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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">
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<qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- X3</fieldentry></qtimetadatafield>
</qtimetadata>
<item ident="X3-3676" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3676"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}} \hspace{2em} x( -6 )= -5" alt="y'= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}} \hspace{2em} x( -6 )= -5" title="y'= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}} \hspace{2em} x( -6 )= -5" data-latex="y'= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}} \hspace{2em} x( -6 )= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%204%20%5C,%20t%20+%2039%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-6%20)=%20-5" alt="y'= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}} \hspace{2em} x( -6 )= -5" title="y'= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}} \hspace{2em} x( -6 )= -5" data-latex="y'= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}} \hspace{2em} x( -6 )= -5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}}" alt="F(t,y)= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}}" title="F(t,y)= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{4}{{\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{2}{3}}}" alt="F_y= \frac{4}{{\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{2}{3}}}" title="F_y= \frac{4}{{\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{4}{{\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{2}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%204%20%5C,%20t%20+%2039%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F(t,y)= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}}" title="F(t,y)= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{1}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B4%7D%7B%7B%5Cleft(3%20%5C,%20%7By%7D%20+%204%20%5C,%20t%20+%2039%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D" alt="F_y= \frac{4}{{\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{2}{3}}}" title="F_y= \frac{4}{{\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{4}{{\left(3 \, {y} + 4 \, t + 39\right)}^{\frac{2}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-2079" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 2079"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= 3" alt="y'= 4 \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= 3" title="y'= 4 \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= 3" data-latex="y'= 4 \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%203" alt="y'= 4 \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= 3" title="y'= 4 \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= 3" data-latex="y'= 4 \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= 3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{7}{3}}" alt="F(t,y)= 4 \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{7}{3}}" title="F(t,y)= 4 \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 4 \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{7}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{56}{3} \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{4}{3}}" alt="F_y= \frac{56}{3} \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{4}{3}}" title="F_y= \frac{56}{3} \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{4}{3}}" data-latex="F_y= \frac{56}{3} \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{4}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D" alt="F(t,y)= 4 \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{7}{3}}" title="F(t,y)= 4 \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 4 \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{7}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B56%7D%7B3%7D%20%5C,%20%7B%5Cleft(3%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F_y= \frac{56}{3} \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{4}{3}}" title="F_y= \frac{56}{3} \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{4}{3}}" data-latex="F_y= \frac{56}{3} \, {\left(3 \, t + 2 \, {y} - 18\right)}^{\frac{4}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-6689" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 6689"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{8}{5}} \hspace{2em} x( -2 )= 3" alt="y'= 6 \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{8}{5}} \hspace{2em} x( -2 )= 3" title="y'= 6 \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{8}{5}} \hspace{2em} x( -2 )= 3" data-latex="y'= 6 \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{8}{5}} \hspace{2em} x( -2 )= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%203" alt="y'= 6 \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{8}{5}} \hspace{2em} x( -2 )= 3" title="y'= 6 \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{8}{5}} \hspace{2em} x( -2 )= 3" data-latex="y'= 6 \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{8}{5}} \hspace{2em} x( -2 )= 3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{8}{5}}" alt="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{8}{5}}" title="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{8}{5}}" data-latex="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{8}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{96}{5} \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{3}{5}}" alt="F_y= \frac{96}{5} \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{3}{5}}" title="F_y= \frac{96}{5} \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{3}{5}}" data-latex="F_y= \frac{96}{5} \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{3}{5}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B5%7D%7D" alt="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{8}{5}}" title="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{8}{5}}" data-latex="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{8}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B96%7D%7B5%7D%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B3%7D%7B5%7D%7D" alt="F_y= \frac{96}{5} \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{3}{5}}" title="F_y= \frac{96}{5} \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{3}{5}}" data-latex="F_y= \frac{96}{5} \, {\left(2 \, {y} - 3 \, t - 12\right)}^{\frac{3}{5}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9253" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9253"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}} \hspace{2em} x( 2 )= 3" alt="y'= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}} \hspace{2em} x( 2 )= 3" title="y'= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}} \hspace{2em} x( 2 )= 3" data-latex="y'= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}} \hspace{2em} x( 2 )= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%2017%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%202%20)=%203" alt="y'= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}} \hspace{2em} x( 2 )= 3" title="y'= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}} \hspace{2em} x( 2 )= 3" data-latex="y'= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}} \hspace{2em} x( 2 )= 3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}}" alt="F(t,y)= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}}" title="F(t,y)= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 40 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{5}{3}}" alt="F_y= 40 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{5}{3}}" title="F_y= 40 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{5}{3}}" data-latex="F_y= 40 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{5}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%2017%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}}" title="F(t,y)= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2040%20%5C,%20%7B%5Cleft(4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%2017%5Cright)%7D%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D" alt="F_y= 40 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{5}{3}}" title="F_y= 40 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{5}{3}}" data-latex="F_y= 40 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{5}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3300" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3300"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{4}{3}} \hspace{2em} x( -2 )= -3" alt="y'= 4 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{4}{3}} \hspace{2em} x( -2 )= -3" title="y'= 4 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{4}{3}} \hspace{2em} x( -2 )= -3" data-latex="y'= 4 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{4}{3}} \hspace{2em} x( -2 )= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%203%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-3" alt="y'= 4 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{4}{3}} \hspace{2em} x( -2 )= -3" title="y'= 4 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{4}{3}} \hspace{2em} x( -2 )= -3" data-latex="y'= 4 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{4}{3}} \hspace{2em} x( -2 )= -3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{4}{3}}" alt="F(t,y)= 4 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{4}{3}}" title="F(t,y)= 4 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{4}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 16 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{1}{3}}" alt="F_y= 16 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{1}{3}}" title="F_y= 16 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{1}{3}}" data-latex="F_y= 16 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{1}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%203%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F(t,y)= 4 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{4}{3}}" title="F(t,y)= 4 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{4}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2016%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%203%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F_y= 16 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{1}{3}}" title="F_y= 16 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{1}{3}}" data-latex="F_y= 16 \, {\left(3 \, {y} - 3 \, t + 3\right)}^{\frac{1}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9394" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9394"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= -2" alt="y'= 6 \, {\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= -2" title="y'= 6 \, {\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= -2" data-latex="y'= 6 \, {\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= -2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(5%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%2031%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%20-2" alt="y'= 6 \, {\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= -2" title="y'= 6 \, {\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= -2" data-latex="y'= 6 \, {\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= -2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{2}{3}}" alt="F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{2}{3}}" title="F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{2}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{12}{{\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{1}{3}}}" alt="F_y= \frac{12}{{\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{1}{3}}}" title="F_y= \frac{12}{{\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{12}{{\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{1}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(5%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%2031%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D" alt="F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{2}{3}}" title="F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{2}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B12%7D%7B%7B%5Cleft(5%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%2031%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D" alt="F_y= \frac{12}{{\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{1}{3}}}" title="F_y= \frac{12}{{\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{12}{{\left(5 \, t + 3 \, {y} + 31\right)}^{\frac{1}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-5429" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 5429"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -5" alt="y'= 6 \, {\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -5" title="y'= 6 \, {\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -5" data-latex="y'= 6 \, {\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%205%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%20-5" alt="y'= 6 \, {\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -5" title="y'= 6 \, {\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -5" data-latex="y'= 6 \, {\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{1}{3}}" alt="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{1}{3}}" title="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{1}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{4}{{\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{2}{3}}}" alt="F_y= \frac{4}{{\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{2}{3}}}" title="F_y= \frac{4}{{\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{4}{{\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{2}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%205%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{1}{3}}" title="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{1}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B4%7D%7B%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%205%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D" alt="F_y= \frac{4}{{\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{2}{3}}}" title="F_y= \frac{4}{{\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{4}{{\left(2 \, {y} - 3 \, t - 5\right)}^{\frac{2}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-8387" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 8387"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}} \hspace{2em} x( 3 )= -5" alt="y'= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}} \hspace{2em} x( 3 )= -5" title="y'= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}} \hspace{2em} x( 3 )= -5" data-latex="y'= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}} \hspace{2em} x( 3 )= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(3%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%203%20)=%20-5" alt="y'= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}} \hspace{2em} x( 3 )= -5" title="y'= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}} \hspace{2em} x( 3 )= -5" data-latex="y'= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}} \hspace{2em} x( 3 )= -5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}}" alt="F(t,y)= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}}" title="F(t,y)= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 48 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{5}{3}}" alt="F_y= 48 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{5}{3}}" title="F_y= 48 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{5}{3}}" data-latex="F_y= 48 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{5}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(3%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D" alt="F(t,y)= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}}" title="F(t,y)= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2048%20%5C,%20%7B%5Cleft(3%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D" alt="F_y= 48 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{5}{3}}" title="F_y= 48 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{5}{3}}" data-latex="F_y= 48 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{5}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-7896" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 7896"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}} \hspace{2em} x( -4 )= -2" alt="y'= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}} \hspace{2em} x( -4 )= -2" title="y'= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}} \hspace{2em} x( -4 )= -2" data-latex="y'= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}} \hspace{2em} x( -4 )= -2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(5%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%2026%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%20-2" alt="y'= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}} \hspace{2em} x( -4 )= -2" title="y'= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}} \hspace{2em} x( -4 )= -2" data-latex="y'= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}} \hspace{2em} x( -4 )= -2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}}" alt="F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}}" title="F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{72}{5 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{1}{5}}}" alt="F_y= \frac{72}{5 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{1}{5}}}" title="F_y= \frac{72}{5 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{72}{5 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{1}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(5%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%2026%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D" alt="F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}}" title="F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B72%7D%7B5%20%5C,%20%7B%5Cleft(5%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%2026%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%7D" alt="F_y= \frac{72}{5 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{1}{5}}}" title="F_y= \frac{72}{5 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{72}{5 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{1}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-6353" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 