5 \, t^{2} + y^{2}" data-latex="-t y {y'} = -5 \, t^{2} + y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20+%20%7By'%7D%20+%20%5Cfrac%7B4%7D%7Bt%20+%20y%20+%205%7D%20-%206%20=%20y" alt="-t + {y'} + \frac{4}{t + y + 5} - 6 = y" title="-t + {y'} + \frac{4}{t + y + 5} - 6 = y" data-latex="-t + {y'} + \frac{4}{t + y + 5} - 6 = y"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = 1" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-0451" title="F5 | Substitution strategies | ver. 0451"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t - {y'} + \frac{2}{4 \, t - y + 1} = -y + 5" alt="-4 \, t - {y'} + \frac{2}{4 \, t - y + 1} = -y + 5" title="-4 \, t - {y'} + \frac{2}{4 \, t - y + 1} = -y + 5" data-latex="-4 \, t - {y'} + \frac{2}{4 \, t - y + 1} = -y + 5"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} = t y {y'} + y^{2}" alt="-4 \, t^{2} = t y {y'} + y^{2}" title="-4 \, t^{2} = t y {y'} + y^{2}" data-latex="-4 \, t^{2} = t y {y'} + y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%20-%20%7By'%7D%20+%20%5Cfrac%7B2%7D%7B4%20%5C,%20t%20-%20y%20+%201%7D%20=%20-y%20+%205" alt="-4 \, t - {y'} + \frac{2}{4 \, t - y + 1} = -y + 5" title="-4 \, t - {y'} + \frac{2}{4 \, t - y + 1} = -y + 5" data-latex="-4 \, t - {y'} + \frac{2}{4 \, t - y + 1} = -y + 5"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%5E%7B2%7D%20=%20t%20y%20%7By'%7D%20+%20y%5E%7B2%7D" alt="-4 \, t^{2} = t y {y'} + y^{2}" title="-4 \, t^{2} = t y {y'} + y^{2}" data-latex="-4 \, t^{2} = t y {y'} + y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = 1" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-9024" title="F5 | Substitution strategies | ver. 9024"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} = 2 \, t^{2} + y^{2}" alt="-t y {y'} = 2 \, t^{2} + y^{2}" title="-t y {y'} = 2 \, t^{2} + y^{2}" data-latex="-t y {y'} = 2 \, t^{2} + y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y + {y'} + 5 = -t - \frac{1}{t - y + 4}" alt="-y + {y'} + 5 = -t - \frac{1}{t - y + 4}" title="-y + {y'} + 5 = -t - \frac{1}{t - y + 4}" data-latex="-y + {y'} + 5 = -t - \frac{1}{t - y + 4}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20=%202%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="-t y {y'} = 2 \, t^{2} + y^{2}" title="-t y {y'} = 2 \, t^{2} + y^{2}" data-latex="-t y {y'} = 2 \, t^{2} + y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20+%20%7By'%7D%20+%205%20=%20-t%20-%20%5Cfrac%7B1%7D%7Bt%20-%20y%20+%204%7D" alt="-y + {y'} + 5 = -t - \frac{1}{t - y + 4}" title="-y + {y'} + 5 = -t - \frac{1}{t - y + 4}" data-latex="-y + {y'} + 5 = -t - \frac{1}{t - y + 4}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = 1" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-9669" title="F5 | Substitution strategies | ver. 9669"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{2} - y^{2} = t y {y'}" alt="-t^{2} - y^{2} = t y {y'}" title="-t^{2} - y^{2} = t y {y'}" data-latex="-t^{2} - y^{2} = t y {y'}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, t - \frac{3}{5 \, t - y - 4} = y - {y'} - 1" alt="5 \, t - \frac{3}{5 \, t - y - 4} = y - {y'} - 1" title="5 \, t - \frac{3}{5 \, t - y - 4} = y - {y'} - 1" data-latex="5 \, t - \frac{3}{5 \, t - y - 4} = y - {y'} - 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%5E%7B2%7D%20-%20y%5E%7B2%7D%20=%20t%20y%20%7By'%7D" alt="-t^{2} - y^{2} = t y {y'}" title="-t^{2} - y^{2} = t y {y'}" data-latex="-t^{2} - y^{2} = t y {y'}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20t%20-%20%5Cfrac%7B3%7D%7B5%20%5C,%20t%20-%20y%20-%204%7D%20=%20y%20-%20%7By'%7D%20-%201" alt="5 \, t - \frac{3}{5 \, t - y - 4} = y - {y'} - 1" title="5 \, t - \frac{3}{5 \, t - y - 4} = y - {y'} - 1" data-latex="5 \, t - \frac{3}{5 \, t - y - 4} = y - {y'} - 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = 1" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-5469" title="F5 | Substitution strategies | ver. 5469"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, t - y - 5 = -{y'} - \frac{5}{2 \, t + y + 3}" alt="-2 \, t - y - 5 = -{y'} - \frac{5}{2 \, t + y + 3}" title="-2 \, t - y - 5 = -{y'} - \frac{5}{2 \, t + y + 3}" data-latex="-2 \, t - y - 5 = -{y'} - \frac{5}{2 \, t + y + 3}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} + y^{2} = -5 \, t^{2}" alt="t y {y'} + y^{2} = -5 \, t^{2}" title="t y {y'} + y^{2} = -5 \, t^{2}" data-latex="t y {y'} + y^{2} = -5 \, t^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20t%20-%20y%20-%205%20=%20-%7By'%7D%20-%20%5Cfrac%7B5%7D%7B2%20%5C,%20t%20+%20y%20+%203%7D" alt="-2 \, t - y - 5 = -{y'} - \frac{5}{2 \, t + y + 3}" title="-2 \, t - y - 5 = -{y'} - \frac{5}{2 \, t + y + 3}" data-latex="-2 \, t - y - 5 = -{y'} - \frac{5}{2 \, t + y + 3}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20+%20y%5E%7B2%7D%20=%20-5%20%5C,%20t%5E%7B2%7D" alt="t y {y'} + y^{2} = -5 \, t^{2}" title="t y {y'} + y^{2} = -5 \, t^{2}" data-latex="t