X2.qti

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<questestinterop xmlns="http://www.imsglobal.org/xsd/ims_qtiasiv1p2" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.imsglobal.org/xsd/ims_qtiasiv1p2 http://www.imsglobal.org/xsd/ims_qtiasiv1p2p1.xsd">
  <objectbank ident="X2">
    <qtimetadata>
      <qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- X2</fieldentry></qtimetadatafield>
    </qtimetadata>
  <item ident="X2-0368" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 0368"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 4\right)} {\left(t - 5\right)} {y'''} - {\left(t - 1\right)} y e^{t} = t^{2} + 5 \, {y''} + 25 \hspace{2em} x( 1 )= 7" alt="-{\left(t + 4\right)} {\left(t - 5\right)} {y'''} - {\left(t - 1\right)} y e^{t} = t^{2} + 5 \, {y''} + 25 \hspace{2em} x( 1 )= 7" title="-{\left(t + 4\right)} {\left(t - 5\right)} {y'''} - {\left(t - 1\right)} y e^{t} = t^{2} + 5 \, {y''} + 25 \hspace{2em} x( 1 )= 7" data-latex="-{\left(t + 4\right)} {\left(t - 5\right)} {y'''} - {\left(t - 1\right)} y e^{t} = t^{2} + 5 \, {y''} + 25 \hspace{2em} x( 1 )= 7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%204%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20-%20%7B%5Cleft(t%20-%201%5Cright)%7D%20y%20e%5E%7Bt%7D%20=%20t%5E%7B2%7D%20+%205%20%5C,%20%7By''%7D%20+%2025%20%5Chspace%7B2em%7D%20x(%201%20)=%207" alt="-{\left(t + 4\right)} {\left(t - 5\right)} {y'''} - {\left(t - 1\right)} y e^{t} = t^{2} + 5 \, {y''} + 25 \hspace{2em} x( 1 )= 7" title="-{\left(t + 4\right)} {\left(t - 5\right)} {y'''} - {\left(t - 1\right)} y e^{t} = t^{2} + 5 \, {y''} + 25 \hspace{2em} x( 1 )= 7" data-latex="-{\left(t + 4\right)} {\left(t - 5\right)} {y'''} - {\left(t - 1\right)} y e^{t} = t^{2} + 5 \, {y''} + 25 \hspace{2em} x( 1 )= 7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,5)" alt="(-4,5)" title="(-4,5)" data-latex="(-4,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,5)" alt="(-4,5)" title="(-4,5)" data-latex="(-4,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-0503" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 0503"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t - 5\right)} {y'''} e^{t} = {\left(t + 4\right)} {\left(t + 2\right)} y + 3 \, {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 9 )= 2" alt="-{\left(t - 5\right)} {y'''} e^{t} = {\left(t + 4\right)} {\left(t + 2\right)} y + 3 \, {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 9 )= 2" title="-{\left(t - 5\right)} {y'''} e^{t} = {\left(t + 4\right)} {\left(t + 2\right)} y + 3 \, {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 9 )= 2" data-latex="-{\left(t - 5\right)} {y'''} e^{t} = {\left(t + 4\right)} {\left(t + 2\right)} y + 3 \, {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 9 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20=%20%7B%5Cleft(t%20+%204%5Cright)%7D%20%7B%5Cleft(t%20+%202%5Cright)%7D%20y%20+%203%20%5C,%20%7By''%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%209%20)=%202" alt="-{\left(t - 5\right)} {y'''} e^{t} = {\left(t + 4\right)} {\left(t + 2\right)} y + 3 \, {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 9 )= 2" title="-{\left(t - 5\right)} {y'''} e^{t} = {\left(t + 4\right)} {\left(t + 2\right)} y + 3 \, {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 9 )= 2" data-latex="-{\left(t - 5\right)} {y'''} e^{t} = {\left(t + 4\right)} {\left(t + 2\right)} y + 3 \, {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 9 )= 2"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+\infty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+%5Cinfty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-7892" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 7892"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 5\right)} t {y'''} = -t^{2} - {\left(t - 5\right)} y - 4 \, {y''} - 16 \hspace{2em} x( -1 )= 1" alt="{\left(t + 5\right)} t {y'''} = -t^{2} - {\left(t - 5\right)} y - 4 \, {y''} - 16 \hspace{2em} x( -1 )= 1" title="{\left(t + 5\right)} t {y'''} = -t^{2} - {\left(t - 5\right)} y - 4 \, {y''} - 16 \hspace{2em} x( -1 )= 1" data-latex="{\left(t + 5\right)} t {y'''} = -t^{2} - {\left(t - 5\right)} y - 4 \, {y''} - 16 \hspace{2em} x( -1 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%205%5Cright)%7D%20t%20%7By'''%7D%20=%20-t%5E%7B2%7D%20-%20%7B%5Cleft(t%20-%205%5Cright)%7D%20y%20-%204%20%5C,%20%7By''%7D%20-%2016%20%5Chspace%7B2em%7D%20x(%20-1%20)=%201" alt="{\left(t + 5\right)} t {y'''} = -t^{2} - {\left(t - 5\right)} y - 4 \, {y''} - 16 \hspace{2em} x( -1 )= 1" title="{\left(t + 5\right)} t {y'''} = -t^{2} - {\left(t - 5\right)} y - 4 \, {y''} - 16 \hspace{2em} x( -1 )= 1" data-latex="{\left(t + 5\right)} t {y'''} = -t^{2} - {\left(t - 5\right)} y - 4 \, {y''} - 16 \hspace{2em} x( -1 )= 1"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,0)" alt="(-5,0)" title="(-5,0)" data-latex="(-5,0)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,0)" alt="(-5,0)" title="(-5,0)" data-latex="(-5,0)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-0316" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 0316"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} + 5 \, {y''} + 25 = -{\left(t + 6\right)} {\left(t - 6\right)} {y'''} - t y e^{t} \hspace{2em} x( -1 )= -10" alt="t^{2} + 5 \, {y''} + 25 = -{\left(t + 6\right)} {\left(t - 6\right)} {y'''} - t y e^{t} \hspace{2em} x( -1 )= -10" title="t^{2} + 5 \, {y''} + 25 = -{\left(t + 6\right)} {\left(t - 6\right)} {y'''} - t y e^{t} \hspace{2em} x( -1 )= -10" data-latex="t^{2} + 5 \, {y''} + 25 = -{\left(t + 6\right)} {\left(t - 6\right)} {y'''} - t y e^{t} \hspace{2em} x( -1 )= -10"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&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+%205%20%5C,%20%7By''%7D%20+%2025%20=%20-%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By'''%7D%20-%20t%20y%20e%5E%7Bt%7D%20%5Chspace%7B2em%7D%20x(%20-1%20)=%20-10" alt="t^{2} + 5 \, {y''} + 25 = -{\left(t + 6\right)} {\left(t - 6\right)} {y'''} - t y e^{t} \hspace{2em} x( -1 )= -10" title="t^{2} + 5 \, {y''} + 25 = -{\left(t + 6\right)} {\left(t - 6\right)} {y'''} - t y e^{t} \hspace{2em} x( -1 )= -10" data-latex="t^{2} + 5 \, {y''} + 25 = -{\left(t + 6\right)} {\left(t - 6\right)} {y'''} - t y e^{t} \hspace{2em} x( -1 )= -10"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,6)" alt="(-6,6)" title="(-6,6)" data-latex="(-6,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,6)" alt="(-6,6)" title="(-6,6)" data-latex="(-6,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-3195" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 3195"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{\left(t + 2\right)} {\left(t - 5\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 5 \, {y''} - e^{\left(2 \, t\right)} \hspace{2em} x( -7 )= -1" alt="0 = -{\left(t + 2\right)} {\left(t - 5\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 5 \, {y''} - e^{\left(2 \, t\right)} \hspace{2em} x( -7 )= -1" title="0 = -{\left(t + 2\right)} {\left(t - 5\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 5 \, {y''} - e^{\left(2 \, t\right)} \hspace{2em} x( -7 )= -1" data-latex="0 = -{\left(t + 2\right)} {\left(t - 5\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 5 \, {y''} - e^{\left(2 \, t\right)} \hspace{2em} x( -7 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7B%5Cleft(t%20+%202%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20y%20-%20%7B%5Cleft(t%20+%205%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20+%205%20%5C,%20%7By''%7D%20-%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%20-7%20)=%20-1" alt="0 = -{\left(t + 2\right)} {\left(t - 5\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 5 \, {y''} - e^{\left(2 \, t\right)} \hspace{2em} x( -7 )= -1" title="0 = -{\left(t + 2\right)} {\left(t - 5\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 5 \, {y''} - e^{\left(2 \, t\right)} \hspace{2em} x( -7 )= -1" data-latex="0 = -{\left(t + 2\right)} {\left(t - 5\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 5 \, {y''} - e^{\left(2 \, t\right)} \hspace{2em} x( -7 )= -1"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,-5)" alt="(-\infty,-5)" title="(-\infty,-5)" data-latex="(-\infty,-5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,-5)" alt="(-\infty,-5)" title="(-\infty,-5)" data-latex="(-\infty,-5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-3351" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 3351"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 4\right)} {\left(t - 5\right)} y - {\left(t - 2\right)} {y''} - e^{\left(-5 \, t\right)} = -{y'} \hspace{2em} x( 0 )= 5" alt="-{\left(t + 4\right)} {\left(t - 5\right)} y - {\left(t - 2\right)} {y''} - e^{\left(-5 \, t\right)} = -{y'} \hspace{2em} x( 0 )= 5" title="-{\left(t + 4\right)} {\left(t - 5\right)} y - {\left(t - 2\right)} {y''} - e^{\left(-5 \, t\right)} = -{y'} \hspace{2em} x( 0 )= 5" data-latex="-{\left(t + 4\right)} {\left(t - 5\right)} y - {\left(t - 2\right)} {y''} - e^{\left(-5 \, t\right)} = -{y'} \hspace{2em} x( 0 )= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%204%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20y%20-%20%7B%5Cleft(t%20-%202%5Cright)%7D%20%7By''%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20=%20-%7By'%7D%20%5Chspace%7B2em%7D%20x(%200%20)=%205" alt="-{\left(t + 4\right)} {\left(t - 5\right)} y - {\left(t - 2\right)} {y''} - e^{\left(-5 \, t\right)} = -{y'} \hspace{2em} x( 0 )= 5" title="-{\left(t + 4\right)} {\left(t - 5\right)} y - {\left(t - 2\right)} {y''} - e^{\left(-5 \, t\right)} = -{y'} \hspace{2em} x( 0 )= 5" data-latex="-{\left(t + 4\right)} {\left(t - 5\right)} y - {\left(t - 2\right)} {y''} - e^{\left(-5 \, t\right)} = -{y'} \hspace{2em} x( 0 )= 5"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,2)" alt="(-\infty,2)" title="(-\infty,2)" data-latex="(-\infty,2)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,2)" alt="(-\infty,2)" title="(-\infty,2)" data-latex="(-\infty,2)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8294" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8294"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 5\right)} {\left(t - 2\right)} - {y''} = {\left(t - 5\right)} {y'''} e^{t} + y e^{t} \hspace{2em} x( 2 )= 8" alt="-{\left(t + 5\right)} {\left(t - 2\right)} - {y''} = {\left(t - 5\right)} {y'''} e^{t} + y e^{t} \hspace{2em} x( 2 )= 8" title="-{\left(t + 5\right)} {\left(t - 2\right)} - {y''} = {\left(t - 5\right)} {y'''} e^{t} + y e^{t} \hspace{2em} x( 2 )= 8" data-latex="-{\left(t + 5\right)} {\left(t - 2\right)} - {y''} = {\left(t - 5\right)} {y'''} e^{t} + y e^{t} \hspace{2em} x( 2 )= 8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%202%5Cright)%7D%20-%20%7By''%7D%20=%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20+%20y%20e%5E%7Bt%7D%20%5Chspace%7B2em%7D%20x(%202%20)=%208" alt="-{\left(t + 5\right)} {\left(t - 2\right)} - {y''} = {\left(t - 5\right)} {y'''} e^{t} + y e^{t} \hspace{2em} x( 2 )= 8" title="-{\left(t + 5\right)} {\left(t - 2\right)} - {y''} = {\left(t - 5\right)} {y'''} e^{t} + y e^{t} \hspace{2em} x( 2 )= 8" data-latex="-{\left(t + 5\right)} {\left(t - 2\right)} - {y''} = {\left(t - 5\right)} {y'''} e^{t} + y e^{t} \hspace{2em} x( 2 )= 8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,5)" alt="(-\infty,5)" title="(-\infty,5)" data-latex="(-\infty,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,5)" alt="(-\infty,5)" title="(-\infty,5)" data-latex="(-\infty,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5707" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5707"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y e^{t} - 2 \, {y''} = -{\left(t + 5\right)} {\left(t - 1\right)} {y'''} - t + 5 \hspace{2em} x( -3 )= 3" alt="y e^{t} - 2 \, {y''} = -{\left(t + 5\right)} {\left(t - 1\right)} {y'''} - t + 5 \hspace{2em} x( -3 )= 3" title="y e^{t} - 2 \, {y''} = -{\left(t + 5\right)} {\left(t - 1\right)} {y'''} - t + 5 \hspace{2em} x( -3 )= 3" data-latex="y e^{t} - 2 \, {y''} = -{\left(t + 5\right)} {\left(t - 1\right)} {y'''} - t + 5 \hspace{2em} x( -3 )= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y%20e%5E%7Bt%7D%20-%202%20%5C,%20%7By''%7D%20=%20-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%201%5Cright)%7D%20%7By'''%7D%20-%20t%20+%205%20%5Chspace%7B2em%7D%20x(%20-3%20)=%203" alt="y e^{t} - 2 \, {y''} = -{\left(t + 5\right)} {\left(t - 1\right)} {y'''} - t + 5 \hspace{2em} x( -3 )= 3" title="y e^{t} - 2 \, {y''} = -{\left(t + 5\right)} {\left(t - 1\right)} {y'''} - t + 5 \hspace{2em} x( -3 )= 3" data-latex="y e^{t} - 2 \, {y''} = -{\left(t + 5\right)} {\left(t - 1\right)} {y'''} - t + 5 \hspace{2em} x( -3 )= 3"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,1)" alt="(-5,1)" title="(-5,1)" data-latex="(-5,1)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,1)" alt="(-5,1)" title="(-5,1)" data-latex="(-5,1)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8751" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8751"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 1\right)} {\left(t - 5\right)} {y'''} + 2 \, {y''} = {\left(t^{2} + 9\right)} y + t + 5 \hspace{2em} x( 4 )= -3" alt="-{\left(t + 1\right)} {\left(t - 5\right)} {y'''} + 2 \, {y''} = {\left(t^{2} + 9\right)} y + t + 5 \hspace{2em} x( 4 )= -3" title="-{\left(t + 1\right)} {\left(t - 5\right)} {y'''} + 2 \, {y''} = {\left(t^{2} + 9\right)} y + t + 5 \hspace{2em} x( 4 )= -3" data-latex="-{\left(t + 1\right)} {\left(t - 5\right)} {y'''} + 2 \, {y''} = {\left(t^{2} + 9\right)} y + t + 5 \hspace{2em} x( 4 )= -3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%201%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20+%202%20%5C,%20%7By''%7D%20=%20%7B%5Cleft(t%5E%7B2%7D%20+%209%5Cright)%7D%20y%20+%20t%20+%205%20%5Chspace%7B2em%7D%20x(%204%20)=%20-3" alt="-{\left(t + 1\right)} {\left(t - 5\right)} {y'''} + 2 \, {y''} = {\left(t^{2} + 9\right)} y + t + 5 \hspace{2em} x( 4 )= -3" title="-{\left(t + 1\right)} {\left(t - 5\right)} {y'''} + 2 \, {y''} = {\left(t^{2} + 9\right)} y + t + 5 \hspace{2em} x( 4 )= -3" data-latex="-{\left(t + 1\right)} {\left(t - 5\right)} {y'''} + 2 \, {y''} = {\left(t^{2} + 9\right)} y + t + 5 \hspace{2em} x( 4 )= -3"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-1,5)" alt="(-1,5)" title="(-1,5)" data-latex="(-1,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-1,5)" alt="(-1,5)" title="(-1,5)" data-latex="(-1,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-7628" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 7628"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 5\right)} {\left(t - 2\right)} y + 2 \, {y''} - e^{\left(3 \, t\right)} = {\left(t - 5\right)} {y'''} e^{t} \hspace{2em} x( 6 )= 3" alt="-{\left(t + 5\right)} {\left(t - 2\right)} y + 2 \, {y''} - e^{\left(3 \, t\right)} = {\left(t - 5\right)} {y'''} e^{t} \hspace{2em} x( 6 )= 3" title="-{\left(t + 5\right)} {\left(t - 2\right)} y + 2 \, {y''} - e^{\left(3 \, t\right)} = {\left(t - 5\right)} {y'''} e^{t} \hspace{2em} x( 6 )= 3" data-latex="-{\left(t + 5\right)} {\left(t - 2\right)} y + 2 \, {y''} - e^{\left(3 \, t\right)} = {\left(t - 5\right)} {y'''} e^{t} \hspace{2em} x( 6 )= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%202%5Cright)%7D%20y%20+%202%20%5C,%20%7By''%7D%20-%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20=%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20%5Chspace%7B2em%7D%20x(%206%20)=%203" alt="-{\left(t + 5\right)} {\left(t - 2\right)} y + 2 \, {y''} - e^{\left(3 \, t\right)} = {\left(t - 5\right)} {y'''} e^{t} \hspace{2em} x( 6 )= 3" title="-{\left(t + 5\right)} {\left(t - 2\right)} y + 2 \, {y''} - e^{\left(3 \, t\right)} = {\left(t - 5\right)} {y'''} e^{t} \hspace{2em} x( 6 )= 3" data-latex="-{\left(t + 5\right)} {\left(t - 2\right)} y + 2 \, {y''} - e^{\left(3 \, t\right)} = {\left(t - 5\right)} {y'''} e^{t} \hspace{2em} x( 6 )= 3"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+\infty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+%5Cinfty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8879" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8879"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t - 5\right)} {y'''} e^{t} - 5 \, {y''} - e^{t} = {\left(t + 4\right)} t y \hspace{2em} x( 8 )= 4" alt="-{\left(t - 5\right)} {y'''} e^{t} - 5 \, {y''} - e^{t} = {\left(t + 4\right)} t y \hspace{2em} x( 8 )= 4" title="-{\left(t - 5\right)} {y'''} e^{t} - 5 \, {y''} - e^{t} = {\left(t + 4\right)} t y \hspace{2em} x( 8 )= 4" data-latex="-{\left(t - 5\right)} {y'''} e^{t} - 5 \, {y''} - e^{t} = {\left(t + 4\right)} t y \hspace{2em} x( 8 )= 4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20-%205%20%5C,%20%7By''%7D%20-%20e%5E%7Bt%7D%20=%20%7B%5Cleft(t%20+%204%5Cright)%7D%20t%20y%20%5Chspace%7B2em%7D%20x(%208%20)=%204" alt="-{\left(t - 5\right)} {y'''} e^{t} - 5 \, {y''} - e^{t} = {\left(t + 4\right)} t y \hspace{2em} x( 8 )= 4" title="-{\left(t - 5\right)} {y'''} e^{t} - 5 \, {y''} - e^{t} = {\left(t + 4\right)} t y \hspace{2em} x( 8 )= 4" data-latex="-{\left(t - 5\right)} {y'''} e^{t} - 5 \, {y''} - e^{t} = {\left(t + 4\right)} t y \hspace{2em} x( 8 )= 4"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+\infty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+%5Cinfty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-9528" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 9528"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3 \, {y''} = {\left(t - 2\right)} {\left(t - 5\right)} {y'''} + {\left(t + 5\right)} y e^{t} + e^{\left(-2 \, t\right)} \hspace{2em} x( 3 )= 8" alt="3 \, {y''} = {\left(t - 2\right)} {\left(t - 5\right)} {y'''} + {\left(t + 5\right)} y e^{t} + e^{\left(-2 \, t\right)} \hspace{2em} x( 3 )= 8" title="3 \, {y''} = {\left(t - 2\right)} {\left(t - 5\right)} {y'''} + {\left(t + 5\right)} y e^{t} + e^{\left(-2 \, t\right)} \hspace{2em} x( 3 )= 8" data-latex="3 \, {y''} = {\left(t - 2\right)} {\left(t - 5\right)} {y'''} + {\left(t + 5\right)} y e^{t} + e^{\left(-2 \, t\right)} \hspace{2em} x( 3 )= 8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?3%20%5C,%20%7By''%7D%20=%20%7B%5Cleft(t%20-%202%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20+%20%7B%5Cleft(t%20+%205%5Cright)%7D%20y%20e%5E%7Bt%7D%20+%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%203%20)=%208" alt="3 \, {y''} = {\left(t - 2\right)} {\left(t - 5\right)} {y'''} + {\left(t + 5\right)} y e^{t} + e^{\left(-2 \, t\right)} \hspace{2em} x( 3 )= 8" title="3 \, {y''} = {\left(t - 2\right)} {\left(t - 5\right)} {y'''} + {\left(t + 5\right)} y e^{t} + e^{\left(-2 \, t\right)} \hspace{2em} x( 3 )= 8" data-latex="3 \, {y''} = {\left(t - 2\right)} {\left(t - 5\right)} {y'''} + {\left(t + 5\right)} y e^{t} + e^{\left(-2 \, t\right)} \hspace{2em} x( 3 )= 8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(2,5)" alt="(2,5)" title="(2,5)" data-latex="(2,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(2,5)" alt="(2,5)" title="(2,5)" data-latex="(2,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-9346" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 9346"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 6\right)} {\left(t - 1\right)} {y''} + {\left(t - 6\right)} y + 4 \, {y'} = -t^{2} - 25 \hspace{2em} x( 0 )= -8" alt="{\left(t + 6\right)} {\left(t - 1\right)} {y''} + {\left(t - 6\right)} y + 4 \, {y'} = -t^{2} - 25 \hspace{2em} x( 0 )= -8" title="{\left(t + 6\right)} {\left(t - 1\right)} {y''} + {\left(t - 6\right)} y + 4 \, {y'} = -t^{2} - 25 \hspace{2em} x( 0 )= -8" data-latex="{\left(t + 6\right)} {\left(t - 1\right)} {y''} + {\left(t - 6\right)} y + 4 \, {y'} = -t^{2} - 25 \hspace{2em} x( 0 )= -8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%201%5Cright)%7D%20%7By''%7D%20+%20%7B%5Cleft(t%20-%206%5Cright)%7D%20y%20+%204%20%5C,%20%7By'%7D%20=%20-t%5E%7B2%7D%20-%2025%20%5Chspace%7B2em%7D%20x(%200%20)=%20-8" alt="{\left(t + 6\right)} {\left(t - 1\right)} {y''} + {\left(t - 6\right)} y + 4 \, {y'} = -t^{2} - 25 \hspace{2em} x( 0 )= -8" title="{\left(t + 6\right)} {\left(t - 1\right)} {y''} + {\left(t - 6\right)} y + 4 \, {y'} = -t^{2} - 25 \hspace{2em} x( 0 )= -8" data-latex="{\left(t + 6\right)} {\left(t - 1\right)} {y''} + {\left(t - 6\right)} y + 4 \, {y'} = -t^{2} - 25 \hspace{2em} x( 0 )= -8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,1)" alt="(-6,1)" title="(-6,1)" data-latex="(-6,1)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,1)" alt="(-6,1)" title="(-6,1)" data-latex="(-6,1)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-6701" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 6701"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t - 5\right)} y + 3 \, {y''} = -{\left(t + 6\right)} t {y'''} - e^{\left(-4 \, t\right)} \hspace{2em} x( -1 )= -8" alt="{\left(t - 5\right)} y + 3 \, {y''} = -{\left(t + 6\right)} t {y'''} - e^{\left(-4 \, t\right)} \hspace{2em} x( -1 )= -8" title="{\left(t - 5\right)} y + 3 \, {y''} = -{\left(t + 6\right)} t {y'''} - e^{\left(-4 \, t\right)} \hspace{2em} x( -1 )= -8" data-latex="{\left(t - 5\right)} y + 3 \, {y''} = -{\left(t + 6\right)} t {y'''} - e^{\left(-4 \, t\right)} \hspace{2em} x( -1 )= -8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20-%205%5Cright)%7D%20y%20+%203%20%5C,%20%7By''%7D%20=%20-%7B%5Cleft(t%20+%206%5Cright)%7D%20t%20%7By'''%7D%20-%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%20-1%20)=%20-8" alt="{\left(t - 5\right)} y + 3 \, {y''} = -{\left(t + 6\right)} t {y'''} - e^{\left(-4 \, t\right)} \hspace{2em} x( -1 )= -8" title="{\left(t - 5\right)} y + 3 \, {y''} = -{\left(t + 6\right)} t {y'''} - e^{\left(-4 \, t\right)} \hspace{2em} x( -1 )= -8" data-latex="{\left(t - 5\right)} y + 3 \, {y''} = -{\left(t + 6\right)} t {y'''} - e^{\left(-4 \, t\right)} \hspace{2em} x( -1 )= -8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,0)" alt="(-6,0)" title="(-6,0)" data-latex="(-6,0)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,0)" alt="(-6,0)" title="(-6,0)" data-latex="(-6,0)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5899" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5899"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 2\right)} {\left(t - 6\right)} {y'''} + e^{\left(4 \, t\right)} = -{\left(t + 6\right)} y + 5 \, {y''} \hspace{2em} x( 1 )= 8" alt="{\left(t + 2\right)} {\left(t - 6\right)} {y'''} + e^{\left(4 \, t\right)} = -{\left(t + 6\right)} y + 5 \, {y''} \hspace{2em} x( 1 )= 8" title="{\left(t + 2\right)} {\left(t - 6\right)} {y'''} + e^{\left(4 \, t\right)} = -{\left(t + 6\right)} y + 5 \, {y''} \hspace{2em} x( 1 )= 8" data-latex="{\left(t + 2\right)} {\left(t - 6\right)} {y'''} + e^{\left(4 \, t\right)} = -{\left(t + 6\right)} y + 5 \, {y''} \hspace{2em} x( 1 )= 8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%202%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By'''%7D%20+%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20=%20-%7B%5Cleft(t%20+%206%5Cright)%7D%20y%20+%205%20%5C,%20%7By''%7D%20%5Chspace%7B2em%7D%20x(%201%20)=%208" alt="{\left(t + 2\right)} {\left(t - 6\right)} {y'''} + e^{\left(4 \, t\right)} = -{\left(t + 6\right)} y + 5 \, {y''} \hspace{2em} x( 1 )= 8" title="{\left(t + 2\right)} {\left(t - 6\right)} {y'''} + e^{\left(4 \, t\right)} = -{\left(t + 6\right)} y + 5 \, {y''} \hspace{2em} x( 1 )= 8" data-latex="{\left(t + 2\right)} {\left(t - 6\right)} {y'''} + e^{\left(4 \, t\right)} = -{\left(t + 6\right)} y + 5 \, {y''} \hspace{2em} x( 1 )= 8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-2,6)" alt="(-2,6)" title="(-2,6)" data-latex="(-2,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-2,6)" alt="(-2,6)" title="(-2,6)" data-latex="(-2,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-1744" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 1744"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 4\right)} {y'''} e^{t} + {\left(t - 1\right)} {\left(t - 6\right)} + {\left(t^{2} + 4\right)} y - {y''} = 0 \hspace{2em} x( -2 )= -6" alt="{\left(t + 4\right)} {y'''} e^{t} + {\left(t - 1\right)} {\left(t - 6\right)} + {\left(t^{2} + 4\right)} y - {y''} = 0 \hspace{2em} x( -2 )= -6" title="{\left(t + 4\right)} {y'''} e^{t} + {\left(t - 1\right)} {\left(t - 6\right)} + {\left(t^{2} + 4\right)} y - {y''} = 0 \hspace{2em} x( -2 )= -6" data-latex="{\left(t + 4\right)} {y'''} e^{t} + {\left(t - 1\right)} {\left(t - 6\right)} + {\left(t^{2} + 4\right)} y - {y''} = 0 \hspace{2em} x( -2 )= -6"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%204%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20+%20%7B%5Cleft(t%20-%201%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20+%20%7B%5Cleft(t%5E%7B2%7D%20+%204%5Cright)%7D%20y%20-%20%7By''%7D%20=%200%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-6" alt="{\left(t + 4\right)} {y'''} e^{t} + {\left(t - 1\right)} {\left(t - 6\right)} + {\left(t^{2} + 4\right)} y - {y''} = 0 \hspace{2em} x( -2 )= -6" title="{\left(t + 4\right)} {y'''} e^{t} + {\left(t - 1\right)} {\left(t - 6\right)} + {\left(t^{2} + 4\right)} y - {y''} = 0 \hspace{2em} x( -2 )= -6" data-latex="{\left(t + 4\right)} {y'''} e^{t} + {\left(t - 1\right)} {\left(t - 6\right)} + {\left(t^{2} + 4\right)} y - {y''} = 0 \hspace{2em} x( -2 )= -6"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,+\infty)" alt="(-4,+\infty)" title="(-4,+\infty)" data-latex="(-4,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,+%5Cinfty)" alt="(-4,+\infty)" title="(-4,+\infty)" data-latex="(-4,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-2556" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 2556"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 2\right)} {\left(t - 6\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 3 \, {y''} = e^{t} \hspace{2em} x( -8 )= -4" alt="-{\left(t + 2\right)} {\left(t - 6\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 3 \, {y''} = e^{t} \hspace{2em} x( -8 )= -4" title="-{\left(t + 2\right)} {\left(t - 6\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 3 \, {y''} = e^{t} \hspace{2em} x( -8 )= -4" data-latex="-{\left(t + 2\right)} {\left(t - 6\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 3 \, {y''} = e^{t} \hspace{2em} x( -8 )= -4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%202%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20y%20-%20%7B%5Cleft(t%20+%205%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20+%203%20%5C,%20%7By''%7D%20=%20e%5E%7Bt%7D%20%5Chspace%7B2em%7D%20x(%20-8%20)=%20-4" alt="-{\left(t + 2\right)} {\left(t - 6\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 3 \, {y''} = e^{t} \hspace{2em} x( -8 )= -4" title="-{\left(t + 2\right)} {\left(t - 6\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 3 \, {y''} = e^{t} \hspace{2em} x( -8 )= -4" data-latex="-{\left(t + 2\right)} {\left(t - 6\right)} y - {\left(t + 5\right)} {y'''} e^{t} + 3 \, {y''} = e^{t} \hspace{2em} x( -8 )= -4"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,-5)" alt="(-\infty,-5)" title="(-\infty,-5)" data-latex="(-\infty,-5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,-5)" alt="(-\infty,-5)" title="(-\infty,-5)" data-latex="(-\infty,-5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-9383" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 9383"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t - 2\right)} {\left(t - 5\right)} = -{\left(t + 5\right)} {y''} e^{t} - {\left(t^{2} + 25\right)} y - 4 \, {y'} \hspace{2em} x( -4 )= -7" alt="{\left(t - 2\right)} {\left(t - 5\right)} = -{\left(t + 5\right)} {y''} e^{t} - {\left(t^{2} + 25\right)} y - 4 \, {y'} \hspace{2em} x( -4 )= -7" title="{\left(t - 2\right)} {\left(t - 5\right)} = -{\left(t + 5\right)} {y''} e^{t} - {\left(t^{2} + 25\right)} y - 4 \, {y'} \hspace{2em} x( -4 )= -7" data-latex="{\left(t - 2\right)} {\left(t - 5\right)} = -{\left(t + 5\right)} {y''} e^{t} - {\left(t^{2} + 25\right)} y - 4 \, {y'} \hspace{2em} x( -4 )= -7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20-%202%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20=%20-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20-%20%7B%5Cleft(t%5E%7B2%7D%20+%2025%5Cright)%7D%20y%20-%204%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%20-7" alt="{\left(t - 2\right)} {\left(t - 5\right)} = -{\left(t + 5\right)} {y''} e^{t} - {\left(t^{2} + 25\right)} y - 4 \, {y'} \hspace{2em} x( -4 )= -7" title="{\left(t - 2\right)} {\left(t - 5\right)} = -{\left(t + 5\right)} {y''} e^{t} - {\left(t^{2} + 25\right)} y - 4 \, {y'} \hspace{2em} x( -4 )= -7" data-latex="{\left(t - 2\right)} {\left(t - 5\right)} = -{\left(t + 5\right)} {y''} e^{t} - {\left(t^{2} + 25\right)} y - 4 \, {y'} \hspace{2em} x( -4 )= -7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,+\infty)" alt="(-5,+\infty)" title="(-5,+\infty)" data-latex="(-5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,+%5Cinfty)" alt="(-5,+\infty)" title="(-5,+\infty)" data-latex="(-5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5040" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5040"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 5\right)} {\left(t - 6\right)} y - {\left(t + 2\right)} {y''} e^{t} = 5 \, {y'} + e^{t} \hspace{2em} x( -3 )= 1" alt="-{\left(t + 5\right)} {\left(t - 6\right)} y - {\left(t + 2\right)} {y''} e^{t} = 5 \, {y'} + e^{t} \hspace{2em} x( -3 )= 1" title="-{\left(t + 5\right)} {\left(t - 6\right)} y - {\left(t + 2\right)} {y''} e^{t} = 5 \, {y'} + e^{t} \hspace{2em} x( -3 )= 1" data-latex="-{\left(t + 5\right)} {\left(t - 6\right)} y - {\left(t + 2\right)} {y''} e^{t} = 5 \, {y'} + e^{t} \hspace{2em} x( -3 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20y%20-%20%7B%5Cleft(t%20+%202%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20=%205%20%5C,%20%7By'%7D%20+%20e%5E%7Bt%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%201" alt="-{\left(t + 5\right)} {\left(t - 6\right)} y - {\left(t + 2\right)} {y''} e^{t} = 5 \, {y'} + e^{t} \hspace{2em} x( -3 )= 1" title="-{\left(t + 5\right)} {\left(t - 6\right)} y - {\left(t + 2\right)} {y''} e^{t} = 5 \, {y'} + e^{t} \hspace{2em} x( -3 )= 1" data-latex="-{\left(t + 5\right)} {\left(t - 6\right)} y - {\left(t + 2\right)} {y''} e^{t} = 5 \, {y'} + e^{t} \hspace{2em} x( -3 )= 1"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,-2)" alt="(-\infty,-2)" title="(-\infty,-2)" data-latex="(-\infty,-2)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,-2)" alt="(-\infty,-2)" title="(-\infty,-2)" data-latex="(-\infty,-2)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-1429" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 1429"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y e^{\left(-4 \, t\right)} + 4 \, {y'} = {\left(t + 4\right)} {\left(t - 2\right)} {y''} + t - 5 \hspace{2em} x( -3 )= 5" alt="-y e^{\left(-4 \, t\right)} + 4 \, {y'} = {\left(t + 4\right)} {\left(t - 2\right)} {y''} + t - 5 \hspace{2em} x( -3 )= 5" title="-y e^{\left(-4 \, t\right)} + 4 \, {y'} = {\left(t + 4\right)} {\left(t - 2\right)} {y''} + t - 5 \hspace{2em} x( -3 )= 5" data-latex="-y e^{\left(-4 \, t\right)} + 4 \, {y'} = {\left(t + 4\right)} {\left(t - 2\right)} {y''} + t - 5 \hspace{2em} x( -3 )= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20%7By'%7D%20=%20%7B%5Cleft(t%20+%204%5Cright)%7D%20%7B%5Cleft(t%20-%202%5Cright)%7D%20%7By''%7D%20+%20t%20-%205%20%5Chspace%7B2em%7D%20x(%20-3%20)=%205" alt="-y e^{\left(-4 \, t\right)} + 4 \, {y'} = {\left(t + 4\right)} {\left(t - 2\right)} {y''} + t - 5 \hspace{2em} x( -3 )= 5" title="-y e^{\left(-4 \, t\right)} + 4 \, {y'} = {\left(t + 4\right)} {\left(t - 2\right)} {y''} + t - 5 \hspace{2em} x( -3 )= 5" data-latex="-y e^{\left(-4 \, t\right)} + 4 \, {y'} = {\left(t + 4\right)} {\left(t - 2\right)} {y''} + t - 5 \hspace{2em} x( -3 )= 5"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,2)" alt="(-4,2)" title="(-4,2)" data-latex="(-4,2)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,2)" alt="(-4,2)" title="(-4,2)" data-latex="(-4,2)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-6046" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 6046"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t - 5\right)} {y''} e^{t} + y e^{\left(-3 \, t\right)} = -{\left(t + 4\right)} {\left(t + 2\right)} + 3 \, {y'} \hspace{2em} x( 8 )= 2" alt="{\left(t - 5\right)} {y''} e^{t} + y e^{\left(-3 \, t\right)} = -{\left(t + 4\right)} {\left(t + 2\right)} + 3 \, {y'} \hspace{2em} x( 8 )= 2" title="{\left(t - 5\right)} {y''} e^{t} + y e^{\left(-3 \, t\right)} = -{\left(t + 4\right)} {\left(t + 2\right)} + 3 \, {y'} \hspace{2em} x( 8 )= 2" data-latex="{\left(t - 5\right)} {y''} e^{t} + y e^{\left(-3 \, t\right)} = -{\left(t + 4\right)} {\left(t + 2\right)} + 3 \, {y'} \hspace{2em} x( 8 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20+%20y%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20=%20-%7B%5Cleft(t%20+%204%5Cright)%7D%20%7B%5Cleft(t%20+%202%5Cright)%7D%20+%203%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20x(%208%20)=%202" alt="{\left(t - 5\right)} {y''} e^{t} + y e^{\left(-3 \, t\right)} = -{\left(t + 4\right)} {\left(t + 2\right)} + 3 \, {y'} \hspace{2em} x( 8 )= 2" title="{\left(t - 5\right)} {y''} e^{t} + y e^{\left(-3 \, t\right)} = -{\left(t + 4\right)} {\left(t + 2\right)} + 3 \, {y'} \hspace{2em} x( 8 )= 2" data-latex="{\left(t - 5\right)} {y''} e^{t} + y e^{\left(-3 \, t\right)} = -{\left(t + 4\right)} {\left(t + 2\right)} + 3 \, {y'} \hspace{2em} x( 8 )= 2"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+\infty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+%5Cinfty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-6728" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 6728"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t^{2} + 25\right)} y = {\left(t - 6\right)} t {y''} + t + {y'} + 6 \hspace{2em} x( 1 )= -4" alt="-{\left(t^{2} + 25\right)} y = {\left(t - 6\right)} t {y''} + t + {y'} + 6 \hspace{2em} x( 1 )= -4" title="-{\left(t^{2} + 25\right)} y = {\left(t - 6\right)} t {y''} + t + {y'} + 6 \hspace{2em} x( 1 )= -4" data-latex="-{\left(t^{2} + 25\right)} y = {\left(t - 6\right)} t {y''} + t + {y'} + 6 \hspace{2em} x( 1 )= -4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%5E%7B2%7D%20+%2025%5Cright)%7D%20y%20=%20%7B%5Cleft(t%20-%206%5Cright)%7D%20t%20%7By''%7D%20+%20t%20+%20%7By'%7D%20+%206%20%5Chspace%7B2em%7D%20x(%201%20)=%20-4" alt="-{\left(t^{2} + 25\right)} y = {\left(t - 6\right)} t {y''} + t + {y'} + 6 \hspace{2em} x( 1 )= -4" title="-{\left(t^{2} + 25\right)} y = {\left(t - 6\right)} t {y''} + t + {y'} + 6 \hspace{2em} x( 1 )= -4" data-latex="-{\left(t^{2} + 25\right)} y = {\left(t - 6\right)} t {y''} + t + {y'} + 6 \hspace{2em} x( 1 )= -4"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(0,6)" alt="(0,6)" title="(0,6)" data-latex="(0,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(0,6)" alt="(0,6)" title="(0,6)" data-latex="(0,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8374" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8374"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t - 6\right)} y e^{t} + t^{2} + 3 \, {y''} + 4 = -{\left(t + 6\right)} {\left(t + 1\right)} {y'''} \hspace{2em} x( -3 )= -8" alt="{\left(t - 6\right)} y e^{t} + t^{2} + 3 \, {y''} + 4 = -{\left(t + 6\right)} {\left(t + 1\right)} {y'''} \hspace{2em} x( -3 )= -8" title="{\left(t - 6\right)} y e^{t} + t^{2} + 3 \, {y''} + 4 = -{\left(t + 6\right)} {\left(t + 1\right)} {y'''} \hspace{2em} x( -3 )= -8" data-latex="{\left(t - 6\right)} y e^{t} + t^{2} + 3 \, {y''} + 4 = -{\left(t + 6\right)} {\left(t + 1\right)} {y'''} \hspace{2em} x( -3 )= -8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20-%206%5Cright)%7D%20y%20e%5E%7Bt%7D%20+%20t%5E%7B2%7D%20+%203%20%5C,%20%7By''%7D%20+%204%20=%20-%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20+%201%5Cright)%7D%20%7By'''%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-8" alt="{\left(t - 6\right)} y e^{t} + t^{2} + 3 \, {y''} + 4 = -{\left(t + 6\right)} {\left(t + 1\right)} {y'''} \hspace{2em} x( -3 )= -8" title="{\left(t - 6\right)} y e^{t} + t^{2} + 3 \, {y''} + 4 = -{\left(t + 6\right)} {\left(t + 1\right)} {y'''} \hspace{2em} x( -3 )= -8" data-latex="{\left(t - 6\right)} y e^{t} + t^{2} + 3 \, {y''} + 4 = -{\left(t + 6\right)} {\left(t + 1\right)} {y'''} \hspace{2em} x( -3 )= -8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,-1)" alt="(-6,-1)" title="(-6,-1)" data-latex="(-6,-1)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,-1)" alt="(-6,-1)" title="(-6,-1)" data-latex="(-6,-1)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-7405" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 7405"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 6\right)} {\left(t + 2\right)} {y''} + 3 \, {y'} = -{\left(t^{2} + 16\right)} y - t + 6 \hspace{2em} x( -4 )= 2" alt="{\left(t + 6\right)} {\left(t + 2\right)} {y''} + 3 \, {y'} = -{\left(t^{2} + 16\right)} y - t + 6 \hspace{2em} x( -4 )= 2" title="{\left(t + 6\right)} {\left(t + 2\right)} {y''} + 3 \, {y'} = -{\left(t^{2} + 16\right)} y - t + 6 \hspace{2em} x( -4 )= 2" data-latex="{\left(t + 6\right)} {\left(t + 2\right)} {y''} + 3 \, {y'} = -{\left(t^{2} + 16\right)} y - t + 6 \hspace{2em} x( -4 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20+%202%5Cright)%7D%20%7By''%7D%20+%203%20%5C,%20%7By'%7D%20=%20-%7B%5Cleft(t%5E%7B2%7D%20+%2016%5Cright)%7D%20y%20-%20t%20+%206%20%5Chspace%7B2em%7D%20x(%20-4%20)=%202" alt="{\left(t + 6\right)} {\left(t + 2\right)} {y''} + 3 \, {y'} = -{\left(t^{2} + 16\right)} y - t + 6 \hspace{2em} x( -4 )= 2" title="{\left(t + 6\right)} {\left(t + 2\right)} {y''} + 3 \, {y'} = -{\left(t^{2} + 16\right)} y - t + 6 \hspace{2em} x( -4 )= 2" data-latex="{\left(t + 6\right)} {\left(t + 2\right)} {y''} + 3 \, {y'} = -{\left(t^{2} + 16\right)} y - t + 6 \hspace{2em} x( -4 )= 2"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,-2)" alt="(-6,-2)" title="(-6,-2)" data-latex="(-6,-2)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,-2)" alt="(-6,-2)" title="(-6,-2)" data-latex="(-6,-2)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5828" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5828"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {\left(t - 5\right)} {y''} e^{t} + {\left(t + 6\right)} t + {\left(t^{2} + 4\right)} y - 3 \, {y'} \hspace{2em} x( 7 )= 4" alt="0 = {\left(t - 5\right)} {y''} e^{t} + {\left(t + 6\right)} t + {\left(t^{2} + 4\right)} y - 3 \, {y'} \hspace{2em} x( 7 )= 4" title="0 = {\left(t - 5\right)} {y''} e^{t} + {\left(t + 6\right)} t + {\left(t^{2} + 4\right)} y - 3 \, {y'} \hspace{2em} x( 7 )= 4" data-latex="0 = {\left(t - 5\right)} {y''} e^{t} + {\left(t + 6\right)} t + {\left(t^{2} + 4\right)} y - 3 \, {y'} \hspace{2em} x( 7 )= 4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20+%20%7B%5Cleft(t%20+%206%5Cright)%7D%20t%20+%20%7B%5Cleft(t%5E%7B2%7D%20+%204%5Cright)%7D%20y%20-%203%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20x(%207%20)=%204" alt="0 = {\left(t - 5\right)} {y''} e^{t} + {\left(t + 6\right)} t + {\left(t^{2} + 4\right)} y - 3 \, {y'} \hspace{2em} x( 7 )= 4" title="0 = {\left(t - 5\right)} {y''} e^{t} + {\left(t + 6\right)} t + {\left(t^{2} + 4\right)} y - 3 \, {y'} \hspace{2em} x( 7 )= 4" data-latex="0 = {\left(t - 5\right)} {y''} e^{t} + {\left(t + 6\right)} t + {\left(t^{2} + 4\right)} y - 3 \, {y'} \hspace{2em} x( 7 )= 4"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+\infty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+%5Cinfty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-4575" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 4575"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 5\right)} {y'''} - {y''} = {\left(t - 2\right)} {\left(t - 5\right)} + y e^{\left(-t\right)} \hspace{2em} x( -3 )= -9" alt="-{\left(t + 5\right)} {y'''} - {y''} = {\left(t - 2\right)} {\left(t - 5\right)} + y e^{\left(-t\right)} \hspace{2em} x( -3 )= -9" title="-{\left(t + 