6353"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -4" alt="y'= 6 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -4" title="y'= 6 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -4" data-latex="y'= 6 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-4" alt="y'= 6 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -4" title="y'= 6 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -4" data-latex="y'= 6 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{3}}" alt="F(t,y)= 6 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{3}}" title="F(t,y)= 6 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 6 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{6}{{\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{2}{3}}}" alt="F_y= \frac{6}{{\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{2}{3}}}" title="F_y= \frac{6}{{\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{6}{{\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{2}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F(t,y)= 6 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{3}}" title="F(t,y)= 6 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 6 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B6%7D%7B%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D" alt="F_y= \frac{6}{{\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{2}{3}}}" title="F_y= \frac{6}{{\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{6}{{\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{2}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9588" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9588"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{4}{3}} \hspace{2em} x( -6 )= 3" alt="y'= 5 \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{4}{3}} \hspace{2em} x( -6 )= 3" title="y'= 5 \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{4}{3}} \hspace{2em} x( -6 )= 3" data-latex="y'= 5 \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{4}{3}} \hspace{2em} x( -6 )= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-6%20)=%203" alt="y'= 5 \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{4}{3}} \hspace{2em} x( -6 )= 3" title="y'= 5 \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{4}{3}} \hspace{2em} x( -6 )= 3" data-latex="y'= 5 \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{4}{3}} \hspace{2em} x( -6 )= 3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{4}{3}}" alt="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{4}{3}}" title="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{4}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{40}{3} \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{1}{3}}" alt="F_y= \frac{40}{3} \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{1}{3}}" title="F_y= \frac{40}{3} \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{1}{3}}" data-latex="F_y= \frac{40}{3} \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{1}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{4}{3}}" title="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{4}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B40%7D%7B3%7D%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F_y= \frac{40}{3} \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{1}{3}}" title="F_y= \frac{40}{3} \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{1}{3}}" data-latex="F_y= \frac{40}{3} \, {\left(2 \, {y} + 2 \, t + 6\right)}^{\frac{1}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-7273" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 7273"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}} \hspace{2em} x( 4 )= 3" alt="y'= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}} \hspace{2em} x( 4 )= 3" title="y'= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}} \hspace{2em} x( 4 )= 3" data-latex="y'= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}} \hspace{2em} x( 4 )= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%203" alt="y'= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}} \hspace{2em} x( 4 )= 3" title="y'= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}} \hspace{2em} x( 4 )= 3" data-latex="y'= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}} \hspace{2em} x( 4 )= 3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}}" alt="F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}}" title="F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{16}{3 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{1}{3}}}" alt="F_y= \frac{16}{3 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{1}{3}}}" title="F_y= \frac{16}{3 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{16}{3 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{1}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D" alt="F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}}" title="F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B16%7D%7B3%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D" alt="F_y= \frac{16}{3 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{1}{3}}}" title="F_y= \frac{16}{3 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{16}{3 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{1}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3684" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3684"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= -2" alt="y'= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= -2" title="y'= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= -2" data-latex="y'= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= -2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2028%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%20-2" alt="y'= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= -2" title="y'= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= -2" data-latex="y'= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}} \hspace{2em} x( 4 )= -2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}}" alt="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 14 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{4}{3}}" alt="F_y= 14 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{4}{3}}" title="F_y= 14 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{4}{3}}" data-latex="F_y= 14 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{4}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2028%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D" alt="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{7}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2014%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2028%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F_y= 14 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{4}{3}}" title="F_y= 14 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{4}{3}}" data-latex="F_y= 14 \, {\left(2 \, {y} - 6 \, t + 28\right)}^{\frac{4}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-2599" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 2599"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{4}{5}} \hspace{2em} x( 2 )= 4" alt="y'= 6 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{4}{5}} \hspace{2em} x( 2 )= 4" title="y'= 6 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{4}{5}} \hspace{2em} x( 2 )= 4" data-latex="y'= 6 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{4}{5}} \hspace{2em} x( 2 )= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(5%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%202%20)=%204" alt="y'= 6 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{4}{5}} \hspace{2em} x( 2 )= 4" title="y'= 6 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{4}{5}} \hspace{2em} x( 2 )= 4" data-latex="y'= 6 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{4}{5}} \hspace{2em} x( 2 )= 4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{4}{5}}" alt="F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{4}{5}}" title="F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{4}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{48}{5 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{1}{5}}}" alt="F_y= \frac{48}{5 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{1}{5}}}" title="F_y= \frac{48}{5 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{48}{5 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{1}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(5%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D" alt="F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{4}{5}}" title="F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{4}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B48%7D%7B5%20%5C,%20%7B%5Cleft(5%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%7D" alt="F_y= \frac{48}{5 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{1}{5}}}" title="F_y= \frac{48}{5 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{48}{5 \, {\left(5 \, t + 2 \, {y} - 18\right)}^{\frac{1}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-6631" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 6631"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{1}{5}} \hspace{2em} x( -2 )= 2" alt="y'= 2 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{1}{5}} \hspace{2em} x( -2 )= 2" title="y'= 2 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{1}{5}} \hspace{2em} x( -2 )= 2" data-latex="y'= 2 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{1}{5}} \hspace{2em} x( -2 )= 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%202%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%202" alt="y'= 2 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{1}{5}} \hspace{2em} x( -2 )= 2" title="y'= 2 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{1}{5}} \hspace{2em} x( -2 )= 2" data-latex="y'= 2 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{1}{5}} \hspace{2em} x( -2 )= 2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{1}{5}}" alt="F(t,y)= 2 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{1}{5}}" title="F(t,y)= 2 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 2 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{1}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{6}{5 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{4}{5}}}" alt="F_y= \frac{6}{5 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{4}{5}}}" title="F_y= \frac{6}{5 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{6}{5 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{4}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%202%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D" alt="F(t,y)= 2 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{1}{5}}" title="F(t,y)= 2 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 2 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{1}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B6%7D%7B5%20%5C,%20%7B%5Cleft(2%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%202%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%7D" alt="F_y= \frac{6}{5 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{4}{5}}}" title="F_y= \frac{6}{5 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{6}{5 \, {\left(2 \, t + 3 \, {y} - 2\right)}^{\frac{4}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-5502" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 5502"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 3" alt="y'= 2 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 3" title="y'= 2 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 3" data-latex="y'= 2 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%203" alt="y'= 2 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 3" title="y'= 2 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 3" data-latex="y'= 2 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{3}}" alt="F(t,y)= 2 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{3}}" title="F(t,y)= 2 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 2 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{16}{3} \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{3}}" alt="F_y= \frac{16}{3} \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{3}}" title="F_y= \frac{16}{3} \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{3}}" data-latex="F_y= \frac{16}{3} \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F(t,y)= 2 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{3}}" title="F(t,y)= 2 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 2 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B16%7D%7B3%7D%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F_y= \frac{16}{3} \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{3}}" title="F_y= \frac{16}{3} \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{3}}" data-latex="F_y= \frac{16}{3} \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-7906" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 7906"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{1}{3}} \hspace{2em} x( -2 )= -6" alt="y'= 4 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{1}{3}} \hspace{2em} x( -2 )= -6" title="y'= 4 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{1}{3}} \hspace{2em} x( -2 )= -6" data-latex="y'= 4 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{1}{3}} \hspace{2em} x( -2 )= -6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%206%20%5C,%20t%20+%2030%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-6" alt="y'= 4 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{1}{3}} \hspace{2em} x( -2 )= -6" title="y'= 4 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{1}{3}} \hspace{2em} x( -2 )= -6" data-latex="y'= 4 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{1}{3}} \hspace{2em} x( -2 )= -6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{1}{3}}" alt="F(t,y)= 4 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{1}{3}}" title="F(t,y)= 4 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{1}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{4}{{\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{2}{3}}}" alt="F_y= \frac{4}{{\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{2}{3}}}" title="F_y= \frac{4}{{\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{4}{{\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{2}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%206%20%5C,%20t%20+%2030%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F(t,y)= 4 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{1}{3}}" title="F(t,y)= 4 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{1}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B4%7D%7B%7B%5Cleft(3%20%5C,%20%7By%7D%20+%206%20%5C,%20t%20+%2030%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D" alt="F_y= \frac{4}{{\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{2}{3}}}" title="F_y= \frac{4}{{\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{4}{{\left(3 \, {y} + 6 \, t + 30\right)}^{\frac{2}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-1214" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 1214"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= 6" alt="y'= 5 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= 6" title="y'= 5 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= 6" data-latex="y'= 5 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= 6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20-%2024%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%203%20)=%206" alt="y'= 5 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= 6" title="y'= 5 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= 6" data-latex="y'= 5 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= 6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{4}{3}}" alt="F(t,y)= 5 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{4}{3}}" title="F(t,y)= 5 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{4}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 20 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{1}{3}}" alt="F_y= 20 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{1}{3}}" title="F_y= 20 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{1}{3}}" data-latex="F_y= 20 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{1}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20-%2024%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{4}{3}}" title="F(t,y)= 5 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{4}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2020%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20-%2024%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F_y= 20 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{1}{3}}" title="F_y= 20 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{1}{3}}" data-latex="F_y= 20 \, {\left(3 \, {y} + 2 \, t - 24\right)}^{\frac{1}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3204" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3204"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -3" alt="y'= 2 \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -3" title="y'= 2 \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -3" data-latex="y'= 2 \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%206%20)=%20-3" alt="y'= 2 \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -3" title="y'= 2 \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -3" data-latex="y'= 2 \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{4}{3}}" alt="F(t,y)= 2 \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{4}{3}}" title="F(t,y)= 2 \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 2 \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{4}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{16}{3} \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{1}{3}}" alt="F_y= \frac{16}{3} \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{1}{3}}" title="F_y= \frac{16}{3} \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{1}{3}}" data-latex="F_y= \frac{16}{3} \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{1}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F(t,y)= 2 \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{4}{3}}" title="F(t,y)= 