0011.qti

<item ident="F5-7332" title="F5 | Substitution strategies | ver. 7332">
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        <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>
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      <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;

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  <itemfeedback ident="general_fb">
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          <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>
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        <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;

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