5\right)} {y'''} - {y''} = {\left(t - 2\right)} {\left(t - 5\right)} + y e^{\left(-t\right)} \hspace{2em} x( -3 )= -9" data-latex="-{\left(t + 5\right)} {y'''} - {y''} = {\left(t - 2\right)} {\left(t - 5\right)} + y e^{\left(-t\right)} \hspace{2em} x( -3 )= -9"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7By'''%7D%20-%20%7By''%7D%20=%20%7B%5Cleft(t%20-%202%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20+%20y%20e%5E%7B%5Cleft(-t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-9" alt="-{\left(t + 5\right)} {y'''} - {y''} = {\left(t - 2\right)} {\left(t - 5\right)} + y e^{\left(-t\right)} \hspace{2em} x( -3 )= -9" title="-{\left(t + 5\right)} {y'''} - {y''} = {\left(t - 2\right)} {\left(t - 5\right)} + y e^{\left(-t\right)} \hspace{2em} x( -3 )= -9" data-latex="-{\left(t + 5\right)} {y'''} - {y''} = {\left(t - 2\right)} {\left(t - 5\right)} + y e^{\left(-t\right)} \hspace{2em} x( -3 )= -9"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,+\infty)" alt="(-5,+\infty)" title="(-5,+\infty)" data-latex="(-5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,+%5Cinfty)" alt="(-5,+\infty)" title="(-5,+\infty)" data-latex="(-5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5480" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5480"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 5\right)} {\left(t - 6\right)} + {\left(t^{2} + 25\right)} y + {\left(t - 2\right)} {y'''} = 2 \, {y''} \hspace{2em} x( 4 )= -2" alt="{\left(t + 5\right)} {\left(t - 6\right)} + {\left(t^{2} + 25\right)} y + {\left(t - 2\right)} {y'''} = 2 \, {y''} \hspace{2em} x( 4 )= -2" title="{\left(t + 5\right)} {\left(t - 6\right)} + {\left(t^{2} + 25\right)} y + {\left(t - 2\right)} {y'''} = 2 \, {y''} \hspace{2em} x( 4 )= -2" data-latex="{\left(t + 5\right)} {\left(t - 6\right)} + {\left(t^{2} + 25\right)} y + {\left(t - 2\right)} {y'''} = 2 \, {y''} \hspace{2em} x( 4 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20+%20%7B%5Cleft(t%5E%7B2%7D%20+%2025%5Cright)%7D%20y%20+%20%7B%5Cleft(t%20-%202%5Cright)%7D%20%7By'''%7D%20=%202%20%5C,%20%7By''%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%20-2" alt="{\left(t + 5\right)} {\left(t - 6\right)} + {\left(t^{2} + 25\right)} y + {\left(t - 2\right)} {y'''} = 2 \, {y''} \hspace{2em} x( 4 )= -2" title="{\left(t + 5\right)} {\left(t - 6\right)} + {\left(t^{2} + 25\right)} y + {\left(t - 2\right)} {y'''} = 2 \, {y''} \hspace{2em} x( 4 )= -2" data-latex="{\left(t + 5\right)} {\left(t - 6\right)} + {\left(t^{2} + 25\right)} y + {\left(t - 2\right)} {y'''} = 2 \, {y''} \hspace{2em} x( 4 )= -2"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(2,+\infty)" alt="(2,+\infty)" title="(2,+\infty)" data-latex="(2,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(2,+%5Cinfty)" alt="(2,+\infty)" title="(2,+\infty)" data-latex="(2,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-7386" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 7386"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 6\right)} y e^{t} - {y''} = {\left(t - 6\right)} t {y'''} + e^{t} \hspace{2em} x( 4 )= -3" alt="-{\left(t + 6\right)} y e^{t} - {y''} = {\left(t - 6\right)} t {y'''} + e^{t} \hspace{2em} x( 4 )= -3" title="-{\left(t + 6\right)} y e^{t} - {y''} = {\left(t - 6\right)} t {y'''} + e^{t} \hspace{2em} x( 4 )= -3" data-latex="-{\left(t + 6\right)} y e^{t} - {y''} = {\left(t - 6\right)} t {y'''} + e^{t} \hspace{2em} x( 4 )= -3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%206%5Cright)%7D%20y%20e%5E%7Bt%7D%20-%20%7By''%7D%20=%20%7B%5Cleft(t%20-%206%5Cright)%7D%20t%20%7By'''%7D%20+%20e%5E%7Bt%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%20-3" alt="-{\left(t + 6\right)} y e^{t} - {y''} = {\left(t - 6\right)} t {y'''} + e^{t} \hspace{2em} x( 4 )= -3" title="-{\left(t + 6\right)} y e^{t} - {y''} = {\left(t - 6\right)} t {y'''} + e^{t} \hspace{2em} x( 4 )= -3" data-latex="-{\left(t + 6\right)} y e^{t} - {y''} = {\left(t - 6\right)} t {y'''} + e^{t} \hspace{2em} x( 4 )= -3"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(0,6)" alt="(0,6)" title="(0,6)" data-latex="(0,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(0,6)" alt="(0,6)" title="(0,6)" data-latex="(0,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5371" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5371"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 5\right)} {\left(t - 1\right)} - {y''} = {\left(t - 5\right)} {y'''} + y e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 7" alt="-{\left(t + 5\right)} {\left(t - 1\right)} - {y''} = {\left(t - 5\right)} {y'''} + y e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 7" title="-{\left(t + 5\right)} {\left(t - 1\right)} - {y''} = {\left(t - 5\right)} {y'''} + y e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 7" data-latex="-{\left(t + 5\right)} {\left(t - 1\right)} - {y''} = {\left(t - 5\right)} {y'''} + y e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%201%5Cright)%7D%20-%20%7By''%7D%20=%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20+%20y%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%207" alt="-{\left(t + 5\right)} {\left(t - 1\right)} - {y''} = {\left(t - 5\right)} {y'''} + y e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 7" title="-{\left(t + 5\right)} {\left(t - 1\right)} - {y''} = {\left(t - 5\right)} {y'''} + y e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 7" data-latex="-{\left(t + 5\right)} {\left(t - 1\right)} - {y''} = {\left(t - 5\right)} {y'''} + y e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,5)" alt="(-\infty,5)" title="(-\infty,5)" data-latex="(-\infty,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,5)" alt="(-\infty,5)" title="(-\infty,5)" data-latex="(-\infty,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-1829" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 1829"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 6\right)} {\left(t - 2\right)} - {\left(t^{2} + 25\right)} y - {\left(t - 6\right)} {y''} = 3 \, {y'} \hspace{2em} x( 10 )= 3" alt="-{\left(t + 6\right)} {\left(t - 2\right)} - {\left(t^{2} + 25\right)} y - {\left(t - 6\right)} {y''} = 3 \, {y'} \hspace{2em} x( 10 )= 3" title="-{\left(t + 6\right)} {\left(t - 2\right)} - {\left(t^{2} + 25\right)} y - {\left(t - 6\right)} {y''} = 3 \, {y'} \hspace{2em} x( 10 )= 3" data-latex="-{\left(t + 6\right)} {\left(t - 2\right)} - {\left(t^{2} + 25\right)} y - {\left(t - 6\right)} {y''} = 3 \, {y'} \hspace{2em} x( 10 )= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%202%5Cright)%7D%20-%20%7B%5Cleft(t%5E%7B2%7D%20+%2025%5Cright)%7D%20y%20-%20%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By''%7D%20=%203%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20x(%2010%20)=%203" alt="-{\left(t + 6\right)} {\left(t - 2\right)} - {\left(t^{2} + 25\right)} y - {\left(t - 6\right)} {y''} = 3 \, {y'} \hspace{2em} x( 10 )= 3" title="-{\left(t + 6\right)} {\left(t - 2\right)} - {\left(t^{2} + 25\right)} y - {\left(t - 6\right)} {y''} = 3 \, {y'} \hspace{2em} x( 10 )= 3" data-latex="-{\left(t + 6\right)} {\left(t - 2\right)} - {\left(t^{2} + 25\right)} y - {\left(t - 6\right)} {y''} = 3 \, {y'} \hspace{2em} x( 10 )= 3"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(6,+\infty)" alt="(6,+\infty)" title="(6,+\infty)" data-latex="(6,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(6,+%5Cinfty)" alt="(6,+\infty)" title="(6,+\infty)" data-latex="(6,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-0561" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 0561"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 6\right)} {\left(t - 5\right)} {y'''} + {\left(t^{2} + 16\right)} y + {\left(t - 2\right)} e^{t} + 5 \, {y''} = 0 \hspace{2em} x( -2 )= -10" alt="{\left(t + 6\right)} {\left(t - 5\right)} {y'''} + {\left(t^{2} + 16\right)} y + {\left(t - 2\right)} e^{t} + 5 \, {y''} = 0 \hspace{2em} x( -2 )= -10" title="{\left(t + 6\right)} {\left(t - 5\right)} {y'''} + {\left(t^{2} + 16\right)} y + {\left(t - 2\right)} e^{t} + 5 \, {y''} = 0 \hspace{2em} x( -2 )= -10" data-latex="{\left(t + 6\right)} {\left(t - 5\right)} {y'''} + {\left(t^{2} + 16\right)} y + {\left(t - 2\right)} e^{t} + 5 \, {y''} = 0 \hspace{2em} x( -2 )= -10"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20+%20%7B%5Cleft(t%5E%7B2%7D%20+%2016%5Cright)%7D%20y%20+%20%7B%5Cleft(t%20-%202%5Cright)%7D%20e%5E%7Bt%7D%20+%205%20%5C,%20%7By''%7D%20=%200%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-10" alt="{\left(t + 6\right)} {\left(t - 5\right)} {y'''} + {\left(t^{2} + 16\right)} y + {\left(t - 2\right)} e^{t} + 5 \, {y''} = 0 \hspace{2em} x( -2 )= -10" title="{\left(t + 6\right)} {\left(t - 5\right)} {y'''} + {\left(t^{2} + 16\right)} y + {\left(t - 2\right)} e^{t} + 5 \, {y''} = 0 \hspace{2em} x( -2 )= -10" data-latex="{\left(t + 6\right)} {\left(t - 5\right)} {y'''} + {\left(t^{2} + 16\right)} y + {\left(t - 2\right)} e^{t} + 5 \, {y''} = 0 \hspace{2em} x( -2 )= -10"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,5)" alt="(-6,5)" title="(-6,5)" data-latex="(-6,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,5)" alt="(-6,5)" title="(-6,5)" data-latex="(-6,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-2539" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 2539"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} + {y'} + 4 = -{\left(t + 5\right)} t {y''} - {\left(t - 5\right)} y e^{t} \hspace{2em} x( -4 )= -8" alt="t^{2} + {y'} + 4 = -{\left(t + 5\right)} t {y''} - {\left(t - 5\right)} y e^{t} \hspace{2em} x( -4 )= -8" title="t^{2} + {y'} + 4 = -{\left(t + 5\right)} t {y''} - {\left(t - 5\right)} y e^{t} \hspace{2em} x( -4 )= -8" data-latex="t^{2} + {y'} + 4 = -{\left(t + 5\right)} t {y''} - {\left(t - 5\right)} y e^{t} \hspace{2em} x( -4 )= -8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&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%7By'%7D%20+%204%20=%20-%7B%5Cleft(t%20+%205%5Cright)%7D%20t%20%7By''%7D%20-%20%7B%5Cleft(t%20-%205%5Cright)%7D%20y%20e%5E%7Bt%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%20-8" alt="t^{2} + {y'} + 4 = -{\left(t + 5\right)} t {y''} - {\left(t - 5\right)} y e^{t} \hspace{2em} x( -4 )= -8" title="t^{2} + {y'} + 4 = -{\left(t + 5\right)} t {y''} - {\left(t - 5\right)} y e^{t} \hspace{2em} x( -4 )= -8" data-latex="t^{2} + {y'} + 4 = -{\left(t + 5\right)} t {y''} - {\left(t - 5\right)} y e^{t} \hspace{2em} x( -4 )= -8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,0)" alt="(-5,0)" title="(-5,0)" data-latex="(-5,0)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,0)" alt="(-5,0)" title="(-5,0)" data-latex="(-5,0)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5048" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5048"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 4\right)} {\left(t - 6\right)} {y''} + 4 \, {y'} = y e^{\left(-t\right)} + t - 2 \hspace{2em} x( -2 )= 10" alt="-{\left(t + 4\right)} {\left(t - 6\right)} {y''} + 4 \, {y'} = y e^{\left(-t\right)} + t - 2 \hspace{2em} x( -2 )= 10" title="-{\left(t + 4\right)} {\left(t - 6\right)} {y''} + 4 \, {y'} = y e^{\left(-t\right)} + t - 2 \hspace{2em} x( -2 )= 10" data-latex="-{\left(t + 4\right)} {\left(t - 6\right)} {y''} + 4 \, {y'} = y e^{\left(-t\right)} + t - 2 \hspace{2em} x( -2 )= 10"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%204%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By''%7D%20+%204%20%5C,%20%7By'%7D%20=%20y%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20t%20-%202%20%5Chspace%7B2em%7D%20x(%20-2%20)=%2010" alt="-{\left(t + 4\right)} {\left(t - 6\right)} {y''} + 4 \, {y'} = y e^{\left(-t\right)} + t - 2 \hspace{2em} x( -2 )= 10" title="-{\left(t + 4\right)} {\left(t - 6\right)} {y''} + 4 \, {y'} = y e^{\left(-t\right)} + t - 2 \hspace{2em} x( -2 )= 10" data-latex="-{\left(t + 4\right)} {\left(t - 6\right)} {y''} + 4 \, {y'} = y e^{\left(-t\right)} + t - 2 \hspace{2em} x( -2 )= 10"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,6)" alt="(-4,6)" title="(-4,6)" data-latex="(-4,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,6)" alt="(-4,6)" title="(-4,6)" data-latex="(-4,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5070" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5070"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t - 5\right)} e^{t} + 5 \, {y''} = {\left(t + 6\right)} t {y'''} + {\left(t^{2} + 25\right)} y \hspace{2em} x( -2 )= 2" alt="-{\left(t - 5\right)} e^{t} + 5 \, {y''} = {\left(t + 6\right)} t {y'''} + {\left(t^{2} + 25\right)} y \hspace{2em} x( -2 )= 2" title="-{\left(t - 5\right)} e^{t} + 5 \, {y''} = {\left(t + 6\right)} t {y'''} + {\left(t^{2} + 25\right)} y \hspace{2em} x( -2 )= 2" data-latex="-{\left(t - 5\right)} e^{t} + 5 \, {y''} = {\left(t + 6\right)} t {y'''} + {\left(t^{2} + 25\right)} y \hspace{2em} x( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20-%205%5Cright)%7D%20e%5E%7Bt%7D%20+%205%20%5C,%20%7By''%7D%20=%20%7B%5Cleft(t%20+%206%5Cright)%7D%20t%20%7By'''%7D%20+%20%7B%5Cleft(t%5E%7B2%7D%20+%2025%5Cright)%7D%20y%20%5Chspace%7B2em%7D%20x(%20-2%20)=%202" alt="-{\left(t - 5\right)} e^{t} + 5 \, {y''} = {\left(t + 6\right)} t {y'''} + {\left(t^{2} + 25\right)} y \hspace{2em} x( -2 )= 2" title="-{\left(t - 5\right)} e^{t} + 5 \, {y''} = {\left(t + 6\right)} t {y'''} + {\left(t^{2} + 25\right)} y \hspace{2em} x( -2 )= 2" data-latex="-{\left(t - 5\right)} e^{t} + 5 \, {y''} = {\left(t + 6\right)} t {y'''} + {\left(t^{2} + 25\right)} y \hspace{2em} x( -2 )= 2"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,0)" alt="(-6,0)" title="(-6,0)" data-latex="(-6,0)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,0)" alt="(-6,0)" title="(-6,0)" data-latex="(-6,0)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-1751" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 1751"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t^{2} + 16\right)} y = -{\left(t - 5\right)} t {y''} - {\left(t + 4\right)} e^{t} + 3 \, {y'} \hspace{2em} x( 3 )= 7" alt="{\left(t^{2} + 16\right)} y = -{\left(t - 5\right)} t {y''} - {\left(t + 4\right)} e^{t} + 3 \, {y'} \hspace{2em} x( 3 )= 7" title="{\left(t^{2} + 16\right)} y = -{\left(t - 5\right)} t {y''} - {\left(t + 4\right)} e^{t} + 3 \, {y'} \hspace{2em} x( 3 )= 7" data-latex="{\left(t^{2} + 16\right)} y = -{\left(t - 5\right)} t {y''} - {\left(t + 4\right)} e^{t} + 3 \, {y'} \hspace{2em} x( 3 )= 7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%5E%7B2%7D%20+%2016%5Cright)%7D%20y%20=%20-%7B%5Cleft(t%20-%205%5Cright)%7D%20t%20%7By''%7D%20-%20%7B%5Cleft(t%20+%204%5Cright)%7D%20e%5E%7Bt%7D%20+%203%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20x(%203%20)=%207" alt="{\left(t^{2} + 16\right)} y = -{\left(t - 5\right)} t {y''} - {\left(t + 4\right)} e^{t} + 3 \, {y'} \hspace{2em} x( 3 )= 7" title="{\left(t^{2} + 16\right)} y = -{\left(t - 5\right)} t {y''} - {\left(t + 4\right)} e^{t} + 3 \, {y'} \hspace{2em} x( 3 )= 7" data-latex="{\left(t^{2} + 16\right)} y = -{\left(t - 5\right)} t {y''} - {\left(t + 4\right)} e^{t} + 3 \, {y'} \hspace{2em} x( 3 )= 7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(0,5)" alt="(0,5)" title="(0,5)" data-latex="(0,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(0,5)" alt="(0,5)" title="(0,5)" data-latex="(0,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5306" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5306"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{2} - {y''} - 9 = {\left(t - 1\right)} {\left(t - 5\right)} y + {\left(t + 6\right)} {y'''} \hspace{2em} x( -4 )= -7" alt="-t^{2} - {y''} - 9 = {\left(t - 1\right)} {\left(t - 5\right)} y + {\left(t + 6\right)} {y'''} \hspace{2em} x( -4 )= -7" title="-t^{2} - {y''} - 9 = {\left(t - 1\right)} {\left(t - 5\right)} y + {\left(t + 6\right)} {y'''} \hspace{2em} x( -4 )= -7" data-latex="-t^{2} - {y''} - 9 = {\left(t - 1\right)} {\left(t - 5\right)} y + {\left(t + 6\right)} {y'''} \hspace{2em} x( -4 )= -7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&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%7By''%7D%20-%209%20=%20%7B%5Cleft(t%20-%201%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20y%20+%20%7B%5Cleft(t%20+%206%5Cright)%7D%20%7By'''%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%20-7" alt="-t^{2} - {y''} - 9 = {\left(t - 1\right)} {\left(t - 5\right)} y + {\left(t + 6\right)} {y'''} \hspace{2em} x( -4 )= -7" title="-t^{2} - {y''} - 9 = {\left(t - 1\right)} {\left(t - 5\right)} y + {\left(t + 6\right)} {y'''} \hspace{2em} x( -4 )= -7" data-latex="-t^{2} - {y''} - 9 = {\left(t - 1\right)} {\left(t - 5\right)} y + {\left(t + 6\right)} {y'''} \hspace{2em} x( -4 )= -7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,+\infty)" alt="(-6,+\infty)" title="(-6,+\infty)" data-latex="(-6,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,+%5Cinfty)" alt="(-6,+\infty)" title="(-6,+\infty)" data-latex="(-6,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-3956" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 3956"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y e^{t} + t + 5 = -{\left(t - 1\right)} {\left(t - 5\right)} {y'''} + 2 \, {y''} \hspace{2em} x( 2 )= -2" alt="y e^{t} + t + 5 = -{\left(t - 1\right)} {\left(t - 5\right)} {y'''} + 2 \, {y''} \hspace{2em} x( 2 )= -2" title="y e^{t} + t + 5 = -{\left(t - 1\right)} {\left(t - 5\right)} {y'''} + 2 \, {y''} \hspace{2em} x( 2 )= -2" data-latex="y e^{t} + t + 5 = -{\left(t - 1\right)} {\left(t - 5\right)} {y'''} + 2 \, {y''} \hspace{2em} x( 2 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y%20e%5E%7Bt%7D%20+%20t%20+%205%20=%20-%7B%5Cleft(t%20-%201%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20+%202%20%5C,%20%7By''%7D%20%5Chspace%7B2em%7D%20x(%202%20)=%20-2" alt="y e^{t} + t + 5 = -{\left(t - 1\right)} {\left(t - 5\right)} {y'''} + 2 \, {y''} \hspace{2em} x( 2 )= -2" title="y e^{t} + t + 5 = -{\left(t - 1\right)} {\left(t - 5\right)} {y'''} + 2 \, {y''} \hspace{2em} x( 2 )= -2" data-latex="y e^{t} + t + 5 = -{\left(t - 1\right)} {\left(t - 5\right)} {y'''} + 2 \, {y''} \hspace{2em} x( 2 )= -2"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(1,5)" alt="(1,5)" title="(1,5)" data-latex="(1,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(1,5)" alt="(1,5)" title="(1,5)" data-latex="(1,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-9922" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 9922"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {\left(t + 5\right)} {\left(t - 1\right)} y + {\left(t - 5\right)} {y'''} e^{t} + {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 7" alt="0 = {\left(t + 5\right)} {\left(t - 1\right)} y + {\left(t - 5\right)} {y'''} e^{t} + {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 7" title="0 = {\left(t + 5\right)} {\left(t - 1\right)} y + {\left(t - 5\right)} {y'''} e^{t} + {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 7" data-latex="0 = {\left(t + 5\right)} {\left(t - 1\right)} y + {\left(t - 5\right)} {y'''} e^{t} + {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%201%5Cright)%7D%20y%20+%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20+%20%7By''%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%207" alt="0 = {\left(t + 5\right)} {\left(t - 1\right)} y + {\left(t - 5\right)} {y'''} e^{t} + {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 7" title="0 = {\left(t + 5\right)} {\left(t - 1\right)} y + {\left(t - 5\right)} {y'''} e^{t} + {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 7" data-latex="0 = {\left(t + 5\right)} {\left(t - 1\right)} y + {\left(t - 5\right)} {y'''} e^{t} + {y''} + e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,5)" alt="(-\infty,5)" title="(-\infty,5)" data-latex="(-\infty,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,5)" alt="(-\infty,5)" title="(-\infty,5)" data-latex="(-\infty,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-1197" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 1197"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{\left(t - 1\right)} {\left(t - 6\right)} - {\left(t^{2} + 4\right)} y - {\left(t + 4\right)} {y''} - 5 \, {y'} \hspace{2em} x( 0 )= -8" alt="0 = -{\left(t - 1\right)} {\left(t - 6\right)} - {\left(t^{2} + 4\right)} y - {\left(t + 4\right)} {y''} - 5 \, {y'} \hspace{2em} x( 0 )= -8" title="0 = -{\left(t - 1\right)} {\left(t - 6\right)} - {\left(t^{2} + 4\right)} y - {\left(t + 4\right)} {y''} - 5 \, {y'} \hspace{2em} x( 0 )= -8" data-latex="0 = -{\left(t - 1\right)} {\left(t - 6\right)} - {\left(t^{2} + 4\right)} y - {\left(t + 4\right)} {y''} - 5 \, {y'} \hspace{2em} x( 0 )= -8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7B%5Cleft(t%20-%201%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20-%20%7B%5Cleft(t%5E%7B2%7D%20+%204%5Cright)%7D%20y%20-%20%7B%5Cleft(t%20+%204%5Cright)%7D%20%7By''%7D%20-%205%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20x(%200%20)=%20-8" alt="0 = -{\left(t - 1\right)} {\left(t - 6\right)} - {\left(t^{2} + 4\right)} y - {\left(t + 4\right)} {y''} - 5 \, {y'} \hspace{2em} x( 0 )= -8" title="0 = -{\left(t - 1\right)} {\left(t - 6\right)} - {\left(t^{2} + 4\right)} y - {\left(t + 4\right)} {y''} - 5 \, {y'} \hspace{2em} x( 0 )= -8" data-latex="0 = -{\left(t - 1\right)} {\left(t - 6\right)} - {\left(t^{2} + 4\right)} y - {\left(t + 4\right)} {y''} - 5 \, {y'} \hspace{2em} x( 0 )= -8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,+\infty)" alt="(-4,+\infty)" title="(-4,+\infty)" data-latex="(-4,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,+%5Cinfty)" alt="(-4,+\infty)" title="(-4,+\infty)" data-latex="(-4,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-9608" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 9608"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t^{2} + 25\right)} y + {\left(t - 1\right)} e^{t} + {y''} = -{\left(t + 5\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( -4 )= -9" alt="{\left(t^{2} + 25\right)} y + {\left(t - 1\right)} e^{t} + {y''} = -{\left(t + 5\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( -4 )= -9" title="{\left(t^{2} + 25\right)} y + {\left(t - 1\right)} e^{t} + {y''} = -{\left(t + 5\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( -4 )= -9" data-latex="{\left(t^{2} + 25\right)} y + {\left(t - 1\right)} e^{t} + {y''} = -{\left(t + 5\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( -4 )= -9"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%5E%7B2%7D%20+%2025%5Cright)%7D%20y%20+%20%7B%5Cleft(t%20-%201%5Cright)%7D%20e%5E%7Bt%7D%20+%20%7By''%7D%20=%20-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By'''%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%20-9" alt="{\left(t^{2} + 25\right)} y + {\left(t - 1\right)} e^{t} + {y''} = -{\left(t + 5\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( -4 )= -9" title="{\left(t^{2} + 25\right)} y + {\left(t - 1\right)} e^{t} + {y''} = -{\left(t + 5\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( -4 )= -9" data-latex="{\left(t^{2} + 25\right)} y + {\left(t - 1\right)} e^{t} + {y''} = -{\left(t + 5\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( -4 )= -9"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,6)" alt="(-5,6)" title="(-5,6)" data-latex="(-5,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,6)" alt="(-5,6)" title="(-5,6)" data-latex="(-5,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-7564" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 7564"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 6\right)} {\left(t - 5\right)} y + {y'} - e^{\left(2 \, t\right)} = t {y''} e^{t} \hspace{2em} x( -2 )= 3" alt="-{\left(t + 6\right)} {\left(t - 5\right)} y + {y'} - e^{\left(2 \, t\right)} = t {y''} e^{t} \hspace{2em} x( -2 )= 3" title="-{\left(t + 6\right)} {\left(t - 5\right)} y + {y'} - e^{\left(2 \, t\right)} = t {y''} e^{t} \hspace{2em} x( -2 )= 3" data-latex="-{\left(t + 6\right)} {\left(t - 5\right)} y + {y'} - e^{\left(2 \, t\right)} = t {y''} e^{t} \hspace{2em} x( -2 )= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20y%20+%20%7By'%7D%20-%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20=%20t%20%7By''%7D%20e%5E%7Bt%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%203" alt="-{\left(t + 6\right)} {\left(t - 5\right)} y + {y'} - e^{\left(2 \, t\right)} = t {y''} e^{t} \hspace{2em} x( -2 )= 3" title="-{\left(t + 6\right)} {\left(t - 5\right)} y + {y'} - e^{\left(2 \, t\right)} = t {y''} e^{t} \hspace{2em} x( -2 )= 3" data-latex="-{\left(t + 6\right)} {\left(t - 5\right)} y + {y'} - e^{\left(2 \, t\right)} = t {y''} e^{t} \hspace{2em} x( -2 )= 3"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,0)" alt="(-\infty,0)" title="(-\infty,0)" data-latex="(-\infty,0)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,0)" alt="(-\infty,0)" title="(-\infty,0)" data-latex="(-\infty,0)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-0585" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 0585"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 1\right)} {\left(t - 5\right)} - y e^{\left(-3 \, t\right)} + 4 \, {y'} = {\left(t + 5\right)} {y''} e^{t} \hspace{2em} x( -9 )= -4" alt="-{\left(t + 1\right)} {\left(t - 5\right)} - y e^{\left(-3 \, t\right)} + 4 \, {y'} = {\left(t + 5\right)} {y''} e^{t} \hspace{2em} x( -9 )= -4" title="-{\left(t + 1\right)} {\left(t - 5\right)} - y e^{\left(-3 \, t\right)} + 4 \, {y'} = {\left(t + 5\right)} {y''} e^{t} \hspace{2em} x( -9 )= -4" data-latex="-{\left(t + 1\right)} {\left(t - 5\right)} - y e^{\left(-3 \, t\right)} + 4 \, {y'} = {\left(t + 5\right)} {y''} e^{t} \hspace{2em} x( -9 )= -4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%201%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20-%20y%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20%7By'%7D%20=%20%7B%5Cleft(t%20+%205%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20%5Chspace%7B2em%7D%20x(%20-9%20)=%20-4" alt="-{\left(t + 1\right)} {\left(t - 5\right)} - y e^{\left(-3 \, t\right)} + 4 \, {y'} = {\left(t + 5\right)} {y''} e^{t} \hspace{2em} x( -9 )= -4" title="-{\left(t + 1\right)} {\left(t - 5\right)} - y e^{\left(-3 \, t\right)} + 4 \, {y'} = {\left(t + 5\right)} {y''} e^{t} \hspace{2em} x( -9 )= -4" data-latex="-{\left(t + 1\right)} {\left(t - 5\right)} - y e^{\left(-3 \, t\right)} + 4 \, {y'} = {\left(t + 5\right)} {y''} e^{t} \hspace{2em} x( -9 )= -4"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,-5)" alt="(-\infty,-5)" title="(-\infty,-5)" data-latex="(-\infty,-5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,-5)" alt="(-\infty,-5)" title="(-\infty,-5)" data-latex="(-\infty,-5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-2223" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 2223"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 5\right)} {y'''} e^{t} - 2 \, {y''} - e^{\left(-t\right)} = {\left(t - 1\right)} {\left(t - 6\right)} y \hspace{2em} x( -4 )= -7" alt="-{\left(t + 5\right)} {y'''} e^{t} - 2 \, {y''} - e^{\left(-t\right)} = {\left(t - 1\right)} {\left(t - 6\right)} y \hspace{2em} x( -4 )= -7" title="-{\left(t + 5\right)} {y'''} e^{t} - 2 \, {y''} - e^{\left(-t\right)} = {\left(t - 1\right)} {\left(t - 6\right)} y \hspace{2em} x( -4 )= -7" data-latex="-{\left(t + 5\right)} {y'''} e^{t} - 2 \, {y''} - e^{\left(-t\right)} = {\left(t - 1\right)} {\left(t - 6\right)} y \hspace{2em} x( -4 )= -7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20-%202%20%5C,%20%7By''%7D%20-%20e%5E%7B%5Cleft(-t%5Cright)%7D%20=%20%7B%5Cleft(t%20-%201%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20y%20%5Chspace%7B2em%7D%20x(%20-4%20)=%20-7" alt="-{\left(t + 5\right)} {y'''} e^{t} - 2 \, {y''} - e^{\left(-t\right)} = {\left(t - 1\right)} {\left(t - 6\right)} y \hspace{2em} x( -4 )= -7" title="-{\left(t + 5\right)} {y'''} e^{t} - 2 \, {y''} - e^{\left(-t\right)} = {\left(t - 1\right)} {\left(t - 6\right)} y \hspace{2em} x( -4 )= -7" data-latex="-{\left(t + 5\right)} {y'''} e^{t} - 2 \, {y''} - e^{\left(-t\right)} = {\left(t - 1\right)} {\left(t - 6\right)} y \hspace{2em} x( -4 )= -7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,+\infty)" alt="(-5,+\infty)" title="(-5,+\infty)" data-latex="(-5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,+%5Cinfty)" alt="(-5,+\infty)" title="(-5,+\infty)" data-latex="(-5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-3531" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 3531"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} + {\left(t + 5\right)} {y''} + 3 \, {y'} + 4 = -{\left(t - 2\right)} {\left(t - 6\right)} y \hspace{2em} x( -8 )= -4" alt="t^{2} + {\left(t + 5\right)} {y''} + 3 \, {y'} + 4 = -{\left(t - 2\right)} {\left(t - 6\right)} y \hspace{2em} x( -8 )= -4" title="t^{2} + {\left(t + 5\right)} {y''} + 3 \, {y'} + 4 = -{\left(t - 2\right)} {\left(t - 6\right)} y \hspace{2em} x( -8 )= -4" data-latex="t^{2} + {\left(t + 5\right)} {y''} + 3 \, {y'} + 4 = -{\left(t - 2\right)} {\left(t - 6\right)} y \hspace{2em} x( -8 )= -4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&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%7B%5Cleft(t%20+%205%5Cright)%7D%20%7By''%7D%20+%203%20%5C,%20%7By'%7D%20+%204%20=%20-%7B%5Cleft(t%20-%202%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20y%20%5Chspace%7B2em%7D%20x(%20-8%20)=%20-4" alt="t^{2} + {\left(t + 5\right)} {y''} + 3 \, {y'} + 4 = -{\left(t - 2\right)} {\left(t - 6\right)} y \hspace{2em} x( -8 )= -4" title="t^{2} + {\left(t + 5\right)} {y''} + 3 \, {y'} + 4 = -{\left(t - 2\right)} {\left(t - 6\right)} y \hspace{2em} x( -8 )= -4" data-latex="t^{2} + {\left(t + 5\right)} {y''} + 3 \, {y'} + 4 = -{\left(t - 2\right)} {\left(t - 6\right)} y \hspace{2em} x( -8 )= -4"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,-5)" alt="(-\infty,-5)" title="(-\infty,-5)" data-latex="(-\infty,-5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,-5)" alt="(-\infty,-5)" title="(-\infty,-5)" data-latex="(-\infty,-5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-2928" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 2928"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t^{2} + 16\right)} y = -{\left(t + 5\right)} {\left(t - 6\right)} {y'''} - t - {y''} - 1 \hspace{2em} x( 1 )= 8" alt="{\left(t^{2} + 16\right)} y = -{\left(t + 5\right)} {\left(t - 6\right)} {y'''} - t - {y''} - 1 \hspace{2em} x( 1 )= 8" title="{\left(t^{2} + 16\right)} y = -{\left(t + 5\right)} {\left(t - 6\right)} {y'''} - t - {y''} - 1 \hspace{2em} x( 1 )= 8" data-latex="{\left(t^{2} + 16\right)} y = -{\left(t + 5\right)} {\left(t - 6\right)} {y'''} - t - {y''} - 1 \hspace{2em} x( 1 )= 8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%5E%7B2%7D%20+%2016%5Cright)%7D%20y%20=%20-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By'''%7D%20-%20t%20-%20%7By''%7D%20-%201%20%5Chspace%7B2em%7D%20x(%201%20)=%208" alt="{\left(t^{2} + 16\right)} y = -{\left(t + 5\right)} {\left(t - 6\right)} {y'''} - t - {y''} - 1 \hspace{2em} x( 1 )= 8" title="{\left(t^{2} + 16\right)} y = -{\left(t + 5\right)} {\left(t - 6\right)} {y'''} - t - {y''} - 1 \hspace{2em} x( 1 )= 8" data-latex="{\left(t^{2} + 16\right)} y = -{\left(t + 5\right)} {\left(t - 6\right)} {y'''} - t - {y''} - 1 \hspace{2em} x( 1 )= 8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,6)" alt="(-5,6)" title="(-5,6)" data-latex="(-5,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,6)" alt="(-5,6)" title="(-5,6)" data-latex="(-5,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-1930" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 1930"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 6\right)} {\left(t - 5\right)} {y''} + 4 \, {y'} = {\left(t^{2} + 9\right)} y + {\left(t + 2\right)} e^{t} \hspace{2em} x( 4 )= -8" alt="-{\left(t + 6\right)} {\left(t - 5\right)} {y''} + 4 \, {y'} = {\left(t^{2} + 9\right)} y + {\left(t + 2\right)} e^{t} \hspace{2em} x( 4 )= -8" title="-{\left(t + 6\right)} {\left(t - 5\right)} {y''} + 4 \, {y'} = {\left(t^{2} + 9\right)} y + {\left(t + 2\right)} e^{t} \hspace{2em} x( 4 )= -8" data-latex="-{\left(t + 6\right)} {\left(t - 5\right)} {y''} + 4 \, {y'} = {\left(t^{2} + 9\right)} y + {\left(t + 2\right)} e^{t} \hspace{2em} x( 4 )= -8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By''%7D%20+%204%20%5C,%20%7By'%7D%20=%20%7B%5Cleft(t%5E%7B2%7D%20+%209%5Cright)%7D%20y%20+%20%7B%5Cleft(t%20+%202%5Cright)%7D%20e%5E%7Bt%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%20-8" alt="-{\left(t + 6\right)} {\left(t - 5\right)} {y''} + 4 \, {y'} = {\left(t^{2} + 9\right)} y + {\left(t + 2\right)} e^{t} \hspace{2em} x( 4 )= -8" title="-{\left(t + 6\right)} {\left(t - 5\right)} {y''} + 4 \, {y'} = {\left(t^{2} + 9\right)} y + {\left(t + 2\right)} e^{t} \hspace{2em} x( 4 )= -8" data-latex="-{\left(t + 6\right)} {\left(t - 5\right)} {y''} + 4 \, {y'} = {\left(t^{2} + 9\right)} y + {\left(t + 2\right)} e^{t} \hspace{2em} x( 4 )= -8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,5)" alt="(-6,5)" title="(-6,5)" data-latex="(-6,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,5)" alt="(-6,5)" title="(-6,5)" data-latex="(-6,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-7460" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 7460"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t^{2} + 1\right)} y + {y'} = -{\left(t + 6\right)} {\left(t - 2\right)} {y''} - t + 6 \hspace{2em} x( -4 )= 5" alt="{\left(t^{2} + 1\right)} y + {y'} = -{\left(t + 6\right)} {\left(t - 2\right)} {y''} - t + 6 \hspace{2em} x( -4 )= 5" title="{\left(t^{2} + 1\right)} y + {y'} = -{\left(t + 6\right)} {\left(t - 2\right)} {y''} - t + 6 \hspace{2em} x( -4 )= 5" data-latex="{\left(t^{2} + 1\right)} y + {y'} = -{\left(t + 6\right)} {\left(t - 2\right)} {y''} - t + 6 \hspace{2em} x( -4 )= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%5E%7B2%7D%20+%201%5Cright)%7D%20y%20+%20%7By'%7D%20=%20-%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%202%5Cright)%7D%20%7By''%7D%20-%20t%20+%206%20%5Chspace%7B2em%7D%20x(%20-4%20)=%205" alt="{\left(t^{2} + 1\right)} y + {y'} = -{\left(t + 6\right)} {\left(t - 2\right)} {y''} - t + 6 \hspace{2em} x( -4 )= 5" title="{\left(t^{2} + 1\right)} y + {y'} = -{\left(t + 6\right)} {\left(t - 2\right)} {y''} - t + 6 \hspace{2em} x( -4 )= 5" data-latex="{\left(t^{2} + 1\right)} y + {y'} = -{\left(t + 6\right)} {\left(t - 2\right)} {y''} - t + 6 \hspace{2em} x( -4 )= 5"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,2)" alt="(-6,2)" title="(-6,2)" data-latex="(-6,2)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,2)" alt="(-6,2)" title="(-6,2)" data-latex="(-6,2)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-2674" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 2674"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y e^{\left(-t\right)} - {\left(t - 2\right)} e^{t} - 3 \, {y''} = {\left(t + 5\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( 2 )= -9" alt="-y e^{\left(-t\right)} - {\left(t - 2\right)} e^{t} - 3 \, {y''} = {\left(t + 5\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( 2 )= -9" title="-y e^{\left(-t\right)} - {\left(t - 2\right)} e^{t} - 3 \, {y''} = {\left(t + 5\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( 2 )= -9" data-latex="-y e^{\left(-t\right)} - {\left(t - 2\right)} e^{t} - 3 \, {y''} = {\left(t + 5\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( 2 )= -9"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%20%7B%5Cleft(t%20-%202%5Cright)%7D%20e%5E%7Bt%7D%20-%203%20%5C,%20%7By''%7D%20=%20%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By'''%7D%20%5Chspace%7B2em%7D%20x(%202%20)=%20-9" alt="-y e^{\left(-t\right)} - {\left(t - 2\right)} e^{t} - 3 \, {y''} = {\left(t + 5\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( 2 )= -9" title="-y e^{\left(-t\right)} - {\left(t - 2\right)} e^{t} - 3 \, {y''} = {\left(t + 5\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( 2 )= -9" data-latex="-y e^{\left(-t\right)} - {\left(t - 2\right)} e^{t} - 3 \, {y''} = {\left(t + 5\right)} {\left(t - 6\right)} {y'''} \hspace{2em} x( 2 )= -9"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,6)" alt="(-5,6)" title="(-5,6)" data-latex="(-5,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,6)" alt="(-5,6)" title="(-5,6)" data-latex="(-5,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-2011" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 2011"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -{\left(t + 2\right)} {\left(t - 6\right)} y - {\left(t + 6\right)} {y'''} e^{t} - 5 \, {y''} - e^{\left(-5 \, t\right)} \hspace{2em} x( -5 )= -10" alt="0 = -{\left(t + 2\right)} {\left(t - 6\right)} y - {\left(t + 6\right)} {y'''} e^{t} - 5 \, {y''} - e^{\left(-5 \, t\right)} \hspace{2em} x( -5 )= -10" title="0 = -{\left(t + 2\right)} {\left(t - 6\right)} y - {\left(t + 6\right)} {y'''} e^{t} - 5 \, {y''} - e^{\left(-5 \, t\right)} \hspace{2em} x( -5 )= -10" data-latex="0 = -{\left(t + 2\right)} {\left(t - 6\right)} y - {\left(t + 6\right)} {y'''} e^{t} - 5 \, {y''} - e^{\left(-5 \, t\right)} \hspace{2em} x( -5 )= -10"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-%7B%5Cleft(t%20+%202%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20y%20-%20%7B%5Cleft(t%20+%206%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20-%205%20%5C,%20%7By''%7D%20-%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%20-10" alt="0 = -{\left(t + 2\right)} {\left(t - 6\right)} y - {\left(t + 6\right)} {y'''} e^{t} - 5 \, {y''} - e^{\left(-5 \, t\right)} \hspace{2em} x( -5 )= -10" title="0 = -{\left(t + 2\right)} {\left(t - 6\right)} y - {\left(t + 6\right)} {y'''} e^{t} - 5 \, {y''} - e^{\left(-5 \, t\right)} \hspace{2em} x( -5 )= -10" data-latex="0 = -{\left(t + 2\right)} {\left(t - 6\right)} y - {\left(t + 6\right)} {y'''} e^{t} - 5 \, {y''} - e^{\left(-5 \, t\right)} \hspace{2em} x( -5 )= -10"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,+\infty)" alt="(-6,+\infty)" title="(-6,+\infty)" data-latex="(-6,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,+%5Cinfty)" alt="(-6,+\infty)" title="(-6,+\infty)" data-latex="(-6,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-7161" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 7161"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y e^{\left(-5 \, t\right)} = {\left(t + 1\right)} {y''} e^{t} + {\left(t + 4\right)} {\left(t - 6\right)} + 3 \, {y'} \hspace{2em} x( 3 )= -4" alt="-y e^{\left(-5 \, t\right)} = {\left(t + 1\right)} {y''} e^{t} + {\left(t + 4\right)} {\left(t - 6\right)} + 3 \, {y'} \hspace{2em} x( 3 )= -4" title="-y e^{\left(-5 \, t\right)} = {\left(t + 1\right)} {y''} e^{t} + {\left(t + 4\right)} {\left(t - 6\right)} + 3 \, {y'} \hspace{2em} x( 3 )= -4" data-latex="-y e^{\left(-5 \, t\right)} = {\left(t + 1\right)} {y''} e^{t} + {\left(t + 4\right)} {\left(t - 6\right)} + 3 \, {y'} \hspace{2em} x( 3 )= -4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20=%20%7B%5Cleft(t%20+%201%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20+%20%7B%5Cleft(t%20+%204%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20+%203%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20x(%203%20)=%20-4" alt="-y e^{\left(-5 \, t\right)} = {\left(t + 1\right)} {y''} e^{t} + {\left(t + 4\right)} {\left(t - 6\right)} + 3 \, {y'} \hspace{2em} x( 3 )= -4" title="-y e^{\left(-5 \, t\right)} = {\left(t + 1\right)} {y''} e^{t} + {\left(t + 4\right)} {\left(t - 6\right)} + 3 \, {y'} \hspace{2em} x( 3 )= -4" data-latex="-y e^{\left(-5 \, t\right)} = {\left(t + 1\right)} {y''} e^{t} + {\left(t + 4\right)} {\left(t - 6\right)} + 3 \, {y'} \hspace{2em} x( 3 )= -4"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-1,+\infty)" alt="(-1,+\infty)" title="(-1,+\infty)" data-latex="(-1,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-1,+%5Cinfty)" alt="(-1,+\infty)" title="(-1,+\infty)" data-latex="(-1,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5708" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5708"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t - 6\right)} {y'''} e^{t} + e^{\left(-4 \, t\right)} = -{\left(t + 5\right)} {\left(t - 1\right)} y + {y''} \hspace{2em} x( 5 )= 7" alt="{\left(t - 6\right)} {y'''} e^{t} + e^{\left(-4 \, t\right)} = -{\left(t + 5\right)} {\left(t - 1\right)} y + {y''} \hspace{2em} x( 5 )= 7" title="{\left(t - 6\right)} {y'''} e^{t} + e^{\left(-4 \, t\right)} = -{\left(t + 5\right)} {\left(t - 1\right)} y + {y''} \hspace{2em} x( 5 )= 7" data-latex="{\left(t - 6\right)} {y'''} e^{t} + e^{\left(-4 \, t\right)} = -{\left(t + 5\right)} {\left(t - 1\right)} y + {y''} \hspace{2em} x( 5 )= 7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20+%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20=%20-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%201%5Cright)%7D%20y%20+%20%7By''%7D%20%5Chspace%7B2em%7D%20x(%205%20)=%207" alt="{\left(t - 6\right)} {y'''} e^{t} + e^{\left(-4 \, t\right)} = -{\left(t + 5\right)} {\left(t - 1\right)} y + {y''} \hspace{2em} x( 5 )= 7" title="{\left(t - 6\right)} {y'''} e^{t} + e^{\left(-4 \, t\right)} = -{\left(t + 5\right)} {\left(t - 1\right)} y + {y''} \hspace{2em} x( 5 )= 7" data-latex="{\left(t - 6\right)} {y'''} e^{t} + e^{\left(-4 \, t\right)} = -{\left(t + 5\right)} {\left(t - 1\right)} y + {y''} \hspace{2em} x( 5 )= 7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,6)" alt="(-\infty,6)" title="(-\infty,6)" data-latex="(-\infty,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,6)" alt="(-\infty,6)" title="(-\infty,6)" data-latex="(-\infty,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-2040" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 2040"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 5\right)} {\left(t - 1\right)} + y e^{\left(3 \, t\right)} = -{\left(t - 6\right)} {y'''} + 2 \, {y''} \hspace{2em} x( 3 )= 10" alt="{\left(t + 5\right)} {\left(t - 1\right)} + y e^{\left(3 \, t\right)} = -{\left(t - 6\right)} {y'''} + 2 \, {y''} \hspace{2em} x( 3 )= 10" title="{\left(t + 5\right)} {\left(t - 1\right)} + y e^{\left(3 \, t\right)} = -{\left(t - 6\right)} {y'''} + 2 \, {y''} \hspace{2em} x( 3 )= 10" data-latex="{\left(t + 5\right)} {\left(t - 1\right)} + y e^{\left(3 \, t\right)} = -{\left(t - 6\right)} {y'''} + 2 \, {y''} \hspace{2em} x( 3 )= 10"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%201%5Cright)%7D%20+%20y%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20=%20-%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By'''%7D%20+%202%20%5C,%20%7By''%7D%20%5Chspace%7B2em%7D%20x(%203%20)=%2010" alt="{\left(t + 5\right)} {\left(t - 1\right)} + y e^{\left(3 \, t\right)} = -{\left(t - 6\right)} {y'''} + 2 \, {y''} \hspace{2em} x( 3 )= 10" title="{\left(t + 5\right)} {\left(t - 1\right)} + y e^{\left(3 \, t\right)} = -{\left(t - 6\right)} {y'''} + 2 \, {y''} \hspace{2em} x( 3 )= 10" data-latex="{\left(t + 5\right)} {\left(t - 1\right)} + y e^{\left(3 \, t\right)} = -{\left(t - 6\right)} {y'''} + 2 \, {y''} \hspace{2em} x( 3 )= 10"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,6)" alt="(-\infty,6)" title="(-\infty,6)" data-latex="(-\infty,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,6)" alt="(-\infty,6)" title="(-\infty,6)" data-latex="(-\infty,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8299" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8299"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y'} + e^{\left(-5 \, t\right)} = -{\left(t + 1\right)} {\left(t - 5\right)} {y''} - {\left(t + 5\right)} y \hspace{2em} x( 1 )= -5" alt="{y'} + e^{\left(-5 \, t\right)} = -{\left(t + 1\right)} {\left(t - 5\right)} {y''} - {\left(t + 5\right)} y \hspace{2em} x( 1 )= -5" title="{y'} + e^{\left(-5 \, t\right)} = -{\left(t + 1\right)} {\left(t - 5\right)} {y''} - {\left(t + 5\right)} y \hspace{2em} x( 1 )= -5" data-latex="{y'} + e^{\left(-5 \, t\right)} = -{\left(t + 1\right)} {\left(t - 5\right)} {y''} - {\left(t + 5\right)} y \hspace{2em} x( 1 )= -5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By'%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20=%20-%7B%5Cleft(t%20+%201%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By''%7D%20-%20%7B%5Cleft(t%20+%205%5Cright)%7D%20y%20%5Chspace%7B2em%7D%20x(%201%20)=%20-5" alt="{y'} + e^{\left(-5 \, t\right)} = -{\left(t + 1\right)} {\left(t - 5\right)} {y''} - {\left(t + 5\right)} y \hspace{2em} x( 1 )= -5" title="{y'} + e^{\left(-5 \, t\right)} = -{\left(t + 1\right)} {\left(t - 5\right)} {y''} - {\left(t + 5\right)} y \hspace{2em} x( 1 )= -5" data-latex="{y'} + e^{\left(-5 \, t\right)} = -{\left(t + 1\right)} {\left(t - 5\right)} {y''} - {\left(t + 5\right)} y \hspace{2em} x( 1 )= -5"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-1,5)" alt="(-1,5)" title="(-1,5)" data-latex="(-1,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-1,5)" alt="(-1,5)" title="(-1,5)" data-latex="(-1,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-3794" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 3794"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {\left(t + 5\right)} {\left(t - 5\right)} {y'''} + {\left(t + 2\right)} y e^{t} + t^{2} + 4 \, {y''} + 16 \hspace{2em} x( -3 )= -7" alt="0 = {\left(t + 5\right)} {\left(t - 5\right)} {y'''} + {\left(t + 2\right)} y e^{t} + t^{2} + 4 \, {y''} + 16 \hspace{2em} x( -3 )= -7" title="0 = {\left(t + 5\right)} {\left(t - 5\right)} {y'''} + {\left(t + 2\right)} y e^{t} + t^{2} + 4 \, {y''} + 16 \hspace{2em} x( -3 )= -7" data-latex="0 = {\left(t + 5\right)} {\left(t - 5\right)} {y'''} + {\left(t + 2\right)} y e^{t} + t^{2} + 4 \, {y''} + 16 \hspace{2em} x( -3 )= -7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20+%20%7B%5Cleft(t%20+%202%5Cright)%7D%20y%20e%5E%7Bt%7D%20+%20t%5E%7B2%7D%20+%204%20%5C,%20%7By''%7D%20+%2016%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-7" alt="0 = {\left(t + 5\right)} {\left(t - 5\right)} {y'''} + {\left(t + 2\right)} y e^{t} + t^{2} + 4 \, {y''} + 16 \hspace{2em} x( -3 )= -7" title="0 = {\left(t + 5\right)} {\left(t - 5\right)} {y'''} + {\left(t + 2\right)} y e^{t} + t^{2} + 4 \, {y''} + 16 \hspace{2em} x( -3 )= -7" data-latex="0 = {\left(t + 5\right)} {\left(t - 5\right)} {y'''} + {\left(t + 2\right)} y e^{t} + t^{2} + 4 \, {y''} + 16 \hspace{2em} x( -3 )= -7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,5)" alt="(-5,5)" title="(-5,5)" data-latex="(-5,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,5)" alt="(-5,5)" title="(-5,5)" data-latex="(-5,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-9649" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 9649"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 5\right)} {\left(t - 2\right)} {y''} + t^{2} + 3 \, {y'} + 25 = -{\left(t - 6\right)} y \hspace{2em} x( -2 )= 3" alt="{\left(t + 5\right)} {\left(t - 2\right)} {y''} + t^{2} + 3 \, {y'} + 25 = -{\left(t - 6\right)} y \hspace{2em} x( -2 )= 3" title="{\left(t + 5\right)} {\left(t - 2\right)} {y''} + t^{2} + 3 \, {y'} + 25 = -{\left(t - 6\right)} y \hspace{2em} x( -2 )= 3" data-latex="{\left(t + 5\right)} {\left(t - 2\right)} {y''} + t^{2} + 3 \, {y'} + 25 = -{\left(t - 6\right)} y \hspace{2em} x( -2 )= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%202%5Cright)%7D%20%7By''%7D%20+%20t%5E%7B2%7D%20+%203%20%5C,%20%7By'%7D%20+%2025%20=%20-%7B%5Cleft(t%20-%206%5Cright)%7D%20y%20%5Chspace%7B2em%7D%20x(%20-2%20)=%203" alt="{\left(t + 5\right)} {\left(t - 2\right)} {y''} + t^{2} + 3 \, {y'} + 25 = -{\left(t - 6\right)} y \hspace{2em} x( -2 )= 3" title="{\left(t + 5\right)} {\left(t - 2\right)} {y''} + t^{2} + 3 \, {y'} + 25 = -{\left(t - 6\right)} y \hspace{2em} x( -2 )= 3" data-latex="{\left(t + 5\right)} {\left(t - 2\right)} {y''} + t^{2} + 3 \, {y'} + 25 = -{\left(t - 6\right)} y \hspace{2em} x( -2 )= 3"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,2)" alt="(-5,2)" title="(-5,2)" data-latex="(-5,2)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,2)" alt="(-5,2)" title="(-5,2)" data-latex="(-5,2)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-9861" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 9861"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5 \, {y'} - e^{\left(5 \, t\right)} = {\left(t - 6\right)} t y + {\left(t + 4\right)} {y''} e^{t} \hspace{2em} x( -2 )= -7" alt="-5 \, {y'} - e^{\left(5 \, t\right)} = {\left(t - 6\right)} t y + {\left(t + 4\right)} {y''} e^{t} \hspace{2em} x( -2 )= -7" title="-5 \, {y'} - e^{\left(5 \, t\right)} = {\left(t - 6\right)} t y + {\left(t + 4\right)} {y''} e^{t} \hspace{2em} x( -2 )= -7" data-latex="-5 \, {y'} - e^{\left(5 \, t\right)} = {\left(t - 6\right)} t y + {\left(t + 4\right)} {y''} e^{t} \hspace{2em} x( -2 )= -7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-5%20%5C,%20%7By'%7D%20-%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20=%20%7B%5Cleft(t%20-%206%5Cright)%7D%20t%20y%20+%20%7B%5Cleft(t%20+%204%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-7" alt="-5 \, {y'} - e^{\left(5 \, t\right)} = {\left(t - 6\right)} t y + {\left(t + 4\right)} {y''} e^{t} \hspace{2em} x( -2 )= -7" title="-5 \, {y'} - e^{\left(5 \, t\right)} = {\left(t - 6\right)} t y + {\left(t + 4\right)} {y''} e^{t} \hspace{2em} x( -2 )= -7" data-latex="-5 \, {y'} - e^{\left(5 \, t\right)} = {\left(t - 6\right)} t y + {\left(t + 4\right)} {y''} e^{t} \hspace{2em} x( -2 )= -7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,+\infty)" alt="(-4,+\infty)" title="(-4,+\infty)" data-latex="(-4,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,+%5Cinfty)" alt="(-4,+\infty)" title="(-4,+\infty)" data-latex="(-4,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5432" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5432"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t - 5\right)} t y - t^{2} + 2 \, {y''} - 1 = {\left(t + 6\right)} {y'''} \hspace{2em} x( -3 )= -7" alt="-{\left(t - 5\right)} t y - t^{2} + 2 \, {y''} - 1 = {\left(t + 6\right)} {y'''} \hspace{2em} x( -3 )= -7" title="-{\left(t - 5\right)} t y - t^{2} + 2 \, {y''} - 1 = {\left(t + 6\right)} {y'''} \hspace{2em} x( -3 )= -7" data-latex="-{\left(t - 5\right)} t y - t^{2} + 2 \, {y''} - 1 = {\left(t + 6\right)} {y'''} \hspace{2em} x( -3 )= -7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20-%205%5Cright)%7D%20t%20y%20-%20t%5E%7B2%7D%20+%202%20%5C,%20%7By''%7D%20-%201%20=%20%7B%5Cleft(t%20+%206%5Cright)%7D%20%7By'''%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-7" alt="-{\left(t - 5\right)} t y - t^{2} + 2 \, {y''} - 1 = {\left(t + 6\right)} {y'''} \hspace{2em} x( -3 )= -7" title="-{\left(t - 5\right)} t y - t^{2} + 2 \, {y''} - 1 = {\left(t + 6\right)} {y'''} \hspace{2em} x( -3 )= -7" data-latex="-{\left(t - 5\right)} t y - t^{2} + 2 \, {y''} - 1 = {\left(t + 6\right)} {y'''} \hspace{2em} x( -3 )= -7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,+\infty)" alt="(-6,+\infty)" title="(-6,+\infty)" data-latex="(-6,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,+%5Cinfty)" alt="(-6,+\infty)" title="(-6,+\infty)" data-latex="(-6,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-1162" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 1162"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t - 5\right)} t y + {\left(t + 5\right)} {y''} e^{t} + t^{2} - 4 \, {y'} + 9 = 0 \hspace{2em} x( -2 )= -7" alt="{\left(t - 5\right)} t y + {\left(t + 5\right)} {y''} e^{t} + t^{2} - 4 \, {y'} + 9 = 0 \hspace{2em} x( -2 )= -7" title="{\left(t - 5\right)} t y + {\left(t + 5\right)} {y''} e^{t} + t^{2} - 4 \, {y'} + 9 = 0 \hspace{2em} x( -2 )= -7" data-latex="{\left(t - 5\right)} t y + {\left(t + 5\right)} {y''} e^{t} + t^{2} - 4 \, {y'} + 9 = 0 \hspace{2em} x( -2 )= -7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20-%205%5Cright)%7D%20t%20y%20+%20%7B%5Cleft(t%20+%205%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20+%20t%5E%7B2%7D%20-%204%20%5C,%20%7By'%7D%20+%209%20=%200%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-7" alt="{\left(t - 5\right)} t y + {\left(t + 5\right)} {y''} e^{t} + t^{2} - 4 \, {y'} + 9 = 0 \hspace{2em} x( -2 )= -7" title="{\left(t - 5\right)} t y + {\left(t + 5\right)} {y''} e^{t} + t^{2} - 4 \, {y'} + 9 = 0 \hspace{2em} x( -2 )= -7" data-latex="{\left(t - 5\right)} t y + {\left(t + 5\right)} {y''} e^{t} + t^{2} - 4 \, {y'} + 9 = 0 \hspace{2em} x( -2 )= -7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,+\infty)" alt="(-5,+\infty)" title="(-5,+\infty)" data-latex="(-5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,+%5Cinfty)" alt="(-5,+\infty)" title="(-5,+\infty)" data-latex="(-5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-1054" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 1054"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {\left(t + 6\right)} t {y''} + y e^{\left(2 \, t\right)} + {\left(t - 6\right)} e^{t} + {y'} \hspace{2em} x( -3 )= -8" alt="0 = {\left(t + 6\right)} t {y''} + y e^{\left(2 \, t\right)} + {\left(t - 6\right)} e^{t} + {y'} \hspace{2em} x( -3 )= -8" title="0 = {\left(t + 6\right)} t {y''} + y e^{\left(2 \, t\right)} + {\left(t - 6\right)} e^{t} + {y'} \hspace{2em} x( -3 )= -8" data-latex="0 = {\left(t + 6\right)} t {y''} + y e^{\left(2 \, t\right)} + {\left(t - 6\right)} e^{t} + {y'} \hspace{2em} x( -3 )= -8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7B%5Cleft(t%20+%206%5Cright)%7D%20t%20%7By''%7D%20+%20y%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%20%7B%5Cleft(t%20-%206%5Cright)%7D%20e%5E%7Bt%7D%20+%20%7By'%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-8" alt="0 = {\left(t + 6\right)} t {y''} + y e^{\left(2 \, t\right)} + {\left(t - 6\right)} e^{t} + {y'} \hspace{2em} x( -3 )= -8" title="0 = {\left(t + 6\right)} t {y''} + y e^{\left(2 \, t\right)} + {\left(t - 6\right)} e^{t} + {y'} \hspace{2em} x( -3 )= -8" data-latex="0 = {\left(t + 6\right)} t {y''} + y e^{\left(2 \, t\right)} + {\left(t - 6\right)} e^{t} + {y'} \hspace{2em} x( -3 )= -8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,0)" alt="(-6,0)" title="(-6,0)" data-latex="(-6,0)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,0)" alt="(-6,0)" title="(-6,0)" data-latex="(-6,0)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-1739" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 1739"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} + 4 \, {y''} + 4 = -{\left(t + 5\right)} {\left(t + 1\right)} {y'''} - {\left(t - 5\right)} y \hspace{2em} x( -3 )= 2" alt="t^{2} + 4 \, {y''} + 4 = -{\left(t + 5\right)} {\left(t + 1\right)} {y'''} - {\left(t - 5\right)} y \hspace{2em} x( -3 )= 2" title="t^{2} + 4 \, {y''} + 4 = -{\left(t + 5\right)} {\left(t + 1\right)} {y'''} - {\left(t - 5\right)} y \hspace{2em} x( -3 )= 2" data-latex="t^{2} + 4 \, {y''} + 4 = -{\left(t + 5\right)} {\left(t + 1\right)} {y'''} - {\left(t - 5\right)} y \hspace{2em} x( -3 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&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+%204%20%5C,%20%7By''%7D%20+%204%20=%20-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20+%201%5Cright)%7D%20%7By'''%7D%20-%20%7B%5Cleft(t%20-%205%5Cright)%7D%20y%20%5Chspace%7B2em%7D%20x(%20-3%20)=%202" alt="t^{2} + 4 \, {y''} + 4 = -{\left(t + 5\right)} {\left(t + 1\right)} {y'''} - {\left(t - 5\right)} y \hspace{2em} x( -3 )= 2" title="t^{2} + 4 \, {y''} + 4 = -{\left(t + 5\right)} {\left(t + 1\right)} {y'''} - {\left(t - 5\right)} y \hspace{2em} x( -3 )= 2" data-latex="t^{2} + 4 \, {y''} + 4 = -{\left(t + 5\right)} {\left(t + 1\right)} {y'''} - {\left(t - 5\right)} y \hspace{2em} x( -3 )= 2"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,-1)" alt="(-5,-1)" title="(-5,-1)" data-latex="(-5,-1)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,-1)" alt="(-5,-1)" title="(-5,-1)" data-latex="(-5,-1)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-1262" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 1262"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 5\right)} {y''} e^{t} + {\left(t + 2\right)} {\left(t - 6\right)} - 3 \, {y'} = -y e^{\left(5 \, t\right)} \hspace{2em} x( -7 )= -1" alt="{\left(t + 5\right)} {y''} e^{t} + {\left(t + 2\right)} {\left(t - 6\right)} - 3 \, {y'} = -y e^{\left(5 \, t\right)} \hspace{2em} x( -7 )= -1" title="{\left(t + 5\right)} {y''} e^{t} + {\left(t + 2\right)} {\left(t - 6\right)} - 3 \, {y'} = -y e^{\left(5 \, t\right)} \hspace{2em} x( -7 )= -1" data-latex="{\left(t + 5\right)} {y''} e^{t} + {\left(t + 2\right)} {\left(t - 6\right)} - 3 \, {y'} = -y e^{\left(5 \, t\right)} \hspace{2em} x( -7 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%205%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20+%20%7B%5Cleft(t%20+%202%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20-%203%20%5C,%20%7By'%7D%20=%20-y%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%20-7%20)=%20-1" alt="{\left(t + 5\right)} {y''} e^{t} + {\left(t + 2\right)} {\left(t - 6\right)} - 3 \, {y'} = -y e^{\left(5 \, t\right)} \hspace{2em} x( -7 )= -1" title="{\left(t + 5\right)} {y''} e^{t} + {\left(t + 2\right)} {\left(t - 6\right)} - 3 \, {y'} = -y e^{\left(5 \, t\right)} \hspace{2em} x( -7 )= -1" data-latex="{\left(t + 5\right)} {y''} e^{t} + {\left(t + 2\right)} {\left(t - 6\right)} - 3 \, {y'} = -y e^{\left(5 \, t\right)} \hspace{2em} x( -7 )= -1"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,-5)" alt="(-\infty,-5)" title="(-\infty,-5)" data-latex="(-\infty,-5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,-5)" alt="(-\infty,-5)" title="(-\infty,-5)" data-latex="(-\infty,-5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-3462" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 3462"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 2\right)} {y''} e^{t} + {\left(t + 4\right)} {\left(t - 6\right)} + 3 \, {y'} = -y e^{\left(-4 \, t\right)} \hspace{2em} x( 2 )= -6" alt="{\left(t + 2\right)} {y''} e^{t} + {\left(t + 4\right)} {\left(t - 6\right)} + 3 \, {y'} = -y e^{\left(-4 \, t\right)} \hspace{2em} x( 2 )= -6" title="{\left(t + 2\right)} {y''} e^{t} + {\left(t + 4\right)} {\left(t - 6\right)} + 3 \, {y'} = -y e^{\left(-4 \, t\right)} \hspace{2em} x( 2 )= -6" data-latex="{\left(t + 2\right)} {y''} e^{t} + {\left(t + 4\right)} {\left(t - 6\right)} + 3 \, {y'} = -y e^{\left(-4 \, t\right)} \hspace{2em} x( 2 )= -6"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%202%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20+%20%7B%5Cleft(t%20+%204%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20+%203%20%5C,%20%7By'%7D%20=%20-y%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%202%20)=%20-6" alt="{\left(t + 2\right)} {y''} e^{t} + {\left(t + 4\right)} {\left(t - 6\right)} + 3 \, {y'} = -y e^{\left(-4 \, t\right)} \hspace{2em} x( 2 )= -6" title="{\left(t + 2\right)} {y''} e^{t} + {\left(t + 4\right)} {\left(t - 6\right)} + 3 \, {y'} = -y e^{\left(-4 \, t\right)} \hspace{2em} x( 2 )= -6" data-latex="{\left(t + 2\right)} {y''} e^{t} + {\left(t + 4\right)} {\left(t - 6\right)} + 3 \, {y'} = -y e^{\left(-4 \, t\right)} \hspace{2em} x( 2 )= -6"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-2,+\infty)" alt="(-2,+\infty)" title="(-2,+\infty)" data-latex="(-2,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-2,+%5Cinfty)" alt="(-2,+\infty)" title="(-2,+\infty)" data-latex="(-2,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8365" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8365"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t - 5\right)} t {y''} + e^{\left(2 \, t\right)} = -{\left(t + 4\right)} y e^{t} - {y'} \hspace{2em} x( 1 )= 7" alt="{\left(t - 5\right)} t {y''} + e^{\left(2 \, t\right)} = -{\left(t + 4\right)} y e^{t} - {y'} \hspace{2em} x( 1 )= 7" title="{\left(t - 5\right)} t {y''} + e^{\left(2 \, t\right)} = -{\left(t + 4\right)} y e^{t} - {y'} \hspace{2em} x( 1 )= 7" data-latex="{\left(t - 5\right)} t {y''} + e^{\left(2 \, t\right)} = -{\left(t + 4\right)} y e^{t} - {y'} \hspace{2em} x( 1 )= 