2 \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 2 \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{4}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B16%7D%7B3%7D%20%5C,%20%7B%5Cleft(2%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F_y= \frac{16}{3} \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{1}{3}}" title="F_y= \frac{16}{3} \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{1}{3}}" data-latex="F_y= \frac{16}{3} \, {\left(2 \, t + 2 \, {y} - 6\right)}^{\frac{1}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-7197" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 7197"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{4}{5}} \hspace{2em} x( 2 )= -5" alt="y'= 3 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{4}{5}} \hspace{2em} x( 2 )= -5" title="y'= 3 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{4}{5}} \hspace{2em} x( 2 )= -5" data-latex="y'= 3 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{4}{5}} \hspace{2em} x( 2 )= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(-6%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%2022%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%202%20)=%20-5" alt="y'= 3 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{4}{5}} \hspace{2em} x( 2 )= -5" title="y'= 3 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{4}{5}} \hspace{2em} x( 2 )= -5" data-latex="y'= 3 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{4}{5}} \hspace{2em} x( 2 )= -5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{4}{5}}" alt="F(t,y)= 3 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{4}{5}}" title="F(t,y)= 3 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 3 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{4}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{24}{5 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{1}{5}}}" alt="F_y= \frac{24}{5 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{1}{5}}}" title="F_y= \frac{24}{5 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{24}{5 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{1}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(-6%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%2022%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D" alt="F(t,y)= 3 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{4}{5}}" title="F(t,y)= 3 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 3 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{4}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B24%7D%7B5%20%5C,%20%7B%5Cleft(-6%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%2022%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%7D" alt="F_y= \frac{24}{5 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{1}{5}}}" title="F_y= \frac{24}{5 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{24}{5 \, {\left(-6 \, t + 2 \, {y} + 22\right)}^{\frac{1}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9235" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9235"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{7}{3}} \hspace{2em} x( -4 )= 5" alt="y'= 3 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{7}{3}} \hspace{2em} x( -4 )= 5" title="y'= 3 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{7}{3}} \hspace{2em} x( -4 )= 5" data-latex="y'= 3 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{7}{3}} \hspace{2em} x( -4 )= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%205" alt="y'= 3 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{7}{3}} \hspace{2em} x( -4 )= 5" title="y'= 3 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{7}{3}} \hspace{2em} x( -4 )= 5" data-latex="y'= 3 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{7}{3}} \hspace{2em} x( -4 )= 5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{7}{3}}" alt="F(t,y)= 3 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{7}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{7}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 14 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{4}{3}}" alt="F_y= 14 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{4}{3}}" title="F_y= 14 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{4}{3}}" data-latex="F_y= 14 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{4}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D" alt="F(t,y)= 3 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{7}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{7}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2014%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F_y= 14 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{4}{3}}" title="F_y= 14 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{4}{3}}" data-latex="F_y= 14 \, {\left(2 \, {y} - 2 \, t - 18\right)}^{\frac{4}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-7136" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 7136"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{4}{3}} \hspace{2em} x( -2 )= 6" alt="y'= 5 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{4}{3}} \hspace{2em} x( -2 )= 6" title="y'= 5 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{4}{3}} \hspace{2em} x( -2 )= 6" data-latex="y'= 5 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{4}{3}} \hspace{2em} x( -2 )= 6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%206" alt="y'= 5 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{4}{3}} \hspace{2em} x( -2 )= 6" title="y'= 5 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{4}{3}} \hspace{2em} x( -2 )= 6" data-latex="y'= 5 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{4}{3}} \hspace{2em} x( -2 )= 6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{4}{3}}" alt="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{4}{3}}" title="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{4}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 20 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{1}{3}}" alt="F_y= 20 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{1}{3}}" title="F_y= 20 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{1}{3}}" data-latex="F_y= 20 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{1}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{4}{3}}" title="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{4}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2020%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F_y= 20 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{1}{3}}" title="F_y= 20 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{1}{3}}" data-latex="F_y= 20 \, {\left(3 \, {y} + 3 \, t - 12\right)}^{\frac{1}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-7277" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 7277"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= 2" alt="y'= 3 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= 2" title="y'= 3 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= 2" data-latex="y'= 3 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20+%2014%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%202" alt="y'= 3 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= 2" title="y'= 3 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= 2" data-latex="y'= 3 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= 2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{1}{5}}" alt="F(t,y)= 3 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{1}{5}}" title="F(t,y)= 3 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 3 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{1}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{9}{5 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{4}{5}}}" alt="F_y= \frac{9}{5 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{4}{5}}}" title="F_y= \frac{9}{5 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{9}{5 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{4}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20+%2014%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D" alt="F(t,y)= 3 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{1}{5}}" title="F(t,y)= 3 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 3 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{1}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B9%7D%7B5%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20+%2014%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%7D" alt="F_y= \frac{9}{5 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{4}{5}}}" title="F_y= \frac{9}{5 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{9}{5 \, {\left(3 \, {y} + 5 \, t + 14\right)}^{\frac{4}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-6684" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 6684"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{1}{3}} \hspace{2em} x( 5 )= -2" alt="y'= 5 \, {\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{1}{3}} \hspace{2em} x( 5 )= -2" title="y'= 5 \, {\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{1}{3}} \hspace{2em} x( 5 )= -2" data-latex="y'= 5 \, {\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{1}{3}} \hspace{2em} x( 5 )= -2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20-%2019%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%205%20)=%20-2" alt="y'= 5 \, {\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{1}{3}} \hspace{2em} x( 5 )= -2" title="y'= 5 \, {\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{1}{3}} \hspace{2em} x( 5 )= -2" data-latex="y'= 5 \, {\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{1}{3}} \hspace{2em} x( 5 )= -2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{1}{3}}" alt="F(t,y)= 5 \, {\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{1}{3}}" title="F(t,y)= 5 \, {\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{1}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{5}{{\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{2}{3}}}" alt="F_y= \frac{5}{{\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{2}{3}}}" title="F_y= \frac{5}{{\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{5}{{\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{2}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20-%2019%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{1}{3}}" title="F(t,y)= 5 \, {\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{1}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B5%7D%7B%7B%5Cleft(3%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20-%2019%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D" alt="F_y= \frac{5}{{\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{2}{3}}}" title="F_y= \frac{5}{{\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{5}{{\left(3 \, {y} + 5 \, t - 19\right)}^{\frac{2}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3850" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3850"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{8}{5}} \hspace{2em} x( 6 )= -4" alt="y'= 3 \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{8}{5}} \hspace{2em} x( 6 )= -4" title="y'= 3 \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{8}{5}} \hspace{2em} x( 6 )= -4" data-latex="y'= 3 \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{8}{5}} \hspace{2em} x( 6 )= -4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%2024%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%206%20)=%20-4" alt="y'= 3 \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{8}{5}} \hspace{2em} x( 6 )= -4" title="y'= 3 \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{8}{5}} \hspace{2em} x( 6 )= -4" data-latex="y'= 3 \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{8}{5}} \hspace{2em} x( 6 )= -4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{8}{5}}" alt="F(t,y)= 3 \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{8}{5}}" title="F(t,y)= 3 \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{8}{5}}" data-latex="F(t,y)= 3 \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{8}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{72}{5} \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{3}{5}}" alt="F_y= \frac{72}{5} \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{3}{5}}" title="F_y= \frac{72}{5} \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{3}{5}}" data-latex="F_y= \frac{72}{5} \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{3}{5}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%2024%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B5%7D%7D" alt="F(t,y)= 3 \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{8}{5}}" title="F(t,y)= 3 \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{8}{5}}" data-latex="F(t,y)= 3 \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{8}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B72%7D%7B5%7D%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%2024%5Cright)%7D%5E%7B%5Cfrac%7B3%7D%7B5%7D%7D" alt="F_y= \frac{72}{5} \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{3}{5}}" title="F_y= \frac{72}{5} \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{3}{5}}" data-latex="F_y= \frac{72}{5} \, {\left(3 \, {y} - 2 \, t + 24\right)}^{\frac{3}{5}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-2876" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 2876"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{7}{3}} \hspace{2em} x( -3 )= 3" alt="y'= 3 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{7}{3}} \hspace{2em} x( -3 )= 3" title="y'= 3 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{7}{3}} \hspace{2em} x( -3 )= 3" data-latex="y'= 3 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{7}{3}} \hspace{2em} x( -3 )= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20+%209%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%203" alt="y'= 3 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{7}{3}} \hspace{2em} x( -3 )= 3" title="y'= 3 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{7}{3}} \hspace{2em} x( -3 )= 3" data-latex="y'= 3 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{7}{3}} \hspace{2em} x( -3 )= 3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{7}{3}}" alt="F(t,y)= 3 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{7}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{7}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 14 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{4}{3}}" alt="F_y= 14 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{4}{3}}" title="F_y= 14 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{4}{3}}" data-latex="F_y= 14 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{4}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20+%209%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D" alt="F(t,y)= 3 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{7}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{7}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2014%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20+%209%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F_y= 14 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{4}{3}}" title="F_y= 14 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{4}{3}}" data-latex="F_y= 14 \, {\left(2 \, {y} + 5 \, t + 9\right)}^{\frac{4}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-0639" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 0639"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= 5" alt="y'= 3 \, {\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= 5" title="y'= 3 \, {\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= 5" data-latex="y'= 3 \, {\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%206%20%5C,%20t%20+%2020%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%205" alt="y'= 3 \, {\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= 5" title="y'= 3 \, {\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= 5" data-latex="y'= 3 \, {\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= 5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{2}{3}}" alt="F(t,y)= 3 \, {\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{2}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{2}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{4}{{\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{1}{3}}}" alt="F_y= \frac{4}{{\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{1}{3}}}" title="F_y= \frac{4}{{\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{4}{{\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{1}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%206%20%5C,%20t%20+%2020%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D" alt="F(t,y)= 3 \, {\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{2}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{2}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B4%7D%7B%7B%5Cleft(2%20%5C,%20%7By%7D%20+%206%20%5C,%20t%20+%2020%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D" alt="F_y= \frac{4}{{\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{1}{3}}}" title="F_y= \frac{4}{{\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{4}{{\left(2 \, {y} + 6 \, t + 20\right)}^{\frac{1}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9686" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9686"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= 3" alt="y'= 3 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= 3" title="y'= 3 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= 3" data-latex="y'= 3 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2012%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%203%20)=%203" alt="y'= 3 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= 3" title="y'= 3 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= 3" data-latex="y'= 3 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= 3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}}" alt="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 8 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{1}{3}}" alt="F_y= 8 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{1}{3}}" title="F_y= 8 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{1}{3}}" data-latex="F_y= 8 