y {y'} + y^{2} = -5 \, t^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = 1" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-9727" title="F5 | Substitution strategies | ver. 9727"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - t^{2} - y^{2} = 0" alt="-t y {y'} - t^{2} - y^{2} = 0" title="-t y {y'} - t^{2} - y^{2} = 0" data-latex="-t y {y'} - t^{2} - y^{2} = 0"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t - y + {y'} = -\frac{4}{t + y + 2} + 3" alt="-t - y + {y'} = -\frac{4}{t + y + 2} + 3" title="-t - y + {y'} = -\frac{4}{t + y + 2} + 3" data-latex="-t - y + {y'} = -\frac{4}{t + y + 2} + 3"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%20t%5E%7B2%7D%20-%20y%5E%7B2%7D%20=%200" alt="-t y {y'} - t^{2} - y^{2} = 0" title="-t y {y'} - t^{2} - y^{2} = 0" data-latex="-t y {y'} - t^{2} - y^{2} = 0"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20-%20y%20+%20%7By'%7D%20=%20-%5Cfrac%7B4%7D%7Bt%20+%20y%20+%202%7D%20+%203" alt="-t - y + {y'} = -\frac{4}{t + y + 2} + 3" title="-t - y + {y'} = -\frac{4}{t + y + 2} + 3" data-latex="-t - y + {y'} = -\frac{4}{t + y + 2} + 3"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = 1" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-0433" title="F5 | Substitution strategies | ver. 0433"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -t y {y'} - 4 \, t^{2} - y^{2}" alt="0 = -t y {y'} - 4 \, t^{2} - y^{2}" title="0 = -t y {y'} - 4 \, t^{2} - y^{2}" data-latex="0 = -t y {y'} - 4 \, t^{2} - y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y - {y'} = -t + \frac{5}{t + y - 1}" alt="y - {y'} = -t + \frac{5}{t + y - 1}" title="y - {y'} = -t + \frac{5}{t + y - 1}" data-latex="y - {y'} = -t + \frac{5}{t + y - 1}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-t%20y%20%7By'%7D%20-%204%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D" alt="0 = -t y {y'} - 4 \, t^{2} - y^{2}" title="0 = -t y {y'} - 4 \, t^{2} - y^{2}" data-latex="0 = -t y {y'} - 4 \, t^{2} - y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y%20-%20%7By'%7D%20=%20-t%20+%20%5Cfrac%7B5%7D%7Bt%20+%20y%20-%201%7D" alt="y - {y'} = -t + \frac{5}{t + y - 1}" title="y - {y'} = -t + \frac{5}{t + y - 1}" data-latex="y - {y'} = -t + \frac{5}{t + y - 1}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = 1" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-0969" title="F5 | Substitution strategies | ver. 0969"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} = 5 \, t - y + \frac{4}{5 \, t - y - 5}" alt="-{y'} = 5 \, t - y + \frac{4}{5 \, t - y - 5}" title="-{y'} = 5 \, t - y + \frac{4}{5 \, t - y - 5}" data-latex="-{y'} = 5 \, t - y + \frac{4}{5 \, t - y - 5}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t^{2} = -t y {y'} - y^{2}" alt="-4 \, t^{2} = -t y {y'} - y^{2}" title="-4 \, t^{2} = -t y {y'} - y^{2}" data-latex="-4 \, t^{2} = -t y {y'} - y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20=%205%20%5C,%20t%20-%20y%20+%20%5Cfrac%7B4%7D%7B5%20%5C,%20t%20-%20y%20-%205%7D" alt="-{y'} = 5 \, t - y + \frac{4}{5 \, t - y - 5}" title="-{y'} = 5 \, t - y + \frac{4}{5 \, t - y - 5}" data-latex="-{y'} = 5 \, t - y + \frac{4}{5 \, t - y - 5}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%5E%7B2%7D%20=%20-t%20y%20%7By'%7D%20-%20y%5E%7B2%7D" alt="-4 \, t^{2} = -t y {y'} - y^{2}" title="-4 \, t^{2} = -t y {y'} - y^{2}" data-latex="-4 \, t^{2} = -t y {y'} - y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = 1" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-7591" title="F5 | Substitution strategies | ver. 7591"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y + {y'} - \frac{4}{3 \, t - y + 4} = -3 \, t - 7" alt="-y + {y'} - \frac{4}{3 \, t - y + 4} = -3 \, t - 7" title="-y + {y'} - \frac{4}{3 \, t - y + 4} = -3 \, t - 7" data-latex="-y + {y'} - \frac{4}{3 \, t - y + 4} = -3 \, t - 7"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} + 3 \, t^{2} = -y^{2}" alt="t y {y'} + 3 \, t^{2} = -y^{2}" title="t y {y'} + 3 \, t^{2} = -y^{2}" data-latex="t y {y'} + 3 \, t^{2} = -y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20+%20%7By'%7D%20-%20%5Cfrac%7B4%7D%7B3%20%5C,%20t%20-%20y%20+%204%7D%20=%20-3%20%5C,%20t%20-%207" alt="-y + {y'} - \frac{4}{3 \, t - y + 4} = -3 \, t - 7" title="-y + {y'} - \frac{4}{3 \, t - y + 4} = -3 \, t - 7" data-latex="-y + {y'} - \frac{4}{3 \, t - y + 4} = -3 \, t - 7"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20+%203%20%5C,%20t%5E%7B2%7D%20=%20-y%5E%7B2%7D" alt="t y {y'} + 3 \, t^{2} = -y^{2}" title="t y {y'} + 3 \, t^{2} = -y^{2}" data-latex="t y {y'} + 3 \, t^{2} = -y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = 1" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 4} = 1" title="\frac{u {u'}}{u^{2} - 4} = 1" data-latex="\frac{u {u'}}{u^{2} - 4} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 3} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-5996" title="F5 | Substitution