7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20-%205%5Cright)%7D%20t%20%7By''%7D%20+%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20=%20-%7B%5Cleft(t%20+%204%5Cright)%7D%20y%20e%5E%7Bt%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7D%20x(%201%20)=%207" alt="{\left(t - 5\right)} t {y''} + e^{\left(2 \, t\right)} = -{\left(t + 4\right)} y e^{t} - {y'} \hspace{2em} x( 1 )= 7" title="{\left(t - 5\right)} t {y''} + e^{\left(2 \, t\right)} = -{\left(t + 4\right)} y e^{t} - {y'} \hspace{2em} x( 1 )= 7" data-latex="{\left(t - 5\right)} t {y''} + e^{\left(2 \, t\right)} = -{\left(t + 4\right)} y e^{t} - {y'} \hspace{2em} x( 1 )= 7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(0,5)" alt="(0,5)" title="(0,5)" data-latex="(0,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(0,5)" alt="(0,5)" title="(0,5)" data-latex="(0,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-1809" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 1809"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 6\right)} t y - {\left(t - 5\right)} {y'''} e^{t} = t^{2} + {y''} + 1 \hspace{2em} x( 6 )= 2" alt="-{\left(t + 6\right)} t y - {\left(t - 5\right)} {y'''} e^{t} = t^{2} + {y''} + 1 \hspace{2em} x( 6 )= 2" title="-{\left(t + 6\right)} t y - {\left(t - 5\right)} {y'''} e^{t} = t^{2} + {y''} + 1 \hspace{2em} x( 6 )= 2" data-latex="-{\left(t + 6\right)} t y - {\left(t - 5\right)} {y'''} e^{t} = t^{2} + {y''} + 1 \hspace{2em} x( 6 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%206%5Cright)%7D%20t%20y%20-%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20=%20t%5E%7B2%7D%20+%20%7By''%7D%20+%201%20%5Chspace%7B2em%7D%20x(%206%20)=%202" alt="-{\left(t + 6\right)} t y - {\left(t - 5\right)} {y'''} e^{t} = t^{2} + {y''} + 1 \hspace{2em} x( 6 )= 2" title="-{\left(t + 6\right)} t y - {\left(t - 5\right)} {y'''} e^{t} = t^{2} + {y''} + 1 \hspace{2em} x( 6 )= 2" data-latex="-{\left(t + 6\right)} t y - {\left(t - 5\right)} {y'''} e^{t} = t^{2} + {y''} + 1 \hspace{2em} x( 6 )= 2"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+\infty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+%5Cinfty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-7950" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 7950"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t^{2} + {\left(t - 6\right)} {y'''} + 9 = -{\left(t + 5\right)} t y - {y''} \hspace{2em} x( 2 )= 10" alt="t^{2} + {\left(t - 6\right)} {y'''} + 9 = -{\left(t + 5\right)} t y - {y''} \hspace{2em} x( 2 )= 10" title="t^{2} + {\left(t - 6\right)} {y'''} + 9 = -{\left(t + 5\right)} t y - {y''} \hspace{2em} x( 2 )= 10" data-latex="t^{2} + {\left(t - 6\right)} {y'''} + 9 = -{\left(t + 5\right)} t y - {y''} \hspace{2em} x( 2 )= 10"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&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%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By'''%7D%20+%209%20=%20-%7B%5Cleft(t%20+%205%5Cright)%7D%20t%20y%20-%20%7By''%7D%20%5Chspace%7B2em%7D%20x(%202%20)=%2010" alt="t^{2} + {\left(t - 6\right)} {y'''} + 9 = -{\left(t + 5\right)} t y - {y''} \hspace{2em} x( 2 )= 10" title="t^{2} + {\left(t - 6\right)} {y'''} + 9 = -{\left(t + 5\right)} t y - {y''} \hspace{2em} x( 2 )= 10" data-latex="t^{2} + {\left(t - 6\right)} {y'''} + 9 = -{\left(t + 5\right)} t y - {y''} \hspace{2em} x( 2 )= 10"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,6)" alt="(-\infty,6)" title="(-\infty,6)" data-latex="(-\infty,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,6)" alt="(-\infty,6)" title="(-\infty,6)" data-latex="(-\infty,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8082" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8082"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {\left(t + 4\right)} {\left(t - 6\right)} {y''} + {\left(t - 2\right)} y e^{t} + t^{2} + 3 \, {y'} + 25 \hspace{2em} x( 5 )= -5" alt="0 = {\left(t + 4\right)} {\left(t - 6\right)} {y''} + {\left(t - 2\right)} y e^{t} + t^{2} + 3 \, {y'} + 25 \hspace{2em} x( 5 )= -5" title="0 = {\left(t + 4\right)} {\left(t - 6\right)} {y''} + {\left(t - 2\right)} y e^{t} + t^{2} + 3 \, {y'} + 25 \hspace{2em} x( 5 )= -5" data-latex="0 = {\left(t + 4\right)} {\left(t - 6\right)} {y''} + {\left(t - 2\right)} y e^{t} + t^{2} + 3 \, {y'} + 25 \hspace{2em} x( 5 )= -5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7B%5Cleft(t%20+%204%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By''%7D%20+%20%7B%5Cleft(t%20-%202%5Cright)%7D%20y%20e%5E%7Bt%7D%20+%20t%5E%7B2%7D%20+%203%20%5C,%20%7By'%7D%20+%2025%20%5Chspace%7B2em%7D%20x(%205%20)=%20-5" alt="0 = {\left(t + 4\right)} {\left(t - 6\right)} {y''} + {\left(t - 2\right)} y e^{t} + t^{2} + 3 \, {y'} + 25 \hspace{2em} x( 5 )= -5" title="0 = {\left(t + 4\right)} {\left(t - 6\right)} {y''} + {\left(t - 2\right)} y e^{t} + t^{2} + 3 \, {y'} + 25 \hspace{2em} x( 5 )= -5" data-latex="0 = {\left(t + 4\right)} {\left(t - 6\right)} {y''} + {\left(t - 2\right)} y e^{t} + t^{2} + 3 \, {y'} + 25 \hspace{2em} x( 5 )= -5"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,6)" alt="(-4,6)" title="(-4,6)" data-latex="(-4,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,6)" alt="(-4,6)" title="(-4,6)" data-latex="(-4,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-1277" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 1277"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t^{2} + 25\right)} y - {\left(t + 6\right)} {y'''} = {\left(t + 2\right)} {\left(t - 6\right)} - 4 \, {y''} \hspace{2em} x( -9 )= -3" alt="-{\left(t^{2} + 25\right)} y - {\left(t + 6\right)} {y'''} = {\left(t + 2\right)} {\left(t - 6\right)} - 4 \, {y''} \hspace{2em} x( -9 )= -3" title="-{\left(t^{2} + 25\right)} y - {\left(t + 6\right)} {y'''} = {\left(t + 2\right)} {\left(t - 6\right)} - 4 \, {y''} \hspace{2em} x( -9 )= -3" data-latex="-{\left(t^{2} + 25\right)} y - {\left(t + 6\right)} {y'''} = {\left(t + 2\right)} {\left(t - 6\right)} - 4 \, {y''} \hspace{2em} x( -9 )= -3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%5E%7B2%7D%20+%2025%5Cright)%7D%20y%20-%20%7B%5Cleft(t%20+%206%5Cright)%7D%20%7By'''%7D%20=%20%7B%5Cleft(t%20+%202%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20-%204%20%5C,%20%7By''%7D%20%5Chspace%7B2em%7D%20x(%20-9%20)=%20-3" alt="-{\left(t^{2} + 25\right)} y - {\left(t + 6\right)} {y'''} = {\left(t + 2\right)} {\left(t - 6\right)} - 4 \, {y''} \hspace{2em} x( -9 )= -3" title="-{\left(t^{2} + 25\right)} y - {\left(t + 6\right)} {y'''} = {\left(t + 2\right)} {\left(t - 6\right)} - 4 \, {y''} \hspace{2em} x( -9 )= -3" data-latex="-{\left(t^{2} + 25\right)} y - {\left(t + 6\right)} {y'''} = {\left(t + 2\right)} {\left(t - 6\right)} - 4 \, {y''} \hspace{2em} x( -9 )= -3"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,-6)" alt="(-\infty,-6)" title="(-\infty,-6)" data-latex="(-\infty,-6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,-6)" alt="(-\infty,-6)" title="(-\infty,-6)" data-latex="(-\infty,-6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-9284" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 9284"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t - 6\right)} t - 4 \, {y''} = -{\left(t + 4\right)} {y'''} - y e^{\left(-5 \, t\right)} \hspace{2em} x( -1 )= -7" alt="{\left(t - 6\right)} t - 4 \, {y''} = -{\left(t + 4\right)} {y'''} - y e^{\left(-5 \, t\right)} \hspace{2em} x( -1 )= -7" title="{\left(t - 6\right)} t - 4 \, {y''} = -{\left(t + 4\right)} {y'''} - y e^{\left(-5 \, t\right)} \hspace{2em} x( -1 )= -7" data-latex="{\left(t - 6\right)} t - 4 \, {y''} = -{\left(t + 4\right)} {y'''} - y e^{\left(-5 \, t\right)} \hspace{2em} x( -1 )= -7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20-%206%5Cright)%7D%20t%20-%204%20%5C,%20%7By''%7D%20=%20-%7B%5Cleft(t%20+%204%5Cright)%7D%20%7By'''%7D%20-%20y%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%20-1%20)=%20-7" alt="{\left(t - 6\right)} t - 4 \, {y''} = -{\left(t + 4\right)} {y'''} - y e^{\left(-5 \, t\right)} \hspace{2em} x( -1 )= -7" title="{\left(t - 6\right)} t - 4 \, {y''} = -{\left(t + 4\right)} {y'''} - y e^{\left(-5 \, t\right)} \hspace{2em} x( -1 )= -7" data-latex="{\left(t - 6\right)} t - 4 \, {y''} = -{\left(t + 4\right)} {y'''} - y e^{\left(-5 \, t\right)} \hspace{2em} x( -1 )= -7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,+\infty)" alt="(-4,+\infty)" title="(-4,+\infty)" data-latex="(-4,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,+%5Cinfty)" alt="(-4,+\infty)" title="(-4,+\infty)" data-latex="(-4,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8731" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8731"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t^{2} + 1\right)} y = -{\left(t + 6\right)} {\left(t - 5\right)} {y''} - t - {y'} - 1 \hspace{2em} x( -4 )= 7" alt="{\left(t^{2} + 1\right)} y = -{\left(t + 6\right)} {\left(t - 5\right)} {y''} - t - {y'} - 1 \hspace{2em} x( -4 )= 7" title="{\left(t^{2} + 1\right)} y = -{\left(t + 6\right)} {\left(t - 5\right)} {y''} - t - {y'} - 1 \hspace{2em} x( -4 )= 7" data-latex="{\left(t^{2} + 1\right)} y = -{\left(t + 6\right)} {\left(t - 5\right)} {y''} - t - {y'} - 1 \hspace{2em} x( -4 )= 7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%5E%7B2%7D%20+%201%5Cright)%7D%20y%20=%20-%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By''%7D%20-%20t%20-%20%7By'%7D%20-%201%20%5Chspace%7B2em%7D%20x(%20-4%20)=%207" alt="{\left(t^{2} + 1\right)} y = -{\left(t + 6\right)} {\left(t - 5\right)} {y''} - t - {y'} - 1 \hspace{2em} x( -4 )= 7" title="{\left(t^{2} + 1\right)} y = -{\left(t + 6\right)} {\left(t - 5\right)} {y''} - t - {y'} - 1 \hspace{2em} x( -4 )= 7" data-latex="{\left(t^{2} + 1\right)} y = -{\left(t + 6\right)} {\left(t - 5\right)} {y''} - t - {y'} - 1 \hspace{2em} x( -4 )= 7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,5)" alt="(-6,5)" title="(-6,5)" data-latex="(-6,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,5)" alt="(-6,5)" title="(-6,5)" data-latex="(-6,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5965" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5965"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t^{2} + 16\right)} y - t + 4 \, {y'} - 4 = {\left(t + 1\right)} {\left(t - 6\right)} {y''} \hspace{2em} x( 3 )= -5" alt="-{\left(t^{2} + 16\right)} y - t + 4 \, {y'} - 4 = {\left(t + 1\right)} {\left(t - 6\right)} {y''} \hspace{2em} x( 3 )= -5" title="-{\left(t^{2} + 16\right)} y - t + 4 \, {y'} - 4 = {\left(t + 1\right)} {\left(t - 6\right)} {y''} \hspace{2em} x( 3 )= -5" data-latex="-{\left(t^{2} + 16\right)} y - t + 4 \, {y'} - 4 = {\left(t + 1\right)} {\left(t - 6\right)} {y''} \hspace{2em} x( 3 )= -5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%5E%7B2%7D%20+%2016%5Cright)%7D%20y%20-%20t%20+%204%20%5C,%20%7By'%7D%20-%204%20=%20%7B%5Cleft(t%20+%201%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By''%7D%20%5Chspace%7B2em%7D%20x(%203%20)=%20-5" alt="-{\left(t^{2} + 16\right)} y - t + 4 \, {y'} - 4 = {\left(t + 1\right)} {\left(t - 6\right)} {y''} \hspace{2em} x( 3 )= -5" title="-{\left(t^{2} + 16\right)} y - t + 4 \, {y'} - 4 = {\left(t + 1\right)} {\left(t - 6\right)} {y''} \hspace{2em} x( 3 )= -5" data-latex="-{\left(t^{2} + 16\right)} y - t + 4 \, {y'} - 4 = {\left(t + 1\right)} {\left(t - 6\right)} {y''} \hspace{2em} x( 3 )= -5"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-1,6)" alt="(-1,6)" title="(-1,6)" data-latex="(-1,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-1,6)" alt="(-1,6)" title="(-1,6)" data-latex="(-1,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8529" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8529"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 6\right)} {y''} = -{\left(t - 2\right)} {\left(t - 6\right)} y - {y'} - e^{\left(5 \, t\right)} \hspace{2em} x( -10 )= -2" alt="{\left(t + 6\right)} {y''} = -{\left(t - 2\right)} {\left(t - 6\right)} y - {y'} - e^{\left(5 \, t\right)} \hspace{2em} x( -10 )= -2" title="{\left(t + 6\right)} {y''} = -{\left(t - 2\right)} {\left(t - 6\right)} y - {y'} - e^{\left(5 \, t\right)} \hspace{2em} x( -10 )= -2" data-latex="{\left(t + 6\right)} {y''} = -{\left(t - 2\right)} {\left(t - 6\right)} y - {y'} - e^{\left(5 \, t\right)} \hspace{2em} x( -10 )= -2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%206%5Cright)%7D%20%7By''%7D%20=%20-%7B%5Cleft(t%20-%202%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20y%20-%20%7By'%7D%20-%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%20-10%20)=%20-2" alt="{\left(t + 6\right)} {y''} = -{\left(t - 2\right)} {\left(t - 6\right)} y - {y'} - e^{\left(5 \, t\right)} \hspace{2em} x( -10 )= -2" title="{\left(t + 6\right)} {y''} = -{\left(t - 2\right)} {\left(t - 6\right)} y - {y'} - e^{\left(5 \, t\right)} \hspace{2em} x( -10 )= -2" data-latex="{\left(t + 6\right)} {y''} = -{\left(t - 2\right)} {\left(t - 6\right)} y - {y'} - e^{\left(5 \, t\right)} \hspace{2em} x( -10 )= -2"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,-6)" alt="(-\infty,-6)" title="(-\infty,-6)" data-latex="(-\infty,-6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,-6)" alt="(-\infty,-6)" title="(-\infty,-6)" data-latex="(-\infty,-6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-4470" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 4470"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 6\right)} t - {\left(t^{2} + 25\right)} y + 5 \, {y'} = {\left(t - 5\right)} {y''} \hspace{2em} x( 6 )= 2" alt="-{\left(t + 6\right)} t - {\left(t^{2} + 25\right)} y + 5 \, {y'} = {\left(t - 5\right)} {y''} \hspace{2em} x( 6 )= 2" title="-{\left(t + 6\right)} t - {\left(t^{2} + 25\right)} y + 5 \, {y'} = {\left(t - 5\right)} {y''} \hspace{2em} x( 6 )= 2" data-latex="-{\left(t + 6\right)} t - {\left(t^{2} + 25\right)} y + 5 \, {y'} = {\left(t - 5\right)} {y''} \hspace{2em} x( 6 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%206%5Cright)%7D%20t%20-%20%7B%5Cleft(t%5E%7B2%7D%20+%2025%5Cright)%7D%20y%20+%205%20%5C,%20%7By'%7D%20=%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By''%7D%20%5Chspace%7B2em%7D%20x(%206%20)=%202" alt="-{\left(t + 6\right)} t - {\left(t^{2} + 25\right)} y + 5 \, {y'} = {\left(t - 5\right)} {y''} \hspace{2em} x( 6 )= 2" title="-{\left(t + 6\right)} t - {\left(t^{2} + 25\right)} y + 5 \, {y'} = {\left(t - 5\right)} {y''} \hspace{2em} x( 6 )= 2" data-latex="-{\left(t + 6\right)} t - {\left(t^{2} + 25\right)} y + 5 \, {y'} = {\left(t - 5\right)} {y''} \hspace{2em} x( 6 )= 2"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+\infty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+%5Cinfty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5497" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5497"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 5\right)} {\left(t + 2\right)} {y''} - y e^{\left(4 \, t\right)} - t - 3 \, {y'} + 5 = 0 \hspace{2em} x( -4 )= 2" alt="-{\left(t + 5\right)} {\left(t + 2\right)} {y''} - y e^{\left(4 \, t\right)} - t - 3 \, {y'} + 5 = 0 \hspace{2em} x( -4 )= 2" title="-{\left(t + 5\right)} {\left(t + 2\right)} {y''} - y e^{\left(4 \, t\right)} - t - 3 \, {y'} + 5 = 0 \hspace{2em} x( -4 )= 2" data-latex="-{\left(t + 5\right)} {\left(t + 2\right)} {y''} - y e^{\left(4 \, t\right)} - t - 3 \, {y'} + 5 = 0 \hspace{2em} x( -4 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20+%202%5Cright)%7D%20%7By''%7D%20-%20y%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%20t%20-%203%20%5C,%20%7By'%7D%20+%205%20=%200%20%5Chspace%7B2em%7D%20x(%20-4%20)=%202" alt="-{\left(t + 5\right)} {\left(t + 2\right)} {y''} - y e^{\left(4 \, t\right)} - t - 3 \, {y'} + 5 = 0 \hspace{2em} x( -4 )= 2" title="-{\left(t + 5\right)} {\left(t + 2\right)} {y''} - y e^{\left(4 \, t\right)} - t - 3 \, {y'} + 5 = 0 \hspace{2em} x( -4 )= 2" data-latex="-{\left(t + 5\right)} {\left(t + 2\right)} {y''} - y e^{\left(4 \, t\right)} - t - 3 \, {y'} + 5 = 0 \hspace{2em} x( -4 )= 2"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,-2)" alt="(-5,-2)" title="(-5,-2)" data-latex="(-5,-2)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,-2)" alt="(-5,-2)" title="(-5,-2)" data-latex="(-5,-2)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-3156" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 3156"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t - 6\right)} {y'''} e^{t} + y e^{\left(-4 \, t\right)} - 4 \, {y''} = -{\left(t + 4\right)} {\left(t - 1\right)} \hspace{2em} x( 7 )= 5" alt="{\left(t - 6\right)} {y'''} e^{t} + y e^{\left(-4 \, t\right)} - 4 \, {y''} = -{\left(t + 4\right)} {\left(t - 1\right)} \hspace{2em} x( 7 )= 5" title="{\left(t - 6\right)} {y'''} e^{t} + y e^{\left(-4 \, t\right)} - 4 \, {y''} = -{\left(t + 4\right)} {\left(t - 1\right)} \hspace{2em} x( 7 )= 5" data-latex="{\left(t - 6\right)} {y'''} e^{t} + y e^{\left(-4 \, t\right)} - 4 \, {y''} = -{\left(t + 4\right)} {\left(t - 1\right)} \hspace{2em} x( 7 )= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20+%20y%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20%7By''%7D%20=%20-%7B%5Cleft(t%20+%204%5Cright)%7D%20%7B%5Cleft(t%20-%201%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%207%20)=%205" alt="{\left(t - 6\right)} {y'''} e^{t} + y e^{\left(-4 \, t\right)} - 4 \, {y''} = -{\left(t + 4\right)} {\left(t - 1\right)} \hspace{2em} x( 7 )= 5" title="{\left(t - 6\right)} {y'''} e^{t} + y e^{\left(-4 \, t\right)} - 4 \, {y''} = -{\left(t + 4\right)} {\left(t - 1\right)} \hspace{2em} x( 7 )= 5" data-latex="{\left(t - 6\right)} {y'''} e^{t} + y e^{\left(-4 \, t\right)} - 4 \, {y''} = -{\left(t + 4\right)} {\left(t - 1\right)} \hspace{2em} x( 7 )= 5"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(6,+\infty)" alt="(6,+\infty)" title="(6,+\infty)" data-latex="(6,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(6,+%5Cinfty)" alt="(6,+\infty)" title="(6,+\infty)" data-latex="(6,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5260" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5260"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {\left(t - 6\right)} t y + {\left(t + 5\right)} {y'''} e^{t} + t^{2} + 5 \, {y''} + 1 \hspace{2em} x( -3 )= -6" alt="0 = {\left(t - 6\right)} t y + {\left(t + 5\right)} {y'''} e^{t} + t^{2} + 5 \, {y''} + 1 \hspace{2em} x( -3 )= -6" title="0 = {\left(t - 6\right)} t y + {\left(t + 5\right)} {y'''} e^{t} + t^{2} + 5 \, {y''} + 1 \hspace{2em} x( -3 )= -6" data-latex="0 = {\left(t - 6\right)} t y + {\left(t + 5\right)} {y'''} e^{t} + t^{2} + 5 \, {y''} + 1 \hspace{2em} x( -3 )= -6"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7B%5Cleft(t%20-%206%5Cright)%7D%20t%20y%20+%20%7B%5Cleft(t%20+%205%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20+%20t%5E%7B2%7D%20+%205%20%5C,%20%7By''%7D%20+%201%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-6" alt="0 = {\left(t - 6\right)} t y + {\left(t + 5\right)} {y'''} e^{t} + t^{2} + 5 \, {y''} + 1 \hspace{2em} x( -3 )= -6" title="0 = {\left(t - 6\right)} t y + {\left(t + 5\right)} {y'''} e^{t} + t^{2} + 5 \, {y''} + 1 \hspace{2em} x( -3 )= -6" data-latex="0 = {\left(t - 6\right)} t y + {\left(t + 5\right)} {y'''} e^{t} + t^{2} + 5 \, {y''} + 1 \hspace{2em} x( -3 )= -6"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,+\infty)" alt="(-5,+\infty)" title="(-5,+\infty)" data-latex="(-5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,+%5Cinfty)" alt="(-5,+\infty)" title="(-5,+\infty)" data-latex="(-5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-0866" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 0866"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{2} - {y''} - 1 = {\left(t + 1\right)} {\left(t - 5\right)} y + {\left(t + 5\right)} {y'''} e^{t} \hspace{2em} x( -6 )= -1" alt="-t^{2} - {y''} - 1 = {\left(t + 1\right)} {\left(t - 5\right)} y + {\left(t + 5\right)} {y'''} e^{t} \hspace{2em} x( -6 )= -1" title="-t^{2} - {y''} - 1 = {\left(t + 1\right)} {\left(t - 5\right)} y + {\left(t + 5\right)} {y'''} e^{t} \hspace{2em} x( -6 )= -1" data-latex="-t^{2} - {y''} - 1 = {\left(t + 1\right)} {\left(t - 5\right)} y + {\left(t + 5\right)} {y'''} e^{t} \hspace{2em} x( -6 