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{1}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2012%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%208%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2012%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F_y= 8 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{1}{3}}" title="F_y= 8 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{1}{3}}" data-latex="F_y= 8 \, {\left(2 \, {y} - 6 \, t + 12\right)}^{\frac{1}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-8970" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 8970"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= -6" alt="y'= 4 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= -6" title="y'= 4 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= -6" data-latex="y'= 4 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= -6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%2022%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-6" alt="y'= 4 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= -6" title="y'= 4 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= -6" data-latex="y'= 4 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= -6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{4}{5}}" alt="F(t,y)= 4 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{4}{5}}" title="F(t,y)= 4 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{4}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{48}{5 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{1}{5}}}" alt="F_y= \frac{48}{5 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{1}{5}}}" title="F_y= \frac{48}{5 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{48}{5 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{1}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%2022%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D" alt="F(t,y)= 4 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{4}{5}}" title="F(t,y)= 4 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{4}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B48%7D%7B5%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%2022%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%7D" alt="F_y= \frac{48}{5 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{1}{5}}}" title="F_y= \frac{48}{5 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{48}{5 \, {\left(3 \, {y} + 2 \, t + 22\right)}^{\frac{1}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-5552" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 5552"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{8}{3}} \hspace{2em} x( 4 )= -4" alt="y'= 4 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{8}{3}} \hspace{2em} x( 4 )= -4" title="y'= 4 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{8}{3}} \hspace{2em} x( 4 )= -4" data-latex="y'= 4 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{8}{3}} \hspace{2em} x( 4 )= -4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%2032%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%20-4" alt="y'= 4 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{8}{3}} \hspace{2em} x( 4 )= -4" title="y'= 4 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{8}{3}} \hspace{2em} x( 4 )= -4" data-latex="y'= 4 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{8}{3}} \hspace{2em} x( 4 )= -4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{8}{3}}" alt="F(t,y)= 4 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{8}{3}}" title="F(t,y)= 4 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{8}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 32 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{5}{3}}" alt="F_y= 32 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{5}{3}}" title="F_y= 32 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{5}{3}}" data-latex="F_y= 32 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{5}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%2032%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D" alt="F(t,y)= 4 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{8}{3}}" title="F(t,y)= 4 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{8}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2032%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%2032%5Cright)%7D%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D" alt="F_y= 32 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{5}{3}}" title="F_y= 32 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{5}{3}}" data-latex="F_y= 32 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{5}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-5168" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 5168"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{2}{3}} \hspace{2em} x( -4 )= 4" alt="y'= 5 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{2}{3}} \hspace{2em} x( -4 )= 4" title="y'= 5 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{2}{3}} \hspace{2em} x( -4 )= 4" data-latex="y'= 5 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{2}{3}} \hspace{2em} x( -4 )= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(-2%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2016%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%204" alt="y'= 5 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{2}{3}} \hspace{2em} x( -4 )= 4" title="y'= 5 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{2}{3}} \hspace{2em} x( -4 )= 4" data-latex="y'= 5 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{2}{3}} \hspace{2em} x( -4 )= 4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{2}{3}}" alt="F(t,y)= 5 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{2}{3}}" title="F(t,y)= 5 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 5 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{2}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{20}{3 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{1}{3}}}" alt="F_y= \frac{20}{3 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{1}{3}}}" title="F_y= \frac{20}{3 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{20}{3 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{1}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(-2%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2016%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{2}{3}}" title="F(t,y)= 5 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 5 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{2}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B20%7D%7B3%20%5C,%20%7B%5Cleft(-2%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2016%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D" alt="F_y= \frac{20}{3 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{1}{3}}}" title="F_y= \frac{20}{3 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{20}{3 \, {\left(-2 \, t + 2 \, {y} - 16\right)}^{\frac{1}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-8463" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 8463"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -5" alt="y'= 6 \, {\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -5" title="y'= 6 \, {\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -5" data-latex="y'= 6 \, {\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(-4%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%2030%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%205%20)=%20-5" alt="y'= 6 \, {\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -5" title="y'= 6 \, {\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -5" data-latex="y'= 6 \, {\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{2}{3}}" alt="F(t,y)= 6 \, {\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{2}{3}}" title="F(t,y)= 6 \, {\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 6 \, {\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{2}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{8}{{\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{1}{3}}}" alt="F_y= \frac{8}{{\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{1}{3}}}" title="F_y= \frac{8}{{\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{8}{{\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{1}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(-4%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%2030%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D" alt="F(t,y)= 6 \, {\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{2}{3}}" title="F(t,y)= 6 \, {\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 6 \, {\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{2}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B8%7D%7B%7B%5Cleft(-4%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%2030%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D" alt="F_y= \frac{8}{{\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{1}{3}}}" title="F_y= \frac{8}{{\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{8}{{\left(-4 \, t + 2 \, {y} + 30\right)}^{\frac{1}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-8648" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 8648"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{7}{3}} \hspace{2em} x( 3 )= -2" alt="y'= 6 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{7}{3}} \hspace{2em} x( 3 )= -2" title="y'= 6 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{7}{3}} \hspace{2em} x( 3 )= -2" data-latex="y'= 6 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{7}{3}} \hspace{2em} x( 3 )= -2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20-%2011%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%203%20)=%20-2" alt="y'= 6 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{7}{3}} \hspace{2em} x( 3 )= -2" title="y'= 6 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{7}{3}} \hspace{2em} x( 3 )= -2" data-latex="y'= 6 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{7}{3}} \hspace{2em} x( 3 )= -2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{7}{3}}" alt="F(t,y)= 6 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{7}{3}}" title="F(t,y)= 6 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 6 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{7}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 28 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{4}{3}}" alt="F_y= 28 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{4}{3}}" title="F_y= 28 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{4}{3}}" data-latex="F_y= 28 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{4}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20-%2011%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D" alt="F(t,y)= 6 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{7}{3}}" title="F(t,y)= 6 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 6 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{7}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2028%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20-%2011%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F_y= 28 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{4}{3}}" title="F_y= 28 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{4}{3}}" data-latex="F_y= 28 \, {\left(2 \, {y} + 5 \, t - 11\right)}^{\frac{4}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-4877" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 4877"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= 2" alt="y'= 2 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= 2" title="y'= 2 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= 2" data-latex="y'= 2 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%204%20%5C,%20t%20+%2012%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%202" alt="y'= 2 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= 2" title="y'= 2 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= 2" data-latex="y'= 2 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{1}{5}} \hspace{2em} x( -4 )= 2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{1}{5}}" alt="F(t,y)= 2 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{1}{5}}" title="F(t,y)= 2 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 2 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{1}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{4}{5 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{4}{5}}}" alt="F_y= \frac{4}{5 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{4}{5}}}" title="F_y= \frac{4}{5 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{4}{5 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{4}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%204%20%5C,%20t%20+%2012%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D" alt="F(t,y)= 2 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{1}{5}}" title="F(t,y)= 2 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 2 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{1}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B4%7D%7B5%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%204%20%5C,%20t%20+%2012%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%7D" alt="F_y= \frac{4}{5 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{4}{5}}}" title="F_y= \frac{4}{5 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{4}{5 \, {\left(2 \, {y} + 4 \, t + 12\right)}^{\frac{4}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-4207" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 4207"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}} \hspace{2em} x( 4 )= 5" alt="y'= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}} \hspace{2em} x( 4 )= 5" title="y'= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}} \hspace{2em} x( 4 )= 5" data-latex="y'= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}} \hspace{2em} x( 4 )= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20-%2027%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%205" alt="y'= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}} \hspace{2em} x( 4 )= 5" title="y'= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}} \hspace{2em} x( 4 )= 5" data-latex="y'= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}} \hspace{2em} x( 4 )= 5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}}" alt="F(t,y)= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}}" title="F(t,y)= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{18}{5 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{3}{5}}}" alt="F_y= \frac{18}{5 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{3}{5}}}" title="F_y= \frac{18}{5 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{18}{5 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{3}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20-%2027%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D" alt="F(t,y)= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}}" title="F(t,y)= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 3 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{2}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B18%7D%7B5%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20-%2027%5Cright)%7D%5E%7B%5Cfrac%7B3%7D%7B5%7D%7D%7D" alt="F_y= \frac{18}{5 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{3}{5}}}" title="F_y= \frac{18}{5 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{18}{5 \, {\left(3 \, {y} + 3 \, t - 27\right)}^{\frac{3}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-6272" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 6272"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{2}{5}} \hspace{2em} x( -3 )= 4" alt="y'= 2 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{2}{5}} \hspace{2em} x( -3 )= 4" title="y'= 2 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{2}{5}} \hspace{2em} x( -3 )= 4" data-latex="y'= 2 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{2}{5}} \hspace{2em} x( -3 )= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%204" alt="y'= 2 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{2}{5}} \hspace{2em} x( -3 )= 4" title="y'= 2 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{2}{5}} \hspace{2em} x( -3 )= 4" data-latex="y'= 2 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{2}{5}} \hspace{2em} x( -3 )= 4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{2}{5}}" alt="F(t,y)= 2 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{2}{5}}" title="F(t,y)= 2 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 2 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{2}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{12}{5 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{3}{5}}}" alt="F_y= \frac{12}{5 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{3}{5}}}" title="F_y= \frac{12}{5 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{12}{5 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{3}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D" alt="F(t,y)= 2 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{2}{5}}" title="F(t,y)= 2 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 2 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{2}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B12%7D%7B5%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B3%7D%7B5%7D%7D%7D" alt="F_y= \frac{12}{5 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{3}{5}}}" title="F_y= \frac{12}{5 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{12}{5 \, {\left(3 \, {y} - 2 \, t - 18\right)}^{\frac{3}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-0302" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 0302"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{2}{3}} \hspace{2em} x( 6 )= 6" alt="y'= 5 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{2}{3}} \hspace{2em} x( 6 )= 6" title="y'= 5 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{2}{3}} \hspace{2em} x( 6 )= 6" data-latex="y'= 5 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{2}{3}} \hspace{2em} x( 6 )= 6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(-2%20%5C,%20t%20+%202%20%5C,%20%7By%7D%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%206%20)=%206" alt="y'= 5 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{2}{3}} \hspace{2em} x( 