strategies | ver. 5996"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y^{2} = -t y {y'} - 5 \, t^{2}" alt="y^{2} = -t y {y'} - 5 \, t^{2}" title="y^{2} = -t y {y'} - 5 \, t^{2}" data-latex="y^{2} = -t y {y'} - 5 \, t^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, t = -y + {y'} + \frac{5}{3 \, t + y - 3}" alt="3 \, t = -y + {y'} + \frac{5}{3 \, t + y - 3}" title="3 \, t = -y + {y'} + \frac{5}{3 \, t + y - 3}" data-latex="3 \, t = -y + {y'} + \frac{5}{3 \, t + y - 3}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y%5E%7B2%7D%20=%20-t%20y%20%7By'%7D%20-%205%20%5C,%20t%5E%7B2%7D" alt="y^{2} = -t y {y'} - 5 \, t^{2}" title="y^{2} = -t y {y'} - 5 \, t^{2}" data-latex="y^{2} = -t y {y'} - 5 \, t^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20t%20=%20-y%20+%20%7By'%7D%20+%20%5Cfrac%7B5%7D%7B3%20%5C,%20t%20+%20y%20-%203%7D" alt="3 \, t = -y + {y'} + \frac{5}{3 \, t + y - 3}" title="3 \, t = -y + {y'} + \frac{5}{3 \, t + y - 3}" data-latex="3 \, t = -y + {y'} + \frac{5}{3 \, t + y - 3}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = 1" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-1529" title="F5 | Substitution strategies | ver. 1529"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, t + \frac{5}{5 \, t - y - 5} = y - {y'}" alt="5 \, t + \frac{5}{5 \, t - y - 5} = y - {y'}" title="5 \, t + \frac{5}{5 \, t - y - 5} = y - {y'}" data-latex="5 \, t + \frac{5}{5 \, t - y - 5} = y - {y'}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -t y {y'} - 5 \, t^{2} - y^{2}" alt="0 = -t y {y'} - 5 \, t^{2} - y^{2}" title="0 = -t y {y'} - 5 \, t^{2} - y^{2}" data-latex="0 = -t y {y'} - 5 \, t^{2} - y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20t%20+%20%5Cfrac%7B5%7D%7B5%20%5C,%20t%20-%20y%20-%205%7D%20=%20y%20-%20%7By'%7D" alt="5 \, t + \frac{5}{5 \, t - y - 5} = y - {y'}" title="5 \, t + \frac{5}{5 \, t - y - 5} = y - {y'}" data-latex="5 \, t + \frac{5}{5 \, t - y - 5} = y - {y'}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-t%20y%20%7By'%7D%20-%205%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D" alt="0 = -t y {y'} - 5 \, t^{2} - y^{2}" title="0 = -t y {y'} - 5 \, t^{2} - y^{2}" data-latex="0 = -t y {y'} - 5 \, t^{2} - y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = 1" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-3132" title="F5 | Substitution strategies | ver. 3132"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + \frac{5}{3 \, t + y - 3} = 3 \, t + y" alt="{y'} + \frac{5}{3 \, t + y - 3} = 3 \, t + y" title="{y'} + \frac{5}{3 \, t + y - 3} = 3 \, t + y" data-latex="{y'} + \frac{5}{3 \, t + y - 3} = 3 \, t + y"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - 4 \, t^{2} - y^{2} = 0" alt="-t y {y'} - 4 \, t^{2} - y^{2} = 0" title="-t y {y'} - 4 \, t^{2} - y^{2} = 0" data-latex="-t y {y'} - 4 \, t^{2} - y^{2} = 0"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%20%5Cfrac%7B5%7D%7B3%20%5C,%20t%20+%20y%20-%203%7D%20=%203%20%5C,%20t%20+%20y" alt="{y'} + \frac{5}{3 \, t + y - 3} = 3 \, t + y" title="{y'} + \frac{5}{3 \, t + y - 3} = 3 \, t + y" data-latex="{y'} + \frac{5}{3 \, t + y - 3} = 3 \, t + y"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%204%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D%20=%200" alt="-t y {y'} - 4 \, t^{2} - y^{2} = 0" title="-t y {y'} - 4 \, t^{2} - y^{2} = 0" data-latex="-t y {y'} - 4 \, t^{2} - y^{2} = 0"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = 1" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-0341" title="F5 | Substitution strategies | ver. 0341"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, t^{2} = t y {y'} + y^{2}" alt="-2 \, t^{2} = t y {y'} + y^{2}" title="-2 \, t^{2} = t y {y'} + y^{2}" data-latex="-2 \, t^{2} = t y {y'} + y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, t + {y'} - \frac{2}{2 \, t - y - 1} + 1 = y" alt="2 \, t + {y'} - \frac{2}{2 \, t - y - 1} + 1 = y" title="2 \, t + {y'} - \frac{2}{2 \, t - y - 1} + 1 = y" data-latex="2 \, t + {y'} - \frac{2}{2 \, t - y - 1} + 1 = y"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20t%5E%7B2%7D%20=%20t%20y%20%7By'%7D%20+%20y%5E%7B2%7D" alt="-2 \, t^{2} = t y {y'} + y^{2}" title="-2 \, t^{2} = t y {y'} + y^{2}" data-latex="-2 \, t^{2} = t y {y'} + y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20t%20+%20%7By'%7D%20-%20%5Cfrac%7B2%7D%7B2%20%5C,%20t%20-%20y%20-%201%7D%20+%201%20=%20y" alt="2 \, t + {y'} - \frac{2}{2 \, t - y - 1} + 1 = y" title="2 \, t + {y'} - \frac{2}{2 \, t - y - 1} + 1 = y" data-latex="2 \, t + {y'} - \frac{2}{2 \, t - y - 1} + 1 = y"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = 1" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-3163" title="F5 | Substitution strategies | ver. 3163"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} - 5 \, t^{2} = -y^{2}" alt="t y {y'} - 5 \, t^{2} = -y^{2}" title="t y {y'} - 5 \, t^{2} = -y^{2}" data-latex="t y {y'} - 5 \, t^{2} = -y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, t = y - {y'} + \frac{2}{5 \, t + y + 1} + 6" alt="-5 \, t = y - {y'} + \frac{2}{5 \, t + y + 1} + 6" title="-5 \, t = y - {y'} + \frac{2}{5 \, t + y + 1} + 6" data-latex="-5 \, t = y - {y'} + \frac{2}{5 \, t + y + 1} + 6"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20-%205%20%5C,%20t%5E%7B2%7D%20=%20-y%5E%7B2%7D" alt="t y {y'} - 5 \, t^{2} = -y^{2}" title="t y {y'} - 5 \, t^{2} = -y^{2}" data-latex="t y {y'} - 5 \, t^{2} = -y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5%20%5C,%20t%20=%20y%20-%20%7By'%7D%20+%20%5Cfrac%7B2%7D%7B5%20%5C,%20t%20+%20y%20+%201%7D%20+%206" alt="-5 \, t = y - {y'} + \frac{2}{5 \, t + y + 1} + 6" title="-5 \, t = y - {y'} + \frac{2}{5 \, t + y + 1} + 6" data-latex="-5 \, t = y - {y'} + \frac{2}{5 \, t + y + 1} + 6"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = 1" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 2} = 1" title="\frac{u {u'}}{u^{2} + 2} = 1" data-latex="\frac{u {u'}}{u^{2} + 2} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-2685" title="F5 | Substitution strategies | ver. 2685"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -t y {y'} + 5 \, t^{2} - y^{2}" alt="0 = -t y {y'} + 5 \, t^{2} - y^{2}" title="0 = -t y {y'} + 5 \, t^{2} - y^{2}" data-latex="0 = -t y {y'} + 5 \, t^{2} - y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y = -t - {y'} - \frac{3}{t - y - 1}" alt="-y = -t - {y'} - \frac{3}{t - y - 1}" title="-y = -t - {y'} - \frac{3}{t - y - 1}" data-latex="-y = -t - {y'} - \frac{3}{t - y - 1}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-t%20y%20%7By'%7D%20+%205%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D" alt="0 = -t y {y'} + 5 \, t^{2} - y^{2}" title="0 = -t y {y'} + 5 \, t^{2} - y^{2}" data-latex="0 = -t y {y'} + 5 \, t^{2} - y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20=%20-t%20-%20%7By'%7D%20-%20%5Cfrac%7B3%7D%7Bt%20-%20y%20-%201%7D" alt="-y = -t - {y'} - \frac{3}{t - y - 1}" title="-y = -t - {y'} - \frac{3}{t - y - 1}" data-latex="-y = -t - {y'} - \frac{3}{t - y - 1}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = 1" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-4179" title="F5 | Substitution strategies | ver. 4179"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, t - y + 3 = -{y'} + \frac{5}{5 \, t - y - 2}" alt="5 \, t - y + 3 = -{y'} + \frac{5}{5 \, t - y - 2}" title="5 \, t - y + 3 = -{y'} + \frac{5}{5 \, t - y - 2}" data-latex="5 \, t - y + 3 = -{y'} + \frac{5}{5 \, t - y - 2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} = 5 \, t^{2} + y^{2}" alt="-t y {y'} = 5 \, t^{2} + y^{2}" title="-t y {y'} = 5 \, t^{2} + y^{2}" data-latex="-t y {y'} = 5 \, t^{2} + y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20t%20-%20y%20+%203%20=%20-%7By'%7D%20+%20%5Cfrac%7B5%7D%7B5%20%5C,%20t%20-%20y%20-%202%7D" alt="5 \, t - y + 3 = -{y'} + \frac{5}{5 \, t - y - 2}" title="5 \, t - y + 3 = -{y'} + \frac{5}{5 \, t - y - 2}" data-latex="5 \, t - y + 3 = -{y'} + \frac{5}{5 \, t - y - 2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20=%205%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="-t y {y'} = 5 \, t^{2} + y^{2}" title="-t y {y'} = 5 \, t^{2} + y^{2}" data-latex="-t y {y'} = 5 \, t^{2} + y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = 1" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-5872" title="F5 | Substitution strategies | ver. 5872"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + \frac{2}{3 \, t + y - 1} = 3 \, t + y + 2" alt="{y'} + \frac{2}{3 \, t + y - 1} = 3 \, t + y + 2" title="{y'} + \frac{2}{3 \, t + y - 1} = 3 \, t + y + 2" data-latex="{y'} + \frac{2}{3 \, t + y - 1} = 3 \, t + y + 2"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} = t^{2} + y^{2}" alt="-t y {y'} = t^{2} + y^{2}" title="-t y {y'} = t^{2} + y^{2}" data-latex="-t y {y'} = t^{2} + y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%20%5Cfrac%7B2%7D%7B3%20%5C,%20t%20+%20y%20-%201%7D%20=%203%20%5C,%20t%20+%20y%20+%202" alt="{y'} + \frac{2}{3 \, t + y - 1} = 3 \, t + y + 2" title="{y'} + \frac{2}{3 \, t + y - 1} = 3 \, t + y + 2" data-latex="{y'} + \frac{2}{3 \, t + y - 1} = 3 \, t + y + 2"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20=%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="-t y {y'} = t^{2} + y^{2}" title="-t y {y'} = t^{2} + y^{2}" data-latex="-t y {y'} = t^{2} + y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = 1" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-3033" title="F5 | Substitution strategies | ver. 3033"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y - \frac{1}{t - y - 1} = -t - {y'}" alt="-y - \frac{1}{t - y - 1} = -t - {y'}" title="-y - \frac{1}{t - y - 1} = -t - {y'}" data-latex="-y - \frac{1}{t - y - 1} = -t - {y'}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - y^{2} = -t^{2}" alt="-t y {y'} - y^{2} = -t^{2}" title="-t y {y'} - y^{2} = -t^{2}" data-latex="-t y {y'} - y^{2} = -t^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20-%20%5Cfrac%7B1%7D%7Bt%20-%20y%20-%201%7D%20=%20-t%20-%20%7By'%7D" alt="-y - \frac{1}{t - y - 1} = -t - {y'}" title="-y - \frac{1}{t - y - 1} = -t - {y'}" data-latex="-y - \frac{1}{t - y - 1} = -t - {y'}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%20y%5E%7B2%7D%20=%20-t%5E%7B2%7D" alt="-t y {y'} - y^{2} = -t^{2}" title="-t y {y'} - y^{2} = -t^{2}" data-latex="-t y {y'} - y^{2} = -t^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 1} = 1" alt="\frac{u {u'}}{u^{2} - 1} = 1" title="\frac{u {u'}}{u^{2} - 1} = 1" data-latex="\frac{u {u'}}{u^{2} - 1} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 1} = 1" title="\frac{u {u'}}{u^{2} - 1} = 1" data-latex="\frac{u {u'}}{u^{2} - 1} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%201%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 1} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-9024" title="F5 | Substitution strategies | ver. 9024"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} = 2 \, t^{2} + y^{2}" alt="-t y {y'} = 2 \, t^{2} + y^{2}" title="-t y {y'} = 2 \, t^{2} + y^{2}" data-latex="-t y {y'} = 2 \, t^{2} + y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y + {y'} + 5 = -t - \frac{1}{t - y + 4}" alt="-y + {y'} + 5 = -t - \frac{1}{t - y + 4}" title="-y + {y'} + 5 = -t - \frac{1}{t - y + 4}" data-latex="-y + {y'} + 5 = -t - \frac{1}{t - y + 4}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20=%202%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="-t y {y'} = 2 \, t^{2} + y^{2}" title="-t y {y'} = 2 \, t^{2} + y^{2}" data-latex="-t y {y'} = 2 \, t^{2} + y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20+%20%7By'%7D%20+%205%20=%20-t%20-%20%5Cfrac%7B1%7D%7Bt%20-%20y%20+%204%7D" alt="-y + {y'} + 5 = -t - \frac{1}{t - y + 4}" title="-y + {y'} + 5 = -t - \frac{1}{t - y + 4}" data-latex="-y + {y'} + 5 = -t - \frac{1}{t - y + 4}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = 1" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-9282" title="F5 | Substitution strategies | ver. 9282"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, t^{2} = -t y {y'} - y^{2}" alt="2 \, t^{2} = -t y {y'} - y^{2}" title="2 \, t^{2} = -t y {y'} - y^{2}" data-latex="2 \, t^{2} = -t y {y'} - y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, t - {y'} = -y + \frac{3}{2 \, t - y + 4} + 6" alt="-2 \, t - {y'} = -y + \frac{3}{2 \, t - y + 4} + 6" title="-2 \, t - {y'} = -y + \frac{3}{2 \, t - y + 4} + 6" data-latex="-2 \, t - {y'} = -y + \frac{3}{2 \, t - y + 4} + 6"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20t%5E%7B2%7D%20=%20-t%20y%20%7By'%7D%20-%20y%5E%7B2%7D" alt="2 \, t^{2} = -t y {y'} - y^{2}" title="2 \, t^{2} = -t y {y'} - y^{2}" data-latex="2 \, t^{2} = -t y {y'} - y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20t%20-%20%7By'%7D%20=%20-y%20+%20%5Cfrac%7B3%7D%7B2%20%5C,%20t%20-%20y%20+%204%7D%20+%206" alt="-2 \, t - {y'} = -y + \frac{3}{2 \, t - y + 4} + 6" title="-2 \, t - {y'} = -y + \frac{3}{2 \, t - y + 4} + 6" data-latex="-2 \, t - {y'} = -y + \frac{3}{2 \, t - y + 4} + 6"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 3} = 1" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 3} = 1" title="\frac{u {u'}}{u^{2} + 3} = 1" data-latex="\frac{u {u'}}{u^{2} + 3} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-3647" title="F5 | Substitution strategies | ver. 3647"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y + \frac{2}{t - y + 4} - 5 = t + {y'}" alt="y + \frac{2}{t - y + 4} - 5 = t + {y'}" title="y + \frac{2}{t - y + 4} - 5 = t + {y'}" data-latex="y + \frac{2}{t - y + 4} - 5 = t + {y'}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t y {y'} = 3 \, t^{2} - y^{2}" alt="t y {y'} = 3 \, t^{2} - y^{2}" title="t y {y'} = 3 \, t^{2} - y^{2}" data-latex="t y {y'} = 3 \, t^{2} - y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y%20+%20%5Cfrac%7B2%7D%7Bt%20-%20y%20+%204%7D%20-%205%20=%20t%20+%20%7By'%7D" alt="y + \frac{2}{t - y + 4} - 5 = t + {y'}" title="y + \frac{2}{t - y + 4} - 5 = t + {y'}" data-latex="y + \frac{2}{t - y + 4} - 5 = t + {y'}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t%20y%20%7By'%7D%20=%203%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D" alt="t y {y'} = 3 \, t^{2} - y^{2}" title="t y {y'} = 3 \, t^{2} - y^{2}" data-latex="t y {y'} = 3 \, t^{2} - y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = 1" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 2} = 1" title="\frac{u {u'}}{u^{2} - 2} = 1" data-latex="\frac{u {u'}}{u^{2} - 2} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 3} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-3924" title="F5 | Substitution strategies | ver. 3924"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t - {y'} + \frac{5}{4 \, t - y + 5} = -y + 9" alt="-4 \, t - {y'} + \frac{5}{4 \, t - y + 5} = -y + 9" title="-4 \, t - {y'} + \frac{5}{4 \, t - y + 5} = -y + 9" data-latex="-4 \, t - {y'} + \frac{5}{4 \, t - y + 5} = -y + 9"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - 2 \, t^{2} - y^{2} = 0" alt="-t y {y'} - 2 \, t^{2} - y^{2} = 0" title="-t y {y'} - 2 \, t^{2} - y^{2} = 0" data-latex="-t y {y'} - 2 \, t^{2} - y^{2} = 0"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%20-%20%7By'%7D%20+%20%5Cfrac%7B5%7D%7B4%20%5C,%20t%20-%20y%20+%205%7D%20=%20-y%20+%209" alt="-4 \, t - {y'} + \frac{5}{4 \, t - y + 5} = -y + 9" title="-4 \, t - {y'} + \frac{5}{4 \, t - y + 5} = -y + 9" data-latex="-4 \, t - {y'} + \frac{5}{4 \, t - y + 5} = -y + 9"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%202%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D%20=%200" alt="-t y {y'} - 2 \, t^{2} - y^{2} = 0" title="-t y {y'} - 2 \, t^{2} - y^{2} = 0" data-latex="-t y {y'} - 2 \, t^{2} - y^{2} = 0"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = 1" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 5} = 1" title="\frac{u {u'}}{u^{2} - 5} = 1" data-latex="\frac{u {u'}}{u^{2} - 5} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-6991" title="F5 | Substitution strategies | ver. 6991"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - 2 \, t^{2} = y^{2}" alt="-t y {y'} - 2 \, t^{2} = y^{2}" title="-t y {y'} - 2 \, t^{2} = y^{2}" data-latex="-t y {y'} - 2 \, t^{2} = y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, t + y = {y'} + \frac{3}{2 \, t + y - 1} - 1" alt="2 \, t + y = {y'} + \frac{3}{2 \, t + y - 1} - 1" title="2 \, t + y = {y'} + \frac{3}{2 \, t + y - 1} - 1" data-latex="2 \, t + y = {y'} + \frac{3}{2 \, t + y - 1} - 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%202%20%5C,%20t%5E%7B2%7D%20=%20y%5E%7B2%7D" alt="-t y {y'} - 2 \, t^{2} = y^{2}" title="-t y {y'} - 2 \, t^{2} = y^{2}" data-latex="-t y {y'} - 2 \, t^{2} = y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20t%20+%20y%20=%20%7By'%7D%20+%20%5Cfrac%7B3%7D%7B2%20%5C,%20t%20+%20y%20-%201%7D%20-%201" alt="2 \, t + y = {y'} + \frac{3}{2 \, t + y - 1} - 1" title="2 \, t + y = {y'} + \frac{3}{2 \, t + y - 1} - 1" data-latex="2 \, t + y = {y'} + \frac{3}{2 \, t + y - 1} - 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = 1" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-8644" title="F5 | Substitution strategies | ver. 8644"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -2 \, t + y - {y'} - \frac{4}{2 \, t - y - 2}" alt="0 = -2 \, t + y - {y'} - \frac{4}{2 \, t - y - 2}" title="0 = -2 \, t + y - {y'} - \frac{4}{2 \, t - y - 2}" data-latex="0 = -2 \, t + y - {y'} - \frac{4}{2 \, t - y - 2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = t y {y'} + 5 \, t^{2} + y^{2}" alt="0 = t y {y'} + 5 \, t^{2} + y^{2}" title="0 = t y {y'} + 5 \, t^{2} + y^{2}" data-latex="0 = t y {y'} + 5 \, t^{2} + y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-2%20%5C,%20t%20+%20y%20-%20%7By'%7D%20-%20%5Cfrac%7B4%7D%7B2%20%5C,%20t%20-%20y%20-%202%7D" alt="0 = -2 \, t + y - {y'} - \frac{4}{2 \, t - y - 2}" title="0 = -2 \, t + y - {y'} - \frac{4}{2 \, t - y - 2}" data-latex="0 = -2 \, t + y - {y'} - \frac{4}{2 \, t - y - 2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20t%20y%20%7By'%7D%20+%205%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="0 = t y {y'} + 5 \, t^{2} + y^{2}" title="0 = t y {y'} + 5 \, t^{2} + y^{2}" data-latex="0 = t y {y'} + 5 \, t^{2} + y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = 1" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-6924" title="F5 | Substitution strategies | ver. 6924"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t + y - {y'} - \frac{1}{4 \, t - y + 1} - 5 = 0" alt="-4 \, t + y - {y'} - \frac{1}{4 \, t - y + 1} - 5 = 0" title="-4 \, t + y - {y'} - \frac{1}{4 \, t - y + 1} - 5 = 0" data-latex="-4 \, t + y - {y'} - \frac{1}{4 \, t - y + 1} - 5 = 0"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, t^{2} - y^{2} = t y {y'}" alt="5 \, t^{2} - y^{2} = t y {y'}" title="5 \, t^{2} - y^{2} = t y {y'}" data-latex="5 \, t^{2} - y^{2} = t y {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%20+%20y%20-%20%7By'%7D%20-%20%5Cfrac%7B1%7D%7B4%20%5C,%20t%20-%20y%20+%201%7D%20-%205%20=%200" alt="-4 \, t + y - {y'} - \frac{1}{4 \, t - y + 1} - 5 = 0" title="-4 \, t + y - {y'} - \frac{1}{4 \, t - y + 1} - 5 = 0" data-latex="-4 \, t + y - {y'} - \frac{1}{4 \, t - y + 1} - 5 = 0"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D%20=%20t%20y%20%7By'%7D" alt="5 \, t^{2} - y^{2} = t y {y'}" title="5 \, t^{2} - y^{2} = t y {y'}" data-latex="5 \, t^{2} - y^{2} = t y {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = 1" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-4860" title="F5 | Substitution strategies | ver. 4860"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} = -2 \, t^{2} + y^{2}" alt="-t y {y'} = -2 \, t^{2} + y^{2}" title="-t