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&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%7By''%7D%20-%201%20=%20%7B%5Cleft(t%20+%201%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20y%20+%20%7B%5Cleft(t%20+%205%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20%5Chspace%7B2em%7D%20x(%20-6%20)=%20-1" alt="-t^{2} - {y''} - 1 = {\left(t + 1\right)} {\left(t - 5\right)} y + {\left(t + 5\right)} {y'''} e^{t} \hspace{2em} x( -6 )= -1" title="-t^{2} - {y''} - 1 = {\left(t + 1\right)} {\left(t - 5\right)} y + {\left(t + 5\right)} {y'''} e^{t} \hspace{2em} x( -6 )= -1" data-latex="-t^{2} - {y''} - 1 = {\left(t + 1\right)} {\left(t - 5\right)} y + {\left(t + 5\right)} {y'''} e^{t} \hspace{2em} x( -6 )= -1"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,-5)" alt="(-\infty,-5)" title="(-\infty,-5)" data-latex="(-\infty,-5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,-5)" alt="(-\infty,-5)" title="(-\infty,-5)" data-latex="(-\infty,-5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5377" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5377"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 6\right)} {\left(t + 1\right)} + {\left(t^{2} + 9\right)} y = -{\left(t - 5\right)} {y''} e^{t} - 2 \, {y'} \hspace{2em} x( 9 )= 4" alt="{\left(t + 6\right)} {\left(t + 1\right)} + {\left(t^{2} + 9\right)} y = -{\left(t - 5\right)} {y''} e^{t} - 2 \, {y'} \hspace{2em} x( 9 )= 4" title="{\left(t + 6\right)} {\left(t + 1\right)} + {\left(t^{2} + 9\right)} y = -{\left(t - 5\right)} {y''} e^{t} - 2 \, {y'} \hspace{2em} x( 9 )= 4" data-latex="{\left(t + 6\right)} {\left(t + 1\right)} + {\left(t^{2} + 9\right)} y = -{\left(t - 5\right)} {y''} e^{t} - 2 \, {y'} \hspace{2em} x( 9 )= 4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20+%201%5Cright)%7D%20+%20%7B%5Cleft(t%5E%7B2%7D%20+%209%5Cright)%7D%20y%20=%20-%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20-%202%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20x(%209%20)=%204" alt="{\left(t + 6\right)} {\left(t + 1\right)} + {\left(t^{2} + 9\right)} y = -{\left(t - 5\right)} {y''} e^{t} - 2 \, {y'} \hspace{2em} x( 9 )= 4" title="{\left(t + 6\right)} {\left(t + 1\right)} + {\left(t^{2} + 9\right)} y = -{\left(t - 5\right)} {y''} e^{t} - 2 \, {y'} \hspace{2em} x( 9 )= 4" data-latex="{\left(t + 6\right)} {\left(t + 1\right)} + {\left(t^{2} + 9\right)} y = -{\left(t - 5\right)} {y''} e^{t} - 2 \, {y'} \hspace{2em} x( 9 )= 4"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+\infty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+%5Cinfty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-0617" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 0617"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-t^{2} - 4 = {\left(t + 4\right)} {\left(t - 2\right)} {y'''} + {\left(t - 5\right)} y - 5 \, {y''} \hspace{2em} x( 1 )= 4" alt="-t^{2} - 4 = {\left(t + 4\right)} {\left(t - 2\right)} {y'''} + {\left(t - 5\right)} y - 5 \, {y''} \hspace{2em} x( 1 )= 4" title="-t^{2} - 4 = {\left(t + 4\right)} {\left(t - 2\right)} {y'''} + {\left(t - 5\right)} y - 5 \, {y''} \hspace{2em} x( 1 )= 4" data-latex="-t^{2} - 4 = {\left(t + 4\right)} {\left(t - 2\right)} {y'''} + {\left(t - 5\right)} y - 5 \, {y''} \hspace{2em} x( 1 )= 4"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&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-%204%20=%20%7B%5Cleft(t%20+%204%5Cright)%7D%20%7B%5Cleft(t%20-%202%5Cright)%7D%20%7By'''%7D%20+%20%7B%5Cleft(t%20-%205%5Cright)%7D%20y%20-%205%20%5C,%20%7By''%7D%20%5Chspace%7B2em%7D%20x(%201%20)=%204" alt="-t^{2} - 4 = {\left(t + 4\right)} {\left(t - 2\right)} {y'''} + {\left(t - 5\right)} y - 5 \, {y''} \hspace{2em} x( 1 )= 4" title="-t^{2} - 4 = {\left(t + 4\right)} {\left(t - 2\right)} {y'''} + {\left(t - 5\right)} y - 5 \, {y''} \hspace{2em} x( 1 )= 4" data-latex="-t^{2} - 4 = {\left(t + 4\right)} {\left(t - 2\right)} {y'''} + {\left(t - 5\right)} y - 5 \, {y''} \hspace{2em} x( 1 )= 4"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,2)" alt="(-4,2)" title="(-4,2)" data-latex="(-4,2)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,2)" alt="(-4,2)" title="(-4,2)" data-latex="(-4,2)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-6696" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 6696"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 6\right)} {\left(t + 1\right)} - {\left(t - 5\right)} {y'''} + 4 \, {y''} = {\left(t^{2} + 4\right)} y \hspace{2em} x( 3 )= 9" alt="-{\left(t + 6\right)} {\left(t + 1\right)} - {\left(t - 5\right)} {y'''} + 4 \, {y''} = {\left(t^{2} + 4\right)} y \hspace{2em} x( 3 )= 9" title="-{\left(t + 6\right)} {\left(t + 1\right)} - {\left(t - 5\right)} {y'''} + 4 \, {y''} = {\left(t^{2} + 4\right)} y \hspace{2em} x( 3 )= 9" data-latex="-{\left(t + 6\right)} {\left(t + 1\right)} - {\left(t - 5\right)} {y'''} + 4 \, {y''} = {\left(t^{2} + 4\right)} y \hspace{2em} x( 3 )= 9"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20+%201%5Cright)%7D%20-%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20+%204%20%5C,%20%7By''%7D%20=%20%7B%5Cleft(t%5E%7B2%7D%20+%204%5Cright)%7D%20y%20%5Chspace%7B2em%7D%20x(%203%20)=%209" alt="-{\left(t + 6\right)} {\left(t + 1\right)} - {\left(t - 5\right)} {y'''} + 4 \, {y''} = {\left(t^{2} + 4\right)} y \hspace{2em} x( 3 )= 9" title="-{\left(t + 6\right)} {\left(t + 1\right)} - {\left(t - 5\right)} {y'''} + 4 \, {y''} = {\left(t^{2} + 4\right)} y \hspace{2em} x( 3 )= 9" data-latex="-{\left(t + 6\right)} {\left(t + 1\right)} - {\left(t - 5\right)} {y'''} + 4 \, {y''} = {\left(t^{2} + 4\right)} y \hspace{2em} x( 3 )= 9"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,5)" alt="(-\infty,5)" title="(-\infty,5)" data-latex="(-\infty,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,5)" alt="(-\infty,5)" title="(-\infty,5)" data-latex="(-\infty,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-4064" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 4064"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t - 5\right)} {y'''} e^{t} + 3 \, {y''} = -{\left(t + 4\right)} {\left(t + 1\right)} y - t^{2} - 4 \hspace{2em} x( 9 )= 3" alt="{\left(t - 5\right)} {y'''} e^{t} + 3 \, {y''} = -{\left(t + 4\right)} {\left(t + 1\right)} y - t^{2} - 4 \hspace{2em} x( 9 )= 3" title="{\left(t - 5\right)} {y'''} e^{t} + 3 \, {y''} = -{\left(t + 4\right)} {\left(t + 1\right)} y - t^{2} - 4 \hspace{2em} x( 9 )= 3" data-latex="{\left(t - 5\right)} {y'''} e^{t} + 3 \, {y''} = -{\left(t + 4\right)} {\left(t + 1\right)} y - t^{2} - 4 \hspace{2em} x( 9 )= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20+%203%20%5C,%20%7By''%7D%20=%20-%7B%5Cleft(t%20+%204%5Cright)%7D%20%7B%5Cleft(t%20+%201%5Cright)%7D%20y%20-%20t%5E%7B2%7D%20-%204%20%5Chspace%7B2em%7D%20x(%209%20)=%203" alt="{\left(t - 5\right)} {y'''} e^{t} + 3 \, {y''} = -{\left(t + 4\right)} {\left(t + 1\right)} y - t^{2} - 4 \hspace{2em} x( 9 )= 3" title="{\left(t - 5\right)} {y'''} e^{t} + 3 \, {y''} = -{\left(t + 4\right)} {\left(t + 1\right)} y - t^{2} - 4 \hspace{2em} x( 9 )= 3" data-latex="{\left(t - 5\right)} {y'''} e^{t} + 3 \, {y''} = -{\left(t + 4\right)} {\left(t + 1\right)} y - t^{2} - 4 \hspace{2em} x( 9 )= 3"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+\infty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+%5Cinfty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5720" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5720"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 6\right)} {\left(t - 1\right)} - {\left(t - 6\right)} {y''} - {y'} = y e^{t} \hspace{2em} x( 4 )= 8" alt="-{\left(t + 6\right)} {\left(t - 1\right)} - {\left(t - 6\right)} {y''} - {y'} = y e^{t} \hspace{2em} x( 4 )= 8" title="-{\left(t + 6\right)} {\left(t - 1\right)} - {\left(t - 6\right)} {y''} - {y'} = y e^{t} \hspace{2em} x( 4 )= 8" data-latex="-{\left(t + 6\right)} {\left(t - 1\right)} - {\left(t - 6\right)} {y''} - {y'} = y e^{t} \hspace{2em} x( 4 )= 8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%201%5Cright)%7D%20-%20%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By''%7D%20-%20%7By'%7D%20=%20y%20e%5E%7Bt%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%208" alt="-{\left(t + 6\right)} {\left(t - 1\right)} - {\left(t - 6\right)} {y''} - {y'} = y e^{t} \hspace{2em} x( 4 )= 8" title="-{\left(t + 6\right)} {\left(t - 1\right)} - {\left(t - 6\right)} {y''} - {y'} = y e^{t} \hspace{2em} x( 4 )= 8" data-latex="-{\left(t + 6\right)} {\left(t - 1\right)} - {\left(t - 6\right)} {y''} - {y'} = y e^{t} \hspace{2em} x( 4 )= 8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,6)" alt="(-\infty,6)" title="(-\infty,6)" data-latex="(-\infty,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,6)" alt="(-\infty,6)" title="(-\infty,6)" data-latex="(-\infty,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-7219" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 7219"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t - 6\right)} t - {\left(t^{2} + 1\right)} y = {\left(t + 5\right)} {y''} e^{t} - 4 \, {y'} \hspace{2em} x( -9 )= -1" alt="-{\left(t - 6\right)} t - {\left(t^{2} + 1\right)} y = {\left(t + 5\right)} {y''} e^{t} - 4 \, {y'} \hspace{2em} x( -9 )= -1" title="-{\left(t - 6\right)} t - {\left(t^{2} + 1\right)} y = {\left(t + 5\right)} {y''} e^{t} - 4 \, {y'} \hspace{2em} x( -9 )= -1" data-latex="-{\left(t - 6\right)} t - {\left(t^{2} + 1\right)} y = {\left(t + 5\right)} {y''} e^{t} - 4 \, {y'} \hspace{2em} x( -9 )= -1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20-%206%5Cright)%7D%20t%20-%20%7B%5Cleft(t%5E%7B2%7D%20+%201%5Cright)%7D%20y%20=%20%7B%5Cleft(t%20+%205%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20-%204%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20x(%20-9%20)=%20-1" alt="-{\left(t - 6\right)} t - {\left(t^{2} + 1\right)} y = {\left(t + 5\right)} {y''} e^{t} - 4 \, {y'} \hspace{2em} x( -9 )= -1" title="-{\left(t - 6\right)} t - {\left(t^{2} + 1\right)} y = {\left(t + 5\right)} {y''} e^{t} - 4 \, {y'} \hspace{2em} x( -9 )= -1" data-latex="-{\left(t - 6\right)} t - {\left(t^{2} + 1\right)} y = {\left(t + 5\right)} {y''} e^{t} - 4 \, {y'} \hspace{2em} x( -9 )= -1"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,-5)" alt="(-\infty,-5)" title="(-\infty,-5)" data-latex="(-\infty,-5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,-5)" alt="(-\infty,-5)" title="(-\infty,-5)" data-latex="(-\infty,-5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-7591" title="X2 | Existence/uniqueness theorem for linear IVPs | 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>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t - 2\right)} y - 2 \, {y''} = {\left(t + 6\right)} {\left(t - 6\right)} {y'''} + t^{2} + 1 \hspace{2em} x( -3 )= -10" alt="-{\left(t - 2\right)} y - 2 \, {y''} = {\left(t + 6\right)} {\left(t - 6\right)} {y'''} + t^{2} + 1 \hspace{2em} x( -3 )= -10" title="-{\left(t - 2\right)} y - 2 \, {y''} = {\left(t + 6\right)} {\left(t - 6\right)} {y'''} + t^{2} + 1 \hspace{2em} x( -3 )= -10" data-latex="-{\left(t - 2\right)} y - 2 \, {y''} = {\left(t + 6\right)} {\left(t - 6\right)} {y'''} + t^{2} + 1 \hspace{2em} x( -3 )= -10"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20-%202%5Cright)%7D%20y%20-%202%20%5C,%20%7By''%7D%20=%20%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By'''%7D%20+%20t%5E%7B2%7D%20+%201%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-10" alt="-{\left(t - 2\right)} y - 2 \, {y''} = {\left(t + 6\right)} {\left(t - 6\right)} {y'''} + t^{2} + 1 \hspace{2em} x( -3 )= -10" title="-{\left(t - 2\right)} y - 2 \, {y''} = {\left(t + 6\right)} {\left(t - 6\right)} {y'''} + t^{2} + 1 \hspace{2em} x( -3 )= -10" data-latex="-{\left(t - 2\right)} y - 2 \, {y''} = {\left(t + 6\right)} {\left(t - 6\right)} {y'''} + t^{2} + 1 \hspace{2em} x( -3 )= -10"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,6)" alt="(-6,6)" title="(-6,6)" data-latex="(-6,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,6)" alt="(-6,6)" title="(-6,6)" data-latex="(-6,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8160" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8160"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t - 1\right)} {\left(t - 6\right)} y - 2 \, {y'} = {\left(t + 6\right)} {y''} e^{t} + t^{2} + 4 \hspace{2em} x( -5 )= -10" alt="-{\left(t - 1\right)} {\left(t - 6\right)} y - 2 \, {y'} = {\left(t + 6\right)} {y''} e^{t} + t^{2} + 4 \hspace{2em} x( -5 )= -10" title="-{\left(t - 1\right)} {\left(t - 6\right)} y - 2 \, {y'} = {\left(t + 6\right)} {y''} e^{t} + t^{2} + 4 \hspace{2em} x( -5 )= -10" data-latex="-{\left(t - 1\right)} {\left(t - 6\right)} y - 2 \, {y'} = {\left(t + 6\right)} {y''} e^{t} + t^{2} + 4 \hspace{2em} x( -5 )= -10"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20-%201%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20y%20-%202%20%5C,%20%7By'%7D%20=%20%7B%5Cleft(t%20+%206%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20+%20t%5E%7B2%7D%20+%204%20%5Chspace%7B2em%7D%20x(%20-5%20)=%20-10" alt="-{\left(t - 1\right)} {\left(t - 6\right)} y - 2 \, {y'} = {\left(t + 6\right)} {y''} e^{t} + t^{2} + 4 \hspace{2em} x( -5 )= -10" title="-{\left(t - 1\right)} {\left(t - 6\right)} y - 2 \, {y'} = {\left(t + 6\right)} {y''} e^{t} + t^{2} + 4 \hspace{2em} x( -5 )= -10" data-latex="-{\left(t - 1\right)} {\left(t - 6\right)} y - 2 \, {y'} = {\left(t + 6\right)} {y''} e^{t} + t^{2} + 4 \hspace{2em} x( -5 )= -10"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,+\infty)" alt="(-6,+\infty)" title="(-6,+\infty)" data-latex="(-6,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,+%5Cinfty)" alt="(-6,+\infty)" title="(-6,+\infty)" data-latex="(-6,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5197" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5197"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t - 1\right)} e^{t} - 4 \, {y''} = -{\left(t + 6\right)} {\left(t - 5\right)} {y'''} - y e^{\left(-3 \, t\right)} \hspace{2em} x( -5 )= 9" alt="{\left(t - 1\right)} e^{t} - 4 \, {y''} = -{\left(t + 6\right)} {\left(t - 5\right)} {y'''} - y e^{\left(-3 \, t\right)} \hspace{2em} x( -5 )= 9" title="{\left(t - 1\right)} e^{t} - 4 \, {y''} = -{\left(t + 6\right)} {\left(t - 5\right)} {y'''} - y e^{\left(-3 \, t\right)} \hspace{2em} x( -5 )= 9" data-latex="{\left(t - 1\right)} e^{t} - 4 \, {y''} = -{\left(t + 6\right)} {\left(t - 5\right)} {y'''} - y e^{\left(-3 \, t\right)} \hspace{2em} x( -5 )= 9"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20-%201%5Cright)%7D%20e%5E%7Bt%7D%20-%204%20%5C,%20%7By''%7D%20=%20-%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20-%20y%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%209" alt="{\left(t - 1\right)} e^{t} - 4 \, {y''} = -{\left(t + 6\right)} {\left(t - 5\right)} {y'''} - y e^{\left(-3 \, t\right)} \hspace{2em} x( -5 )= 9" title="{\left(t - 1\right)} e^{t} - 4 \, {y''} = -{\left(t + 6\right)} {\left(t - 5\right)} {y'''} - y e^{\left(-3 \, t\right)} \hspace{2em} x( -5 )= 9" data-latex="{\left(t - 1\right)} e^{t} - 4 \, {y''} = -{\left(t + 6\right)} {\left(t - 5\right)} {y'''} - y e^{\left(-3 \, t\right)} \hspace{2em} x( -5 )= 9"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,5)" alt="(-6,5)" title="(-6,5)" data-latex="(-6,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,5)" alt="(-6,5)" title="(-6,5)" data-latex="(-6,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-5034" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 5034"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = {\left(t + 5\right)} {\left(t + 2\right)} y + {\left(t - 5\right)} {y''} e^{t} + 4 \, {y'} + e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 8" alt="0 = {\left(t + 5\right)} {\left(t + 2\right)} y + {\left(t - 5\right)} {y''} e^{t} + 4 \, {y'} + e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 8" title="0 = {\left(t + 5\right)} {\left(t + 2\right)} y + {\left(t - 5\right)} {y''} e^{t} + 4 \, {y'} + e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 8" data-latex="0 = {\left(t + 5\right)} {\left(t + 2\right)} y + {\left(t - 5\right)} {y''} e^{t} + 4 \, {y'} + e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20+%202%5Cright)%7D%20y%20+%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20+%204%20%5C,%20%7By'%7D%20+%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%208" alt="0 = {\left(t + 5\right)} {\left(t + 2\right)} y + {\left(t - 5\right)} {y''} e^{t} + 4 \, {y'} + e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 8" title="0 = {\left(t + 5\right)} {\left(t + 2\right)} y + {\left(t - 5\right)} {y''} e^{t} + 4 \, {y'} + e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 8" data-latex="0 = {\left(t + 5\right)} {\left(t + 2\right)} y + {\left(t - 5\right)} {y''} e^{t} + 4 \, {y'} + e^{\left(-5 \, t\right)} \hspace{2em} x( 4 )= 8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,5)" alt="(-\infty,5)" title="(-\infty,5)" data-latex="(-\infty,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,5)" alt="(-\infty,5)" title="(-\infty,5)" data-latex="(-\infty,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-7572" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 7572"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t - 5\right)} e^{t} = {\left(t + 6\right)} {\left(t - 2\right)} {y'''} + {\left(t^{2} + 16\right)} y - 4 \, {y''} \hspace{2em} x( -5 )= 3" alt="-{\left(t - 5\right)} e^{t} = {\left(t + 6\right)} {\left(t - 2\right)} {y'''} + {\left(t^{2} + 16\right)} y - 4 \, {y''} \hspace{2em} x( -5 )= 3" title="-{\left(t - 5\right)} e^{t} = {\left(t + 6\right)} {\left(t - 2\right)} {y'''} + {\left(t^{2} + 16\right)} y - 4 \, {y''} \hspace{2em} x( -5 )= 3" data-latex="-{\left(t - 5\right)} e^{t} = {\left(t + 6\right)} {\left(t - 2\right)} {y'''} + {\left(t^{2} + 16\right)} y - 4 \, {y''} \hspace{2em} x( -5 )= 3"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20-%205%5Cright)%7D%20e%5E%7Bt%7D%20=%20%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%202%5Cright)%7D%20%7By'''%7D%20+%20%7B%5Cleft(t%5E%7B2%7D%20+%2016%5Cright)%7D%20y%20-%204%20%5C,%20%7By''%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%203" alt="-{\left(t - 5\right)} e^{t} = {\left(t + 6\right)} {\left(t - 2\right)} {y'''} + {\left(t^{2} + 16\right)} y - 4 \, {y''} \hspace{2em} x( -5 )= 3" title="-{\left(t - 5\right)} e^{t} = {\left(t + 6\right)} {\left(t - 2\right)} {y'''} + {\left(t^{2} + 16\right)} y - 4 \, {y''} \hspace{2em} x( -5 )= 3" data-latex="-{\left(t - 5\right)} e^{t} = {\left(t + 6\right)} {\left(t - 2\right)} {y'''} + {\left(t^{2} + 16\right)} y - 4 \, {y''} \hspace{2em} x( -5 )= 3"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,2)" alt="(-6,2)" title="(-6,2)" data-latex="(-6,2)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,2)" alt="(-6,2)" title="(-6,2)" data-latex="(-6,2)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8415" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8415"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 6\right)} {\left(t - 6\right)} {y'''} - {\left(t^{2} + 1\right)} y - t = -2 \, {y''} \hspace{2em} x( 4 )= 7" alt="-{\left(t + 6\right)} {\left(t - 6\right)} {y'''} - {\left(t^{2} + 1\right)} y - t = -2 \, {y''} \hspace{2em} x( 4 )= 7" title="-{\left(t + 6\right)} {\left(t - 6\right)} {y'''} - {\left(t^{2} + 1\right)} y - t = -2 \, {y''} \hspace{2em} x( 4 )= 7" data-latex="-{\left(t + 6\right)} {\left(t - 6\right)} {y'''} - {\left(t^{2} + 1\right)} y - t = -2 \, {y''} \hspace{2em} x( 4 )= 7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By'''%7D%20-%20%7B%5Cleft(t%5E%7B2%7D%20+%201%5Cright)%7D%20y%20-%20t%20=%20-2%20%5C,%20%7By''%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%207" alt="-{\left(t + 6\right)} {\left(t - 6\right)} {y'''} - {\left(t^{2} + 1\right)} y - t = -2 \, {y''} \hspace{2em} x( 4 )= 7" title="-{\left(t + 6\right)} {\left(t - 6\right)} {y'''} - {\left(t^{2} + 1\right)} y - t = -2 \, {y''} \hspace{2em} x( 4 )= 7" data-latex="-{\left(t + 6\right)} {\left(t - 6\right)} {y'''} - {\left(t^{2} + 1\right)} y - t = -2 \, {y''} \hspace{2em} x( 4 )= 7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,6)" alt="(-6,6)" title="(-6,6)" data-latex="(-6,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,6)" alt="(-6,6)" title="(-6,6)" data-latex="(-6,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-0628" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 0628"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 5\right)} {\left(t - 5\right)} + {\left(t^{2} + 1\right)} y + {\left(t - 2\right)} {y''} + 5 \, {y'} = 0 \hspace{2em} x( 1 )= 5" alt="{\left(t + 5\right)} {\left(t - 5\right)} + {\left(t^{2} + 1\right)} y + {\left(t - 2\right)} {y''} + 5 \, {y'} = 0 \hspace{2em} x( 1 )= 5" title="{\left(t + 5\right)} {\left(t - 5\right)} + {\left(t^{2} + 1\right)} y + {\left(t - 2\right)} {y''} + 5 \, {y'} = 0 \hspace{2em} x( 1 )= 5" data-latex="{\left(t + 5\right)} {\left(t - 5\right)} + {\left(t^{2} + 1\right)} y + {\left(t - 2\right)} {y''} + 5 \, {y'} = 0 \hspace{2em} x( 1 )= 5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20+%20%7B%5Cleft(t%5E%7B2%7D%20+%201%5Cright)%7D%20y%20+%20%7B%5Cleft(t%20-%202%5Cright)%7D%20%7By''%7D%20+%205%20%5C,%20%7By'%7D%20=%200%20%5Chspace%7B2em%7D%20x(%201%20)=%205" alt="{\left(t + 5\right)} {\left(t - 5\right)} + {\left(t^{2} + 1\right)} y + {\left(t - 2\right)} {y''} + 5 \, {y'} = 0 \hspace{2em} x( 1 )= 5" title="{\left(t + 5\right)} {\left(t - 5\right)} + {\left(t^{2} + 1\right)} y + {\left(t - 2\right)} {y''} + 5 \, {y'} = 0 \hspace{2em} x( 1 )= 5" data-latex="{\left(t + 5\right)} {\left(t - 5\right)} + {\left(t^{2} + 1\right)} y + {\left(t - 2\right)} {y''} + 5 \, {y'} = 0 \hspace{2em} x( 1 )= 5"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,2)" alt="(-\infty,2)" title="(-\infty,2)" data-latex="(-\infty,2)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,2)" alt="(-\infty,2)" title="(-\infty,2)" data-latex="(-\infty,2)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8499" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8499"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 2\right)} {\left(t - 5\right)} + {\left(t + 4\right)} {y'''} + {y''} = -{\left(t^{2} + 4\right)} y \hspace{2em} x( -3 )= -6" alt="{\left(t + 2\right)} {\left(t - 5\right)} + {\left(t + 4\right)} {y'''} + {y''} = -{\left(t^{2} + 4\right)} y \hspace{2em} x( -3 )= -6" title="{\left(t + 2\right)} {\left(t - 5\right)} + {\left(t + 4\right)} {y'''} + {y''} = -{\left(t^{2} + 4\right)} y \hspace{2em} x( -3 )= -6" data-latex="{\left(t + 2\right)} {\left(t - 5\right)} + {\left(t + 4\right)} {y'''} + {y''} = -{\left(t^{2} + 4\right)} y \hspace{2em} x( -3 )= -6"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%202%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20+%20%7B%5Cleft(t%20+%204%5Cright)%7D%20%7By'''%7D%20+%20%7By''%7D%20=%20-%7B%5Cleft(t%5E%7B2%7D%20+%204%5Cright)%7D%20y%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-6" alt="{\left(t + 2\right)} {\left(t - 5\right)} + {\left(t + 4\right)} {y'''} + {y''} = -{\left(t^{2} + 4\right)} y \hspace{2em} x( -3 )= -6" title="{\left(t + 2\right)} {\left(t - 5\right)} + {\left(t + 4\right)} {y'''} + {y''} = -{\left(t^{2} + 4\right)} y \hspace{2em} x( -3 )= -6" data-latex="{\left(t + 2\right)} {\left(t - 5\right)} + {\left(t + 4\right)} {y'''} + {y''} = -{\left(t^{2} + 4\right)} y \hspace{2em} x( -3 )= -6"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,+\infty)" alt="(-4,+\infty)" title="(-4,+\infty)" data-latex="(-4,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-4,+%5Cinfty)" alt="(-4,+\infty)" title="(-4,+\infty)" data-latex="(-4,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8097" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8097"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y e^{\left(5 \, t\right)} - 5 \, {y''} = {\left(t - 5\right)} {y'''} e^{t} + {\left(t + 4\right)} {\left(t + 2\right)} \hspace{2em} x( 4 )= 8" alt="-y e^{\left(5 \, t\right)} - 5 \, {y''} = {\left(t - 5\right)} {y'''} e^{t} + {\left(t + 4\right)} {\left(t + 2\right)} \hspace{2em} x( 4 )= 8" title="-y e^{\left(5 \, t\right)} - 5 \, {y''} = {\left(t - 5\right)} {y'''} e^{t} + {\left(t + 4\right)} {\left(t + 2\right)} \hspace{2em} x( 4 )= 8" data-latex="-y e^{\left(5 \, t\right)} - 5 \, {y''} = {\left(t - 5\right)} {y'''} e^{t} + {\left(t + 4\right)} {\left(t + 2\right)} \hspace{2em} x( 4 )= 8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-y%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D%20-%205%20%5C,%20%7By''%7D%20=%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By'''%7D%20e%5E%7Bt%7D%20+%20%7B%5Cleft(t%20+%204%5Cright)%7D%20%7B%5Cleft(t%20+%202%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%208" alt="-y e^{\left(5 \, t\right)} - 5 \, {y''} = {\left(t - 5\right)} {y'''} e^{t} + {\left(t + 4\right)} {\left(t + 2\right)} \hspace{2em} x( 4 )= 8" title="-y e^{\left(5 \, t\right)} - 5 \, {y''} = {\left(t - 5\right)} {y'''} e^{t} + {\left(t + 4\right)} {\left(t + 2\right)} \hspace{2em} x( 4 )= 8" data-latex="-y e^{\left(5 \, t\right)} - 5 \, {y''} = {\left(t - 5\right)} {y'''} e^{t} + {\left(t + 4\right)} {\left(t + 2\right)} \hspace{2em} x( 4 )= 8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,5)" alt="(-\infty,5)" title="(-\infty,5)" data-latex="(-\infty,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,5)" alt="(-\infty,5)" title="(-\infty,5)" data-latex="(-\infty,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8986" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8986"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 1\right)} y e^{t} - 5 \, {y'} = -{\left(t + 5\right)} {\left(t - 6\right)} {y''} - e^{\left(-t\right)} \hspace{2em} x( -2 )= 8" alt="{\left(t + 1\right)} y e^{t} - 5 \, {y'} = -{\left(t + 5\right)} {\left(t - 6\right)} {y''} - e^{\left(-t\right)} \hspace{2em} x( -2 )= 8" title="{\left(t + 1\right)} y e^{t} - 5 \, {y'} = -{\left(t + 5\right)} {\left(t - 6\right)} {y''} - e^{\left(-t\right)} \hspace{2em} x( -2 )= 8" data-latex="{\left(t + 1\right)} y e^{t} - 5 \, {y'} = -{\left(t + 5\right)} {\left(t - 6\right)} {y''} - e^{\left(-t\right)} \hspace{2em} x( -2 )= 8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%201%5Cright)%7D%20y%20e%5E%7Bt%7D%20-%205%20%5C,%20%7By'%7D%20=%20-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20%7By''%7D%20-%20e%5E%7B%5Cleft(-t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%208" alt="{\left(t + 1\right)} y e^{t} - 5 \, {y'} = -{\left(t + 5\right)} {\left(t - 6\right)} {y''} - e^{\left(-t\right)} \hspace{2em} x( -2 )= 8" title="{\left(t + 1\right)} y e^{t} - 5 \, {y'} = -{\left(t + 5\right)} {\left(t - 6\right)} {y''} - e^{\left(-t\right)} \hspace{2em} x( -2 )= 8" data-latex="{\left(t + 1\right)} y e^{t} - 5 \, {y'} = -{\left(t + 5\right)} {\left(t - 6\right)} {y''} - e^{\left(-t\right)} \hspace{2em} x( -2 )= 8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,6)" alt="(-5,6)" title="(-5,6)" data-latex="(-5,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,6)" alt="(-5,6)" title="(-5,6)" data-latex="(-5,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-7938" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 7938"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 6\right)} {\left(t - 5\right)} y + t {y'''} e^{t} - 5 \, {y''} = -e^{\left(2 \, t\right)} \hspace{2em} x( -4 )= 1" alt="{\left(t + 6\right)} {\left(t - 5\right)} y + t {y'''} e^{t} - 5 \, {y''} = -e^{\left(2 \, t\right)} \hspace{2em} x( -4 )= 1" title="{\left(t + 6\right)} {\left(t - 5\right)} y + t {y'''} e^{t} - 5 \, {y''} = -e^{\left(2 \, t\right)} \hspace{2em} x( -4 )= 1" data-latex="{\left(t + 6\right)} {\left(t - 5\right)} y + t {y'''} e^{t} - 5 \, {y''} = -e^{\left(2 \, t\right)} \hspace{2em} x( -4 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20y%20+%20t%20%7By'''%7D%20e%5E%7Bt%7D%20-%205%20%5C,%20%7By''%7D%20=%20-e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%201" alt="{\left(t + 6\right)} {\left(t - 5\right)} y + t {y'''} e^{t} - 5 \, {y''} = -e^{\left(2 \, t\right)} \hspace{2em} x( -4 )= 1" title="{\left(t + 6\right)} {\left(t - 5\right)} y + t {y'''} e^{t} - 5 \, {y''} = -e^{\left(2 \, t\right)} \hspace{2em} x( -4 )= 1" data-latex="{\left(t + 6\right)} {\left(t - 5\right)} y + t {y'''} e^{t} - 5 \, {y''} = -e^{\left(2 \, t\right)} \hspace{2em} x( -4 )= 1"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-\infty,0)" alt="(-\infty,0)" title="(-\infty,0)" data-latex="(-\infty,0)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-%5Cinfty,0)" alt="(-\infty,0)" title="(-\infty,0)" data-latex="(-\infty,0)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-0307" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 0307"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 5\right)} {y'''} + 5 \, {y''} = -{\left(t - 2\right)} {\left(t - 6\right)} y - t^{2} - 4 \hspace{2em} x( -4 )= -7" alt="{\left(t + 5\right)} {y'''} + 5 \, {y''} = -{\left(t - 2\right)} {\left(t - 6\right)} y - t^{2} - 4 \hspace{2em} x( -4 )= -7" title="{\left(t + 5\right)} {y'''} + 5 \, {y''} = -{\left(t - 2\right)} {\left(t - 6\right)} y - t^{2} - 4 \hspace{2em} x( -4 )= -7" data-latex="{\left(t + 5\right)} {y'''} + 5 \, {y''} = -{\left(t - 2\right)} {\left(t - 6\right)} y - t^{2} - 4 \hspace{2em} x( -4 )= -7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%205%5Cright)%7D%20%7By'''%7D%20+%205%20%5C,%20%7By''%7D%20=%20-%7B%5Cleft(t%20-%202%5Cright)%7D%20%7B%5Cleft(t%20-%206%5Cright)%7D%20y%20-%20t%5E%7B2%7D%20-%204%20%5Chspace%7B2em%7D%20x(%20-4%20)=%20-7" alt="{\left(t + 5\right)} {y'''} + 5 \, {y''} = -{\left(t - 2\right)} {\left(t - 6\right)} y - t^{2} - 4 \hspace{2em} x( -4 )= -7" title="{\left(t + 5\right)} {y'''} + 5 \, {y''} = -{\left(t - 2\right)} {\left(t - 6\right)} y - t^{2} - 4 \hspace{2em} x( -4 )= -7" data-latex="{\left(t + 5\right)} {y'''} + 5 \, {y''} = -{\left(t - 2\right)} {\left(t - 6\right)} y - t^{2} - 4 \hspace{2em} x( -4 )= -7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,+\infty)" alt="(-5,+\infty)" title="(-5,+\infty)" data-latex="(-5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,+%5Cinfty)" alt="(-5,+\infty)" title="(-5,+\infty)" data-latex="(-5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-8316" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 8316"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t^{2} + 16\right)} y - {y''} = {\left(t - 6\right)} t {y'''} + {\left(t + 4\right)} e^{t} \hspace{2em} x( 4 )= 8" alt="-{\left(t^{2} + 16\right)} y - {y''} = {\left(t - 6\right)} t {y'''} + {\left(t + 4\right)} e^{t} \hspace{2em} x( 4 )= 8" title="-{\left(t^{2} + 16\right)} y - {y''} = {\left(t - 6\right)} t {y'''} + {\left(t + 4\right)} e^{t} \hspace{2em} x( 4 )= 8" data-latex="-{\left(t^{2} + 16\right)} y - {y''} = {\left(t - 6\right)} t {y'''} + {\left(t + 4\right)} e^{t} \hspace{2em} x( 4 )= 8"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%5E%7B2%7D%20+%2016%5Cright)%7D%20y%20-%20%7By''%7D%20=%20%7B%5Cleft(t%20-%206%5Cright)%7D%20t%20%7By'''%7D%20+%20%7B%5Cleft(t%20+%204%5Cright)%7D%20e%5E%7Bt%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%208" alt="-{\left(t^{2} + 16\right)} y - {y''} = {\left(t - 6\right)} t {y'''} + {\left(t + 4\right)} e^{t} \hspace{2em} x( 4 )= 8" title="-{\left(t^{2} + 16\right)} y - {y''} = {\left(t - 6\right)} t {y'''} + {\left(t + 4\right)} e^{t} \hspace{2em} x( 4 )= 8" data-latex="-{\left(t^{2} + 16\right)} y - {y''} = {\left(t - 6\right)} t {y'''} + {\left(t + 4\right)} e^{t} \hspace{2em} x( 4 )= 8"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(0,6)" alt="(0,6)" title="(0,6)" data-latex="(0,6)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(0,6)" alt="(0,6)" title="(0,6)" data-latex="(0,6)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-3536" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 3536"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 2\right)} {\left(t - 5\right)} {y''} + {\left(t + 6\right)} e^{t} = -y e^{\left(-t\right)} - {y'} \hspace{2em} x( 1 )= -5" alt="{\left(t + 2\right)} {\left(t - 5\right)} {y''} + {\left(t + 6\right)} e^{t} = -y e^{\left(-t\right)} - {y'} \hspace{2em} x( 1 )= -5" title="{\left(t + 2\right)} {\left(t - 5\right)} {y''} + {\left(t + 6\right)} e^{t} = -y e^{\left(-t\right)} - {y'} \hspace{2em} x( 1 )= -5" data-latex="{\left(t + 2\right)} {\left(t - 5\right)} {y''} + {\left(t + 6\right)} e^{t} = -y e^{\left(-t\right)} - {y'} \hspace{2em} x( 1 )= -5"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%202%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By''%7D%20+%20%7B%5Cleft(t%20+%206%5Cright)%7D%20e%5E%7Bt%7D%20=%20-y%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%20%7By'%7D%20%5Chspace%7B2em%7D%20x(%201%20)=%20-5" alt="{\left(t + 2\right)} {\left(t - 5\right)} {y''} + {\left(t + 6\right)} e^{t} = -y e^{\left(-t\right)} - {y'} \hspace{2em} x( 1 )= -5" title="{\left(t + 2\right)} {\left(t - 5\right)} {y''} + {\left(t + 6\right)} e^{t} = -y e^{\left(-t\right)} - {y'} \hspace{2em} x( 1 )= -5" data-latex="{\left(t + 2\right)} {\left(t - 5\right)} {y''} + {\left(t + 6\right)} e^{t} = -y e^{\left(-t\right)} - {y'} \hspace{2em} x( 1 )= -5"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-2,5)" alt="(-2,5)" title="(-2,5)" data-latex="(-2,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-2,5)" alt="(-2,5)" title="(-2,5)" data-latex="(-2,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-4264" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 4264"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t + 5\right)} {\left(t - 1\right)} {y'''} - t - 5 \, {y''} + 5 = y e^{\left(2 \, t\right)} \hspace{2em} x( -2 )= 2" alt="-{\left(t + 5\right)} {\left(t - 1\right)} {y'''} - t - 5 \, {y''} + 5 = y e^{\left(2 \, t\right)} \hspace{2em} x( -2 )= 2" title="-{\left(t + 5\right)} {\left(t - 1\right)} {y'''} - t - 5 \, {y''} + 5 = y e^{\left(2 \, t\right)} \hspace{2em} x( -2 )= 2" data-latex="-{\left(t + 5\right)} {\left(t - 1\right)} {y'''} - t - 5 \, {y''} + 5 = y e^{\left(2 \, t\right)} \hspace{2em} x( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20+%205%5Cright)%7D%20%7B%5Cleft(t%20-%201%5Cright)%7D%20%7By'''%7D%20-%20t%20-%205%20%5C,%20%7By''%7D%20+%205%20=%20y%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%202" alt="-{\left(t + 5\right)} {\left(t - 1\right)} {y'''} - t - 5 \, {y''} + 5 = y e^{\left(2 \, t\right)} \hspace{2em} x( -2 )= 2" title="-{\left(t + 5\right)} {\left(t - 1\right)} {y'''} - t - 5 \, {y''} + 5 = y e^{\left(2 \, t\right)} \hspace{2em} x( -2 )= 2" data-latex="-{\left(t + 5\right)} {\left(t - 1\right)} {y'''} - t - 5 \, {y''} + 5 = y e^{\left(2 \, t\right)} \hspace{2em} x( -2 )= 2"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,1)" alt="(-5,1)" title="(-5,1)" data-latex="(-5,1)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-5,1)" alt="(-5,1)" title="(-5,1)" data-latex="(-5,1)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-3990" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 3990"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-{\left(t - 5\right)} {y''} e^{t} - 2 \, {y'} = {\left(t + 6\right)} {\left(t - 1\right)} y + t^{2} + 1 \hspace{2em} x( 6 )= 1" alt="-{\left(t - 5\right)} {y''} e^{t} - 2 \, {y'} = {\left(t + 6\right)} {\left(t - 1\right)} y + t^{2} + 1 \hspace{2em} x( 6 )= 1" title="-{\left(t - 5\right)} {y''} e^{t} - 2 \, {y'} = {\left(t + 6\right)} {\left(t - 1\right)} y + t^{2} + 1 \hspace{2em} x( 6 )= 1" data-latex="-{\left(t - 5\right)} {y''} e^{t} - 2 \, {y'} = {\left(t + 6\right)} {\left(t - 1\right)} y + t^{2} + 1 \hspace{2em} x( 6 )= 1"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20-%202%20%5C,%20%7By'%7D%20=%20%7B%5Cleft(t%20+%206%5Cright)%7D%20%7B%5Cleft(t%20-%201%5Cright)%7D%20y%20+%20t%5E%7B2%7D%20+%201%20%5Chspace%7B2em%7D%20x(%206%20)=%201" alt="-{\left(t - 5\right)} {y''} e^{t} - 2 \, {y'} = {\left(t + 6\right)} {\left(t - 1\right)} y + t^{2} + 1 \hspace{2em} x( 6 )= 1" title="-{\left(t - 5\right)} {y''} e^{t} - 2 \, {y'} = {\left(t + 6\right)} {\left(t - 1\right)} y + t^{2} + 1 \hspace{2em} x( 6 )= 1" data-latex="-{\left(t - 5\right)} {y''} e^{t} - 2 \, {y'} = {\left(t + 6\right)} {\left(t - 1\right)} y + t^{2} + 1 \hspace{2em} x( 6 )= 1"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+\infty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(5,+%5Cinfty)" alt="(5,+\infty)" title="(5,+\infty)" data-latex="(5,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-9515" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 9515"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2 \, {y'} = -{\left(t - 2\right)} {\left(t - 5\right)} {y''} - {\left(t^{2} + 16\right)} y - t - 6 \hspace{2em} x( 3 )= 7" alt="2 \, {y'} = -{\left(t - 2\right)} {\left(t - 5\right)} {y''} - {\left(t^{2} + 16\right)} y - t - 6 \hspace{2em} x( 3 )= 7" title="2 \, {y'} = -{\left(t - 2\right)} {\left(t - 5\right)} {y''} - {\left(t^{2} + 16\right)} y - t - 6 \hspace{2em} x( 3 )= 7" data-latex="2 \, {y'} = -{\left(t - 2\right)} {\left(t - 5\right)} {y''} - {\left(t^{2} + 16\right)} y - t - 6 \hspace{2em} x( 3 )= 7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?2%20%5C,%20%7By'%7D%20=%20-%7B%5Cleft(t%20-%202%5Cright)%7D%20%7B%5Cleft(t%20-%205%5Cright)%7D%20%7By''%7D%20-%20%7B%5Cleft(t%5E%7B2%7D%20+%2016%5Cright)%7D%20y%20-%20t%20-%206%20%5Chspace%7B2em%7D%20x(%203%20)=%207" alt="2 \, {y'} = -{\left(t - 2\right)} {\left(t - 5\right)} {y''} - {\left(t^{2} + 16\right)} y - t - 6 \hspace{2em} x( 3 )= 7" title="2 \, {y'} = -{\left(t - 2\right)} {\left(t - 5\right)} {y''} - {\left(t^{2} + 16\right)} y - t - 6 \hspace{2em} x( 3 )= 7" data-latex="2 \, {y'} = -{\left(t - 2\right)} {\left(t - 5\right)} {y''} - {\left(t^{2} + 16\right)} y - t - 6 \hspace{2em} x( 3 )= 7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(2,5)" alt="(2,5)" title="(2,5)" data-latex="(2,5)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(2,5)" alt="(2,5)" title="(2,5)" data-latex="(2,5)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="X2-9510" title="X2 | Existence/uniqueness theorem for linear IVPs | ver. 9510"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X2.</strong></p><p> Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{\left(t + 6\right)} {y''} e^{t} = -{\left(t - 6\right)} t - y e^{\left(-5 \, t\right)} - 3 \, {y'} \hspace{2em} x( -5 )= -7" alt="{\left(t + 6\right)} {y''} e^{t} = -{\left(t - 6\right)} t - y e^{\left(-5 \, t\right)} - 3 \, {y'} \hspace{2em} x( -5 )= -7" title="{\left(t + 6\right)} {y''} e^{t} = -{\left(t - 6\right)} t - y e^{\left(-5 \, t\right)} - 3 \, {y'} \hspace{2em} x( -5 )= -7" data-latex="{\left(t + 6\right)} {y''} e^{t} = -{\left(t - 6\right)} t - y e^{\left(-5 \, t\right)} - 3 \, {y'} \hspace{2em} x( -5 )= -7"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X2.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7B%5Cleft(t%20+%206%5Cright)%7D%20%7By''%7D%20e%5E%7Bt%7D%20=%20-%7B%5Cleft(t%20-%206%5Cright)%7D%20t%20-%20y%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%203%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%20-7" alt="{\left(t + 6\right)} {y''} e^{t} = -{\left(t - 6\right)} t - y e^{\left(-5 \, t\right)} - 3 \, {y'} \hspace{2em} x( -5 )= -7" title="{\left(t + 6\right)} {y''} e^{t} = -{\left(t - 6\right)} t - y e^{\left(-5 \, t\right)} - 3 \, {y'} \hspace{2em} x( -5 )= -7" data-latex="{\left(t + 6\right)} {y''} e^{t} = -{\left(t - 6\right)} t - y e^{\left(-5 \, t\right)} - 3 \, {y'} \hspace{2em} x( -5 )= -7"&gt;
  &lt;/p&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><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,+\infty)" alt="(-6,+\infty)" title="(-6,+\infty)" data-latex="(-6,+\infty)"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(-6,+%5Cinfty)" alt="(-6,+\infty)" title="(-6,+\infty)" data-latex="(-6,+\infty)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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