6 )= 6" title="y'= 5 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{2}{3}} \hspace{2em} x( 6 )= 6" data-latex="y'= 5 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{2}{3}} \hspace{2em} x( 6 )= 6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{2}{3}}" alt="F(t,y)= 5 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{2}{3}}" title="F(t,y)= 5 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 5 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{2}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{20}{3 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{1}{3}}}" alt="F_y= \frac{20}{3 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{1}{3}}}" title="F_y= \frac{20}{3 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{20}{3 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{1}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(-2%20%5C,%20t%20+%202%20%5C,%20%7By%7D%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{2}{3}}" title="F(t,y)= 5 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 5 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{2}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B20%7D%7B3%20%5C,%20%7B%5Cleft(-2%20%5C,%20t%20+%202%20%5C,%20%7By%7D%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D" alt="F_y= \frac{20}{3 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{1}{3}}}" title="F_y= \frac{20}{3 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{20}{3 \, {\left(-2 \, t + 2 \, {y}\right)}^{\frac{1}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9076" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9076"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{4}{3}} \hspace{2em} x( -4 )= 4" alt="y'= 2 \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{4}{3}} \hspace{2em} x( -4 )= 4" title="y'= 2 \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{4}{3}} \hspace{2em} x( -4 )= 4" data-latex="y'= 2 \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{4}{3}} \hspace{2em} x( -4 )= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%2028%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%204" alt="y'= 2 \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{4}{3}} \hspace{2em} x( -4 )= 4" title="y'= 2 \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{4}{3}} \hspace{2em} x( -4 )= 4" data-latex="y'= 2 \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{4}{3}} \hspace{2em} x( -4 )= 4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{4}{3}}" alt="F(t,y)= 2 \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{4}{3}}" title="F(t,y)= 2 \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 2 \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{4}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{16}{3} \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{1}{3}}" alt="F_y= \frac{16}{3} \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{1}{3}}" title="F_y= \frac{16}{3} \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{1}{3}}" data-latex="F_y= \frac{16}{3} \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{1}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%2028%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F(t,y)= 2 \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{4}{3}}" title="F(t,y)= 2 \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 2 \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{4}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B16%7D%7B3%7D%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%2028%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F_y= \frac{16}{3} \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{1}{3}}" title="F_y= \frac{16}{3} \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{1}{3}}" data-latex="F_y= \frac{16}{3} \, {\left(2 \, {y} - 5 \, t - 28\right)}^{\frac{1}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-0267" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 0267"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{2}{5}} \hspace{2em} x( -4 )= 2" alt="y'= 4 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{2}{5}} \hspace{2em} x( -4 )= 2" title="y'= 4 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{2}{5}} \hspace{2em} x( -4 )= 2" data-latex="y'= 4 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{2}{5}} \hspace{2em} x( -4 )= 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(-3%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2016%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%202" alt="y'= 4 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{2}{5}} \hspace{2em} x( -4 )= 2" title="y'= 4 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{2}{5}} \hspace{2em} x( -4 )= 2" data-latex="y'= 4 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{2}{5}} \hspace{2em} x( -4 )= 2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{2}{5}}" alt="F(t,y)= 4 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{2}{5}}" title="F(t,y)= 4 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 4 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{2}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{16}{5 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{3}{5}}}" alt="F_y= \frac{16}{5 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{3}{5}}}" title="F_y= \frac{16}{5 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{16}{5 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{3}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(-3%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2016%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D" alt="F(t,y)= 4 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{2}{5}}" title="F(t,y)= 4 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 4 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{2}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B16%7D%7B5%20%5C,%20%7B%5Cleft(-3%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2016%5Cright)%7D%5E%7B%5Cfrac%7B3%7D%7B5%7D%7D%7D" alt="F_y= \frac{16}{5 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{3}{5}}}" title="F_y= \frac{16}{5 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{16}{5 \, {\left(-3 \, t + 2 \, {y} - 16\right)}^{\frac{3}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-5334" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 5334"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{2}{5}} \hspace{2em} x( 4 )= 5" alt="y'= 2 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{2}{5}} \hspace{2em} x( 4 )= 5" title="y'= 2 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{2}{5}} \hspace{2em} x( 4 )= 5" data-latex="y'= 2 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{2}{5}} \hspace{2em} x( 4 )= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%205" alt="y'= 2 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{2}{5}} \hspace{2em} x( 4 )= 5" title="y'= 2 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{2}{5}} \hspace{2em} x( 4 )= 5" data-latex="y'= 2 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{2}{5}} \hspace{2em} x( 4 )= 5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{2}{5}}" alt="F(t,y)= 2 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{2}{5}}" title="F(t,y)= 2 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 2 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{2}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{8}{5 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{3}{5}}}" alt="F_y= \frac{8}{5 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{3}{5}}}" title="F_y= \frac{8}{5 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{8}{5 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{3}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D" alt="F(t,y)= 2 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{2}{5}}" title="F(t,y)= 2 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 2 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{2}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B8%7D%7B5%20%5C,%20%7B%5Cleft(2%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B3%7D%7B5%7D%7D%7D" alt="F_y= \frac{8}{5 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{3}{5}}}" title="F_y= \frac{8}{5 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{8}{5 \, {\left(2 \, t + 2 \, {y} - 18\right)}^{\frac{3}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-6809" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 6809"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{1}{3}} \hspace{2em} x( -2 )= 6" alt="y'= 4 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{1}{3}} \hspace{2em} x( -2 )= 6" title="y'= 4 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{1}{3}} \hspace{2em} x( -2 )= 6" data-latex="y'= 4 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{1}{3}} \hspace{2em} x( -2 )= 6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20-%2016%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%206" alt="y'= 4 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{1}{3}} \hspace{2em} x( -2 )= 6" title="y'= 4 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{1}{3}} \hspace{2em} x( -2 )= 6" data-latex="y'= 4 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{1}{3}} \hspace{2em} x( -2 )= 6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{1}{3}}" alt="F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{1}{3}}" title="F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{1}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{8}{3 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{2}{3}}}" alt="F_y= \frac{8}{3 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{2}{3}}}" title="F_y= \frac{8}{3 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{8}{3 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{2}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20-%2016%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{1}{3}}" title="F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{1}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B8%7D%7B3%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20-%2016%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D" alt="F_y= \frac{8}{3 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{2}{3}}}" title="F_y= \frac{8}{3 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{8}{3 \, {\left(2 \, {y} - 2 \, t - 16\right)}^{\frac{2}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-1264" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 1264"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= 5" alt="y'= 4 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= 5" title="y'= 4 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= 5" data-latex="y'= 4 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%2030%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%205" alt="y'= 4 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= 5" title="y'= 4 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= 5" data-latex="y'= 4 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{2}{3}} \hspace{2em} x( -5 )= 5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{2}{3}}" alt="F(t,y)= 4 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{2}{3}}" title="F(t,y)= 4 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 4 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{2}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{16}{3 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{1}{3}}}" alt="F_y= \frac{16}{3 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{1}{3}}}" title="F_y= \frac{16}{3 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{16}{3 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{1}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%2030%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D" alt="F(t,y)= 4 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{2}{3}}" title="F(t,y)= 4 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 4 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{2}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B16%7D%7B3%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%2030%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D" alt="F_y= \frac{16}{3 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{1}{3}}}" title="F_y= \frac{16}{3 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{16}{3 \, {\left(2 \, {y} - 4 \, t - 30\right)}^{\frac{1}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9181" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9181"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{8}{5}} \hspace{2em} x( -2 )= 4" alt="y'= 5 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{8}{5}} \hspace{2em} x( -2 )= 4" title="y'= 5 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{8}{5}} \hspace{2em} x( -2 )= 4" data-latex="y'= 5 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{8}{5}} \hspace{2em} x( -2 )= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%2022%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%204" alt="y'= 5 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{8}{5}} \hspace{2em} x( -2 )= 4" title="y'= 5 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{8}{5}} \hspace{2em} x( -2 )= 4" data-latex="y'= 5 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{8}{5}} \hspace{2em} x( -2 )= 4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{8}{5}}" alt="F(t,y)= 5 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{8}{5}}" title="F(t,y)= 5 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{8}{5}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{8}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 24 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{3}{5}}" alt="F_y= 24 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{3}{5}}" title="F_y= 24 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{3}{5}}" data-latex="F_y= 24 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{3}{5}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%2022%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B5%7D%7D" alt="F(t,y)= 5 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{8}{5}}" title="F(t,y)= 5 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{8}{5}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{8}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2024%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%2022%5Cright)%7D%5E%7B%5Cfrac%7B3%7D%7B5%7D%7D" alt="F_y= 24 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{3}{5}}" title="F_y= 24 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{3}{5}}" data-latex="F_y= 24 \, {\left(3 \, {y} - 5 \, t - 22\right)}^{\frac{3}{5}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-0982" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 0982"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= -5" alt="y'= 6 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= -5" title="y'= 6 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= -5" data-latex="y'= 6 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(-4%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%20-5" alt="y'= 6 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= -5" title="y'= 6 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= -5" data-latex="y'= 6 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= -5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{8}{3}}" alt="F(t,y)= 6 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{8}{3}}" title="F(t,y)= 6 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 6 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{8}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 32 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{5}{3}}" alt="F_y= 32 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{5}{3}}" title="F_y= 32 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{5}{3}}" data-latex="F_y= 32 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{5}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(-4%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D" alt="F(t,y)= 6 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{8}{3}}" title="F(t,y)= 6 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 6 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{8}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2032%20%5C,%20%7B%5Cleft(-4%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D" alt="F_y= 32 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{5}{3}}" title="F_y= 32 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{5}{3}}" data-latex="F_y= 32 \, {\left(-4 \, t + 2 \, {y} - 6\right)}^{\frac{5}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-4584" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 4584"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{8}{3}} \hspace{2em} x( 4 )= 4" alt="y'= 6 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{8}{3}} \hspace{2em} x( 4 )= 4" title="y'= 6 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{8}{3}} \hspace{2em} x( 4 )= 4" data-latex="y'= 6 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{8}{3}} \hspace{2em} x( 4 )= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%204%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%204" alt="y'= 6 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{8}{3}} \hspace{2em} x( 4 )= 4" title="y'= 6 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{8}{3}} \hspace{2em} x( 4 )= 4" data-latex="y'= 6 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{8}{3}} \hspace{2em} x( 4 )= 4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{8}{3}}" alt="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{8}{3}}" title="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{8}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 32 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{5}{3}}" alt="F_y= 32 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{5}{3}}" title="F_y= 32 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{5}{3}}" data-latex="F_y= 32 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{5}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%204%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D" alt="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{8}{3}}" title="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 