y {y'} = -2 \, t^{2} + y^{2}" data-latex="-t y {y'} = -2 \, t^{2} + y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-\frac{3}{4 \, t - y + 5} = -4 \, t + y - {y'} - 9" alt="-\frac{3}{4 \, t - y + 5} = -4 \, t + y - {y'} - 9" title="-\frac{3}{4 \, t - y + 5} = -4 \, t + y - {y'} - 9" data-latex="-\frac{3}{4 \, t - y + 5} = -4 \, t + y - {y'} - 9"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20=%20-2%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="-t y {y'} = -2 \, t^{2} + y^{2}" title="-t y {y'} = -2 \, t^{2} + y^{2}" data-latex="-t y {y'} = -2 \, t^{2} + y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%5Cfrac%7B3%7D%7B4%20%5C,%20t%20-%20y%20+%205%7D%20=%20-4%20%5C,%20t%20+%20y%20-%20%7By'%7D%20-%209" alt="-\frac{3}{4 \, t - y + 5} = -4 \, t + y - {y'} - 9" title="-\frac{3}{4 \, t - y + 5} = -4 \, t + y - {y'} - 9" data-latex="-\frac{3}{4 \, t - y + 5} = -4 \, t + y - {y'} - 9"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = 1" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-1518" title="F5 | Substitution strategies | ver. 1518"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, t + {y'} - 4 = y + \frac{1}{2 \, t + y + 2}" alt="-2 \, t + {y'} - 4 = y + \frac{1}{2 \, t + y + 2}" title="-2 \, t + {y'} - 4 = y + \frac{1}{2 \, t + y + 2}" data-latex="-2 \, t + {y'} - 4 = y + \frac{1}{2 \, t + y + 2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -t y {y'} + 5 \, t^{2} - y^{2}" alt="0 = -t y {y'} + 5 \, t^{2} - y^{2}" title="0 = -t y {y'} + 5 \, t^{2} - y^{2}" data-latex="0 = -t y {y'} + 5 \, t^{2} - y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20t%20+%20%7By'%7D%20-%204%20=%20y%20+%20%5Cfrac%7B1%7D%7B2%20%5C,%20t%20+%20y%20+%202%7D" alt="-2 \, t + {y'} - 4 = y + \frac{1}{2 \, t + y + 2}" title="-2 \, t + {y'} - 4 = y + \frac{1}{2 \, t + y + 2}" data-latex="-2 \, t + {y'} - 4 = y + \frac{1}{2 \, t + y + 2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-t%20y%20%7By'%7D%20+%205%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D" alt="0 = -t y {y'} + 5 \, t^{2} - y^{2}" title="0 = -t y {y'} + 5 \, t^{2} - y^{2}" data-latex="0 = -t y {y'} + 5 \, t^{2} - y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 1} = 1" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 1} = 1" title="\frac{u {u'}}{u^{2} + 1} = 1" data-latex="\frac{u {u'}}{u^{2} + 1} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-7537" title="F5 | Substitution strategies | ver. 7537"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} - 2 \, t^{2} = y^{2}" alt="-t y {y'} - 2 \, t^{2} = y^{2}" title="-t y {y'} - 2 \, t^{2} = y^{2}" data-latex="-t y {y'} - 2 \, t^{2} = y^{2}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?1 = 3 \, t + y - {y'} - \frac{3}{3 \, t + y - 4}" alt="1 = 3 \, t + y - {y'} - \frac{3}{3 \, t + y - 4}" title="1 = 3 \, t + y - {y'} - \frac{3}{3 \, t + y - 4}" data-latex="1 = 3 \, t + y - {y'} - \frac{3}{3 \, t + y - 4}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20-%202%20%5C,%20t%5E%7B2%7D%20=%20y%5E%7B2%7D" alt="-t y {y'} - 2 \, t^{2} = y^{2}" title="-t y {y'} - 2 \, t^{2} = y^{2}" data-latex="-t y {y'} - 2 \, t^{2} = y^{2}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?1%20=%203%20%5C,%20t%20+%20y%20-%20%7By'%7D%20-%20%5Cfrac%7B3%7D%7B3%20%5C,%20t%20+%20y%20-%204%7D" alt="1 = 3 \, t + y - {y'} - \frac{3}{3 \, t + y - 4}" title="1 = 3 \, t + y - {y'} - \frac{3}{3 \, t + y - 4}" data-latex="1 = 3 \, t + y - {y'} - \frac{3}{3 \, t + y - 4}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 3} = 1" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%203%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 3} = 1" title="\frac{u {u'}}{u^{2} - 3} = 1" data-latex="\frac{u {u'}}{u^{2} - 3} = 1"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-3867" title="F5 | Substitution strategies | ver. 3867"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t - {y'} = -y + \frac{5}{4 \, t - y - 1} + 3" alt="-4 \, t - {y'} = -y + \frac{5}{4 \, t - y - 1} + 3" title="-4 \, t - {y'} = -y + \frac{5}{4 \, t - y - 1} + 3" data-latex="-4 \, t - {y'} = -y + \frac{5}{4 \, t - y - 1} + 3"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t y {y'} + 4 \, t^{2} - y^{2} = 0" alt="-t y {y'} + 4 \, t^{2} - y^{2} = 0" title="-t y {y'} + 4 \, t^{2} - y^{2} = 0" data-latex="-t y {y'} + 4 \, t^{2} - y^{2} = 0"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%20-%20%7By'%7D%20=%20-y%20+%20%5Cfrac%7B5%7D%7B4%20%5C,%20t%20-%20y%20-%201%7D%20+%203" alt="-4 \, t - {y'} = -y + \frac{5}{4 \, t - y - 1} + 3" title="-4 \, t - {y'} = -y + \frac{5}{4 \, t - y - 1} + 3" data-latex="-4 \, t - {y'} = -y + \frac{5}{4 \, t - y - 1} + 3"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t%20y%20%7By'%7D%20+%204%20%5C,%20t%5E%7B2%7D%20-%20y%5E%7B2%7D%20=%200" alt="-t y {y'} + 4 \, t^{2} - y^{2} = 0" title="-t y {y'} + 4 \, t^{2} - y^{2} = 0" data-latex="-t y {y'} + 4 \, t^{2} - y^{2} = 