6 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{8}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2032%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%204%5Cright)%7D%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D" alt="F_y= 32 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{5}{3}}" title="F_y= 32 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{5}{3}}" data-latex="F_y= 32 \, {\left(2 \, {y} - 3 \, t + 4\right)}^{\frac{5}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-8483" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 8483"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{4}{5}} \hspace{2em} x( 4 )= -5" alt="y'= 2 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{4}{5}} \hspace{2em} x( 4 )= -5" title="y'= 2 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{4}{5}} \hspace{2em} x( 4 )= -5" data-latex="y'= 2 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{4}{5}} \hspace{2em} x( 4 )= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%2018%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%20-5" alt="y'= 2 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{4}{5}} \hspace{2em} x( 4 )= -5" title="y'= 2 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{4}{5}} \hspace{2em} x( 4 )= -5" data-latex="y'= 2 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{4}{5}} \hspace{2em} x( 4 )= -5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{4}{5}}" alt="F(t,y)= 2 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{4}{5}}" title="F(t,y)= 2 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 2 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{4}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{16}{5 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{1}{5}}}" alt="F_y= \frac{16}{5 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{1}{5}}}" title="F_y= \frac{16}{5 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{16}{5 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{1}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%2018%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D" alt="F(t,y)= 2 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{4}{5}}" title="F(t,y)= 2 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 2 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{4}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B16%7D%7B5%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%2018%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%7D" alt="F_y= \frac{16}{5 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{1}{5}}}" title="F_y= \frac{16}{5 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{16}{5 \, {\left(2 \, {y} - 2 \, t + 18\right)}^{\frac{1}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-2215" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 2215"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}} \hspace{2em} x( -3 )= -6" alt="y'= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}} \hspace{2em} x( -3 )= -6" title="y'= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}} \hspace{2em} x( -3 )= -6" data-latex="y'= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}} \hspace{2em} x( -3 )= -6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%2018%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-6" alt="y'= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}} \hspace{2em} x( -3 )= -6" title="y'= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}} \hspace{2em} x( -3 )= -6" data-latex="y'= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}} \hspace{2em} x( -3 )= -6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}}" alt="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}}" title="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{8}{{\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}}}" alt="F_y= \frac{8}{{\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}}}" title="F_y= \frac{8}{{\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{8}{{\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%2018%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D" alt="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}}" title="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B8%7D%7B%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%2018%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%7D" alt="F_y= \frac{8}{{\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}}}" title="F_y= \frac{8}{{\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{8}{{\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9877" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9877"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{7}{5}} \hspace{2em} x( 6 )= -3" alt="y'= 3 \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{7}{5}} \hspace{2em} x( 6 )= -3" title="y'= 3 \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{7}{5}} \hspace{2em} x( 6 )= -3" data-latex="y'= 3 \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{7}{5}} \hspace{2em} x( 6 )= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%206%20%5C,%20t%20-%2027%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%206%20)=%20-3" alt="y'= 3 \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{7}{5}} \hspace{2em} x( 6 )= -3" title="y'= 3 \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{7}{5}} \hspace{2em} x( 6 )= -3" data-latex="y'= 3 \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{7}{5}} \hspace{2em} x( 6 )= -3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{7}{5}}" alt="F(t,y)= 3 \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{7}{5}}" title="F(t,y)= 3 \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{7}{5}}" data-latex="F(t,y)= 3 \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{7}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{63}{5} \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{2}{5}}" alt="F_y= \frac{63}{5} \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{2}{5}}" title="F_y= \frac{63}{5} \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{2}{5}}" data-latex="F_y= \frac{63}{5} \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{2}{5}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%206%20%5C,%20t%20-%2027%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B5%7D%7D" alt="F(t,y)= 3 \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{7}{5}}" title="F(t,y)= 3 \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{7}{5}}" data-latex="F(t,y)= 3 \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{7}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B63%7D%7B5%7D%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%206%20%5C,%20t%20-%2027%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D" alt="F_y= \frac{63}{5} \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{2}{5}}" title="F_y= \frac{63}{5} \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{2}{5}}" data-latex="F_y= \frac{63}{5} \, {\left(3 \, {y} + 6 \, t - 27\right)}^{\frac{2}{5}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-5724" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 5724"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{4}{5}} \hspace{2em} x( -4 )= -6" alt="y'= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{4}{5}} \hspace{2em} x( -4 )= -6" title="y'= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{4}{5}} \hspace{2em} x( -4 )= -6" data-latex="y'= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{4}{5}} \hspace{2em} x( -4 )= -6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%20-6" alt="y'= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{4}{5}} \hspace{2em} x( -4 )= -6" title="y'= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{4}{5}} \hspace{2em} x( -4 )= -6" data-latex="y'= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{4}{5}} \hspace{2em} x( -4 )= -6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{4}{5}}" alt="F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{4}{5}}" title="F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{4}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{24}{5 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{5}}}" alt="F_y= \frac{24}{5 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{5}}}" title="F_y= \frac{24}{5 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{24}{5 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D" alt="F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{4}{5}}" title="F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{4}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B24%7D%7B5%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%7D" alt="F_y= \frac{24}{5 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{5}}}" title="F_y= \frac{24}{5 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{24}{5 \, {\left(3 \, {y} - 6 \, t - 6\right)}^{\frac{1}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3942" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3942"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{7}{3}} \hspace{2em} x( -3 )= -6" alt="y'= 5 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{7}{3}} \hspace{2em} x( -3 )= -6" title="y'= 5 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{7}{3}} \hspace{2em} x( -3 )= -6" data-latex="y'= 5 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{7}{3}} \hspace{2em} x( -3 )= -6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(-4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-6" alt="y'= 5 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{7}{3}} \hspace{2em} x( -3 )= -6" title="y'= 5 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{7}{3}} \hspace{2em} x( -3 )= -6" data-latex="y'= 5 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{7}{3}} \hspace{2em} x( -3 )= -6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{7}{3}}" alt="F(t,y)= 5 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{7}{3}}" title="F(t,y)= 5 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 5 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{7}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 35 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{4}{3}}" alt="F_y= 35 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{4}{3}}" title="F_y= 35 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{4}{3}}" data-latex="F_y= 35 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{4}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(-4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{7}{3}}" title="F(t,y)= 5 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 5 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{7}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2035%20%5C,%20%7B%5Cleft(-4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F_y= 35 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{4}{3}}" title="F_y= 35 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{4}{3}}" data-latex="F_y= 35 \, {\left(-4 \, t + 3 \, {y} + 6\right)}^{\frac{4}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9177" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9177"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{7}{3}} \hspace{2em} x( 5 )= 6" alt="y'= 3 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{7}{3}} \hspace{2em} x( 5 )= 6" title="y'= 3 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{7}{3}} \hspace{2em} x( 5 )= 6" data-latex="y'= 3 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{7}{3}} \hspace{2em} x( 5 )= 6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2012%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%205%20)=%206" alt="y'= 3 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{7}{3}} \hspace{2em} x( 5 )= 6" title="y'= 3 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{7}{3}} \hspace{2em} x( 5 )= 6" data-latex="y'= 3 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{7}{3}} \hspace{2em} x( 5 )= 6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{7}{3}}" alt="F(t,y)= 3 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{7}{3}}" title="F(t,y)= 3 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 3 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{7}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 21 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}}" alt="F_y= 21 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}}" title="F_y= 21 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}}" data-latex="F_y= 21 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2012%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D" alt="F(t,y)= 3 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{7}{3}}" title="F(t,y)= 3 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 3 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{7}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2021%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2012%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F_y= 21 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}}" title="F_y= 21 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}}" data-latex="F_y= 21 \, {\left(3 \, {y} - 6 \, t + 12\right)}^{\frac{4}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-4341" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 4341"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{7}{3}} \hspace{2em} x( 2 )= 2" alt="y'= 2 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{7}{3}} \hspace{2em} x( 2 )= 2" title="y'= 2 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{7}{3}} \hspace{2em} x( 2 )= 2" data-latex="y'= 2 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{7}{3}} \hspace{2em} x( 2 )= 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%2014%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%202%20)=%202" alt="y'= 2 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{7}{3}} \hspace{2em} x( 2 )= 2" title="y'= 2 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{7}{3}} \hspace{2em} x( 2 )= 2" data-latex="y'= 2 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{7}{3}} \hspace{2em} x( 2 )= 2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{7}{3}}" alt="F(t,y)= 2 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{7}{3}}" title="F(t,y)= 2 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 2 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{7}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 14 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{4}{3}}" alt="F_y= 14 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{4}{3}}" title="F_y= 14 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{4}{3}}" data-latex="F_y= 14 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{4}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%2014%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D" alt="F(t,y)= 2 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{7}{3}}" title="F(t,y)= 2 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 2 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{7}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2014%20%5C,%20%7B%5Cleft(4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%2014%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F_y= 14 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{4}{3}}" title="F_y= 14 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{4}{3}}" data-latex="F_y= 14 \, {\left(4 \, t + 3 \, {y} - 14\right)}^{\frac{4}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3426" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3426"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= -3" alt="y'= 5 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= -3" title="y'= 5 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= -3" data-latex="y'= 5 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%20-3" alt="y'= 5 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= -3" title="y'= 5 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= -3" data-latex="y'= 5 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -4 )= -3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{8}{3}}" alt="F(t,y)= 5 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{8}{3}}" title="F(t,y)= 5 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{8}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{80}{3} \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{5}{3}}" alt="F_y= \frac{80}{3} \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{5}{3}}" title="F_y= \frac{80}{3} \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{5}{3}}" data-latex="F_y= \frac{80}{3} \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{5}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{8}{3}}" title="F(t,y)= 5 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{8}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B80%7D%7B3%7D%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D" alt="F_y= \frac{80}{3} \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{5}{3}}" title="F_y= \frac{80}{3} \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{5}{3}}" data-latex="F_y= \frac{80}{3} \, {\left(2 \, {y} - 3 \, t - 6\right)}^{\frac{5}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-8644" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 8644"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 6" alt="y'= 5 \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 6" title="y'= 5 \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 6" data-latex="y'= 5 \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%206" alt="y'= 5 \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 6" title="y'= 5 \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 6" data-latex="y'= 5 \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{4}{3}}" alt="F(t,y)= 5 \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{4}{3}}" title="F(t,y)= 5 \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{4}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{40}{3} \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{1}{3}}" alt="F_y= \frac{40}{3} \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{1}{3}}" title="F_y= \frac{40}{3} \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{1}{3}}" data-latex="F_y= \frac{40}{3} \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{1}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{4}{3}}" title="F(t,y)= 