0"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = 1" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 5} = 1" title="\frac{u {u'}}{u^{2} + 5} = 1" data-latex="\frac{u {u'}}{u^{2} + 5} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%204%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} - 4} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-3764" title="F5 | Substitution strategies | ver. 3764"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{y'} - 9 = 5 \, t - y + \frac{4}{5 \, t - y + 4}" alt="-{y'} - 9 = 5 \, t - y + \frac{4}{5 \, t - y + 4}" title="-{y'} - 9 = 5 \, t - y + \frac{4}{5 \, t - y + 4}" data-latex="-{y'} - 9 = 5 \, t - y + \frac{4}{5 \, t - y + 4}"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5 \, t^{2} + y^{2} = -t y {y'}" alt="5 \, t^{2} + y^{2} = -t y {y'}" title="5 \, t^{2} + y^{2} = -t y {y'}" data-latex="5 \, t^{2} + y^{2} = -t y {y'}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7By'%7D%20-%209%20=%205%20%5C,%20t%20-%20y%20+%20%5Cfrac%7B4%7D%7B5%20%5C,%20t%20-%20y%20+%204%7D" alt="-{y'} - 9 = 5 \, t - y + \frac{4}{5 \, t - y + 4}" title="-{y'} - 9 = 5 \, t - y + \frac{4}{5 \, t - y + 4}" data-latex="-{y'} - 9 = 5 \, t - y + \frac{4}{5 \, t - y + 4}"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?5%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D%20=%20-t%20y%20%7By'%7D" alt="5 \, t^{2} + y^{2} = -t y {y'}" title="5 \, t^{2} + y^{2} = -t y {y'}" data-latex="5 \, t^{2} + y^{2} = -t y {y'}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 4} = 1" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%204%7D%20=%201" alt="\frac{u {u'}}{u^{2} + 4} = 1" title="\frac{u {u'}}{u^{2} + 4} = 1" data-latex="\frac{u {u'}}{u^{2} + 4} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%205%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 5} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="F5-7146" title="F5 | Substitution strategies | ver. 7146"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>F5.</strong></p><p> Apply each of the substitutions <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=\frac{y}{t}" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"/> and <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"/> to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) </p><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, t - {y'} + \frac{1}{4 \, t - y + 2} = -y + 6" alt="-4 \, t - {y'} + \frac{1}{4 \, t - y + 2} = -y + 6" title="-4 \, t - {y'} + \frac{1}{4 \, t - y + 2} = -y + 6" data-latex="-4 \, t - {y'} + \frac{1}{4 \, t - y + 2} = -y + 6"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = t y {y'} + 2 \, t^{2} + y^{2}" alt="0 = t y {y'} + 2 \, t^{2} + y^{2}" title="0 = t y {y'} + 2 \, t^{2} + y^{2}" data-latex="0 = t y {y'} + 2 \, t^{2} + y^{2}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;F5.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Apply each of the substitutions &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=%5Cfrac%7By%7D%7Bt%7D" alt="u=\frac{y}{t}" title="u=\frac{y}{t}" data-latex="u=\frac{y}{t}"&gt; and &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?u=at+by+c" alt="u=at+by+c" title="u=at+by+c" data-latex="u=at+by+c"&gt; to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) &lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20t%20-%20%7By'%7D%20+%20%5Cfrac%7B1%7D%7B4%20%5C,%20t%20-%20y%20+%202%7D%20=%20-y%20+%206" alt="-4 \, t - {y'} + \frac{1}{4 \, t - y + 2} = -y + 6" title="-4 \, t - {y'} + \frac{1}{4 \, t - y + 2} = -y + 6" data-latex="-4 \, t - {y'} + \frac{1}{4 \, t - y + 2} = -y + 6"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20t%20y%20%7By'%7D%20+%202%20%5C,%20t%5E%7B2%7D%20+%20y%5E%7B2%7D" alt="0 = t y {y'} + 2 \, t^{2} + y^{2}" title="0 = t y {y'} + 2 \, t^{2} + y^{2}" data-latex="0 = t y {y'} + 2 \, t^{2} + y^{2}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><ol type="a"><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} - 1} = 1" alt="\frac{u {u'}}{u^{2} - 1} = 1" title="\frac{u {u'}}{u^{2} - 1} = 1" data-latex="\frac{u {u'}}{u^{2} - 1} = 1"/></li><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"/></li></ol></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;ol type="a"&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20-%201%7D%20=%201" alt="\frac{u {u'}}{u^{2} - 1} = 1" title="\frac{u {u'}}{u^{2} - 1} = 1" data-latex="\frac{u {u'}}{u^{2} - 1} = 1"&gt;
    &lt;/li&gt;
    &lt;li&gt;
      &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7Bu%20%7Bu'%7D%7D%7Bu%5E%7B2%7D%20+%202%7D%20=%20%5Cfrac%7B1%7D%7Bt%7D" alt="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" title="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}" data-latex="\frac{u {u'}}{u^{2} + 2} = \frac{1}{t}"&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item></objectbank>
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