5 \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{4}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B40%7D%7B3%7D%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F_y= \frac{40}{3} \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{1}{3}}" title="F_y= \frac{40}{3} \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{1}{3}}" data-latex="F_y= \frac{40}{3} \, {\left(2 \, {y} - 3 \, t\right)}^{\frac{1}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9670" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9670"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{7}{3}} \hspace{2em} x( -2 )= 5" alt="y'= 6 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{7}{3}} \hspace{2em} x( -2 )= 5" title="y'= 6 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{7}{3}} \hspace{2em} x( -2 )= 5" data-latex="y'= 6 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{7}{3}} \hspace{2em} x( -2 )= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20-%204%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%205" alt="y'= 6 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{7}{3}} \hspace{2em} x( -2 )= 5" title="y'= 6 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{7}{3}} \hspace{2em} x( -2 )= 5" data-latex="y'= 6 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{7}{3}} \hspace{2em} x( -2 )= 5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{7}{3}}" alt="F(t,y)= 6 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{7}{3}}" title="F(t,y)= 6 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 6 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{7}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 28 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{4}{3}}" alt="F_y= 28 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{4}{3}}" title="F_y= 28 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{4}{3}}" data-latex="F_y= 28 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{4}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20-%204%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D" alt="F(t,y)= 6 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{7}{3}}" title="F(t,y)= 6 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 6 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{7}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2028%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20-%204%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F_y= 28 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{4}{3}}" title="F_y= 28 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{4}{3}}" data-latex="F_y= 28 \, {\left(2 \, {y} + 3 \, t - 4\right)}^{\frac{4}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-8313" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 8313"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{7}{5}} \hspace{2em} x( -3 )= -4" alt="y'= 3 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{7}{5}} \hspace{2em} x( -3 )= -4" title="y'= 3 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{7}{5}} \hspace{2em} x( -3 )= -4" data-latex="y'= 3 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{7}{5}} \hspace{2em} x( -3 )= -4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-4" alt="y'= 3 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{7}{5}} \hspace{2em} x( -3 )= -4" title="y'= 3 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{7}{5}} \hspace{2em} x( -3 )= -4" data-latex="y'= 3 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{7}{5}} \hspace{2em} x( -3 )= -4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{7}{5}}" alt="F(t,y)= 3 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{7}{5}}" title="F(t,y)= 3 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{7}{5}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{7}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{42}{5} \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{2}{5}}" alt="F_y= \frac{42}{5} \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{2}{5}}" title="F_y= \frac{42}{5} \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{2}{5}}" data-latex="F_y= \frac{42}{5} \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{2}{5}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B5%7D%7D" alt="F(t,y)= 3 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{7}{5}}" title="F(t,y)= 3 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{7}{5}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{7}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B42%7D%7B5%7D%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D" alt="F_y= \frac{42}{5} \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{2}{5}}" title="F_y= \frac{42}{5} \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{2}{5}}" data-latex="F_y= \frac{42}{5} \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{2}{5}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-1973" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 1973"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= -3" alt="y'= 4 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= -3" title="y'= 4 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= -3" data-latex="y'= 4 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%20-3" alt="y'= 4 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= -3" title="y'= 4 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= -3" data-latex="y'= 4 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}} \hspace{2em} x( -2 )= -3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}}" alt="F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}}" title="F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{32}{5 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{5}}}" alt="F_y= \frac{32}{5 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{5}}}" title="F_y= \frac{32}{5 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{32}{5 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D" alt="F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}}" title="F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 4 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B32%7D%7B5%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%7D" alt="F_y= \frac{32}{5 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{5}}}" title="F_y= \frac{32}{5 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{32}{5 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-1012" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 1012"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(-4 \, t + 3 \, {y}\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= -4" alt="y'= 3 \, {\left(-4 \, t + 3 \, {y}\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= -4" title="y'= 3 \, {\left(-4 \, t + 3 \, {y}\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= -4" data-latex="y'= 3 \, {\left(-4 \, t + 3 \, {y}\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= -4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(-4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-4" alt="y'= 3 \, {\left(-4 \, t + 3 \, {y}\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= -4" title="y'= 3 \, {\left(-4 \, t + 3 \, {y}\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= -4" data-latex="y'= 3 \, {\left(-4 \, t + 3 \, {y}\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= -4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(-4 \, t + 3 \, {y}\right)}^{\frac{2}{3}}" alt="F(t,y)= 3 \, {\left(-4 \, t + 3 \, {y}\right)}^{\frac{2}{3}}" title="F(t,y)= 3 \, {\left(-4 \, t + 3 \, {y}\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 3 \, {\left(-4 \, t + 3 \, {y}\right)}^{\frac{2}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{6}{{\left(-4 \, t + 3 \, {y}\right)}^{\frac{1}{3}}}" alt="F_y= \frac{6}{{\left(-4 \, t + 3 \, {y}\right)}^{\frac{1}{3}}}" title="F_y= \frac{6}{{\left(-4 \, t + 3 \, {y}\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{6}{{\left(-4 \, t + 3 \, {y}\right)}^{\frac{1}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(-4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D" alt="F(t,y)= 3 \, {\left(-4 \, t + 3 \, {y}\right)}^{\frac{2}{3}}" title="F(t,y)= 3 \, {\left(-4 \, t + 3 \, {y}\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 3 \, {\left(-4 \, t + 3 \, {y}\right)}^{\frac{2}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B6%7D%7B%7B%5Cleft(-4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D" alt="F_y= \frac{6}{{\left(-4 \, t + 3 \, {y}\right)}^{\frac{1}{3}}}" title="F_y= \frac{6}{{\left(-4 \, t + 3 \, {y}\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{6}{{\left(-4 \, t + 3 \, {y}\right)}^{\frac{1}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-6121" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 6121"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 4" alt="y'= 6 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 4" title="y'= 6 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 4" data-latex="y'= 6 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%208%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%204" alt="y'= 6 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 4" title="y'= 6 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 4" data-latex="y'= 6 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{4}{3}}" alt="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{4}{3}}" title="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{4}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 24 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{1}{3}}" alt="F_y= 24 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{1}{3}}" title="F_y= 24 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{1}{3}}" data-latex="F_y= 24 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{1}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%208%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{4}{3}}" title="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{4}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2024%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%208%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F_y= 24 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{1}{3}}" title="F_y= 24 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{1}{3}}" data-latex="F_y= 24 \, {\left(3 \, {y} - 5 \, t + 8\right)}^{\frac{1}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-4019" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 4019"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -4" alt="y'= 3 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -4" title="y'= 3 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -4" data-latex="y'= 3 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%203%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-4" alt="y'= 3 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -4" title="y'= 3 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -4" data-latex="y'= 3 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}}" alt="F(t,y)= 3 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}}" title="F(t,y)= 3 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 3 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{3}{{\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}}}" alt="F_y= \frac{3}{{\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}}}" title="F_y= \frac{3}{{\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{3}{{\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%203%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F(t,y)= 3 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}}" title="F(t,y)= 3 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 3 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B3%7D%7B%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%203%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D" alt="F_y= \frac{3}{{\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}}}" title="F_y= \frac{3}{{\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{3}{{\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-1213" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 1213"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= -5" alt="y'= 3 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= -5" title="y'= 3 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= -5" data-latex="y'= 3 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%203%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%203%20)=%20-5" alt="y'= 3 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= -5" title="y'= 3 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= -5" data-latex="y'= 3 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{4}{3}} \hspace{2em} x( 3 )= -5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{4}{3}}" alt="F(t,y)= 3 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{4}{3}}" title="F(t,y)= 3 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 3 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{4}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 12 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{1}{3}}" alt="F_y= 12 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{1}{3}}" title="F_y= 12 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{1}{3}}" data-latex="F_y= 12 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{1}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%203%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F(t,y)= 3 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{4}{3}}" title="F(t,y)= 3 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 3 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{4}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2012%20%5C,%20%7B%5Cleft(4%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%203%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F_y= 12 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{1}{3}}" title="F_y= 12 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{1}{3}}" data-latex="F_y= 12 \, {\left(4 \, t + 3 \, {y} + 3\right)}^{\frac{1}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3562" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3562"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{7}{3}} \hspace{2em} x( 2 )= 5" alt="y'= 4 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{7}{3}} \hspace{2em} x( 2 )= 5" title="y'= 4 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{7}{3}} \hspace{2em} x( 2 )= 5" data-latex="y'= 4 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{7}{3}} \hspace{2em} x( 2 )= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(-3%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%209%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%202%20)=%205" alt="y'= 4 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{7}{3}} \hspace{2em} x( 2 )= 5" title="y'= 4 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{7}{3}} \hspace{2em} x( 2 )= 5" data-latex="y'= 4 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{7}{3}} \hspace{2em} x( 2 )= 5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{7}{3}}" alt="F(t,y)= 4 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{7}{3}}" title="F(t,y)= 4 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 4 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{7}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 28 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{4}{3}}" alt="F_y= 28 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{4}{3}}" title="F_y= 28 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{4}{3}}" data-latex="F_y= 28 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{4}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(-3%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%209%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D" alt="F(t,y)= 4 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{7}{3}}" title="F(t,y)= 4 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 4 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{7}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2028%20%5C,%20%7B%5Cleft(-3%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%209%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F_y= 28 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{4}{3}}" title="F_y= 28 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{4}{3}}" data-latex="F_y= 28 \, {\left(-3 \, t + 3 \, {y} - 9\right)}^{\frac{4}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3460" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3460"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= 4" alt="y'= 4 \, {\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= 4" title="y'= 4 \, {\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= 4" data-latex="y'= 4 \, {\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20+%203%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%204" alt="y'= 4 \, {\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= 4" title="y'= 4 \, {\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= 4" data-latex="y'= 4 \, {\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= 4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{1}{3}}" alt="F(t,y)= 4 \, {\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{1}{3}}" title="F(t,y)= 4 \, {\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{1}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{4}{{\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{2}{3}}}" alt="F_y= \frac{4}{{\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{2}{3}}}" title="F_y= \frac{4}{{\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{4}{{\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{2}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20+%203%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F(t,y)= 4 \, {\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{1}{3}}" title="F(t,y)= 4 \, {\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{1}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B4%7D%7B%7B%5Cleft(3%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20+%203%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D" alt="F_y= \frac{4}{{\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{2}{3}}}" title="F_y= \frac{4}{{\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{4}{{\left(3 \, {y} + 5 \, t + 3\right)}^{\frac{2}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-5024" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 5024"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 4" alt="y'= 6 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 4" title="y'= 6 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 4" data-latex="y'= 6 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(6%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2032%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%204" alt="y'= 6 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 4" title="y'= 6 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 4" data-latex="y'= 6 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= 4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{4}{3}}" alt="F(t,y)= 6 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{4}{3}}" title="F(t,y)= 6 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 6 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{4}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 16 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{1}{3}}" alt="F_y= 16 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{1}{3}}" title="F_y= 16 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{1}{3}}" data-latex="F_y= 16 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{1}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(6%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2032%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F(t,y)= 6 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{4}{3}}" title="F(t,y)= 6 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 6 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{4}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2016%20%5C,%20%7B%5Cleft(6%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2032%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F_y= 16 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{1}{3}}" title="F_y= 16 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{1}{3}}" data-latex="F_y= 16 \, {\left(6 \, t + 2 \, {y} - 32\right)}^{\frac{1}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-6997" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 6997"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -5" alt="y'= 6 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -5" title="y'= 6 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -5" data-latex="y'= 6 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%202%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%20-5" alt="y'= 6 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -5" title="y'= 6 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -5" data-latex="y'= 6 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}} \hspace{2em} x( -3 )= -5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}}" alt="F(t,y)= 6 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}}" title="F(t,y)= 6 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 6 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{4}{{\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{2}{3}}}" alt="F_y= \frac{4}{{\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{2}{3}}}" title="F_y= \frac{4}{{\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{4}{{\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{2}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%202%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F(t,y)= 6 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}}" title="F(t,y)= 6 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 6 \, {\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{1}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B4%7D%7B%7B%5Cleft(2%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%202%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D" alt="F_y= \frac{4}{{\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{2}{3}}}" title="F_y= \frac{4}{{\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{4}{{\left(2 \, {y} - 4 \, t - 2\right)}^{\frac{2}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-7017" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 7017"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{1}{5}} \hspace{2em} x( 5 )= 3" alt="y'= 4 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{1}{5}} \hspace{2em} x( 5 )= 3" title="y'= 4 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{1}{5}} \hspace{2em} x( 5 )= 3" data-latex="y'= 4 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{1}{5}} \hspace{2em} x( 5 )= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20-%2019%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%205%20)=%203" alt="y'= 4 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{1}{5}} \hspace{2em} x( 5 )= 3" title="y'= 4 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{1}{5}} \hspace{2em} x( 5 )= 3" data-latex="y'= 4 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{1}{5}} \hspace{2em} x( 5 )= 3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{1}{5}}" alt="F(t,y)= 4 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{1}{5}}" title="F(t,y)= 4 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{1}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{12}{5 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{4}{5}}}" alt="F_y= \frac{12}{5 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{4}{5}}}" title="F_y= \frac{12}{5 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{12}{5 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{4}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20-%2019%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D" alt="F(t,y)= 4 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{1}{5}}" title="F(t,y)= 4 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{1}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B12%7D%7B5%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20-%2019%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%7D" alt="F_y= \frac{12}{5 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{4}{5}}}" title="F_y= \frac{12}{5 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{12}{5 \, {\left(3 \, {y} + 2 \, t - 19\right)}^{\frac{4}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-5316" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 5316"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{7}{3}} \hspace{2em} x( 5 )= -6" alt="y'= 4 \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{7}{3}} \hspace{2em} x( 5 )= -6" title="y'= 4 \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{7}{3}} \hspace{2em} x( 5 )= -6" data-latex="y'= 4 \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{7}{3}} \hspace{2em} x( 5 )= -6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%2027%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%205%20)=%20-6" alt="y'= 4 \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{7}{3}} \hspace{2em} x( 5 )= -6" title="y'= 4 \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{7}{3}} \hspace{2em} x( 5 )= -6" data-latex="y'= 4 \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{7}{3}} \hspace{2em} x( 5 )= -6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{7}{3}}" alt="F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{7}{3}}" title="F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{7}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{56}{3} \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{4}{3}}" alt="F_y= \frac{56}{3} \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{4}{3}}" title="F_y= \frac{56}{3} \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{4}{3}}" data-latex="F_y= \frac{56}{3} \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{4}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%2027%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D" alt="F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{7}{3}}" title="F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{7}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B56%7D%7B3%7D%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%2027%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F_y= \frac{56}{3} \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{4}{3}}" title="F_y= \frac{56}{3} \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{4}{3}}" data-latex="F_y= \frac{56}{3} \, {\left(2 \, {y} - 3 \, t + 27\right)}^{\frac{4}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-4860" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 4860"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{1}{5}} \hspace{2em} x( 6 )= -5" alt="y'= 5 \, {\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{1}{5}} \hspace{2em} x( 6 )= -5" title="y'= 5 \, {\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{1}{5}} \hspace{2em} x( 6 )= -5" data-latex="y'= 5 \, {\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{1}{5}} \hspace{2em} x( 6 )= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%2040%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%206%20)=%20-5" alt="y'= 5 \, {\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{1}{5}} \hspace{2em} x( 6 )= -5" title="y'= 5 \, {\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{1}{5}} \hspace{2em} x( 6 )= -5" data-latex="y'= 5 \, {\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{1}{5}} \hspace{2em} x( 6 )= -5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{1}{5}}" alt="F(t,y)= 5 \, {\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{1}{5}}" title="F(t,y)= 5 \, {\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{1}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{2}{{\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{4}{5}}}" alt="F_y= \frac{2}{{\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{4}{5}}}" title="F_y= \frac{2}{{\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{2}{{\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{4}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%2040%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D" alt="F(t,y)= 5 \, {\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{1}{5}}" title="F(t,y)= 5 \, {\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{1}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B2%7D%7B%7B%5Cleft(2%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%2040%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%7D" alt="F_y= \frac{2}{{\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{4}{5}}}" title="F_y= \frac{2}{{\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{2}{{\left(2 \, {y} - 5 \, t + 40\right)}^{\frac{4}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-4968" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 4968"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{8}{5}} \hspace{2em} x( 5 )= -3" alt="y'= 5 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{8}{5}} \hspace{2em} x( 5 )= -3" title="y'= 5 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{8}{5}} \hspace{2em} x( 5 )= -3" data-latex="y'= 5 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{8}{5}} \hspace{2em} x( 5 )= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%2024%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%205%20)=%20-3" alt="y'= 5 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{8}{5}} \hspace{2em} x( 5 )= -3" title="y'= 5 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{8}{5}} \hspace{2em} x( 5 )= -3" data-latex="y'= 5 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{8}{5}} \hspace{2em} x( 5 )= -3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{8}{5}}" alt="F(t,y)= 5 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{8}{5}}" title="F(t,y)= 5 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{8}{5}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{8}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 24 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{3}{5}}" alt="F_y= 24 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{3}{5}}" title="F_y= 24 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{3}{5}}" data-latex="F_y= 24 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{3}{5}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%2024%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B5%7D%7D" alt="F(t,y)= 5 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{8}{5}}" title="F(t,y)= 5 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{8}{5}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{8}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2024%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20+%2024%5Cright)%7D%5E%7B%5Cfrac%7B3%7D%7B5%7D%7D" alt="F_y= 24 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{3}{5}}" title="F_y= 24 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{3}{5}}" data-latex="F_y= 24 \, {\left(3 \, {y} - 3 \, t + 24\right)}^{\frac{3}{5}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-6535" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 6535"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{7}{5}} \hspace{2em} x( -5 )= 4" alt="y'= 4 \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{7}{5}} \hspace{2em} x( -5 )= 4" title="y'= 4 \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{7}{5}} \hspace{2em} x( -5 )= 4" data-latex="y'= 4 \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{7}{5}} \hspace{2em} x( -5 )= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%206%20%5C,%20t%20+%2022%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%204" alt="y'= 4 \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{7}{5}} \hspace{2em} x( -5 )= 4" title="y'= 4 \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{7}{5}} \hspace{2em} x( -5 )= 4" data-latex="y'= 4 \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{7}{5}} \hspace{2em} x( -5 )= 4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{7}{5}}" alt="F(t,y)= 4 \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{7}{5}}" title="F(t,y)= 4 \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{7}{5}}" data-latex="F(t,y)= 4 \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{7}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{56}{5} \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{2}{5}}" alt="F_y= \frac{56}{5} \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{2}{5}}" title="F_y= \frac{56}{5} \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{2}{5}}" data-latex="F_y= \frac{56}{5} \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{2}{5}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%206%20%5C,%20t%20+%2022%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B5%7D%7D" alt="F(t,y)= 4 \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{7}{5}}" title="F(t,y)= 4 \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{7}{5}}" data-latex="F(t,y)= 4 \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{7}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B56%7D%7B5%7D%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%206%20%5C,%20t%20+%2022%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D" alt="F_y= \frac{56}{5} \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{2}{5}}" title="F_y= \frac{56}{5} \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{2}{5}}" data-latex="F_y= \frac{56}{5} \, {\left(2 \, {y} + 6 \, t + 22\right)}^{\frac{2}{5}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-2192" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 2192"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= 5" alt="y'= 5 \, {\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= 5" title="y'= 5 \, {\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= 5" data-latex="y'= 5 \, {\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20-%2033%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%205" alt="y'= 5 \, {\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= 5" title="y'= 5 \, {\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= 5" data-latex="y'= 5 \, {\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{2}{3}} \hspace{2em} x( -3 )= 5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{2}{3}}" alt="F(t,y)= 5 \, {\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{2}{3}}" title="F(t,y)= 5 \, {\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{2}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{10}{{\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{1}{3}}}" alt="F_y= \frac{10}{{\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{1}{3}}}" title="F_y= \frac{10}{{\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{10}{{\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{1}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20-%2033%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{2}{3}}" title="F(t,y)= 5 \, {\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{2}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B10%7D%7B%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20-%2033%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D" alt="F_y= \frac{10}{{\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{1}{3}}}" title="F_y= \frac{10}{{\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{10}{{\left(3 \, {y} - 6 \, t - 33\right)}^{\frac{1}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3151" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3151"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{7}{5}} \hspace{2em} x( -4 )= 2" alt="y'= 2 \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{7}{5}} \hspace{2em} x( -4 )= 2" title="y'= 2 \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{7}{5}} \hspace{2em} x( -4 )= 2" data-latex="y'= 2 \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{7}{5}} \hspace{2em} x( -4 )= 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%2022%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-4%20)=%202" alt="y'= 2 \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{7}{5}} \hspace{2em} x( -4 )= 2" title="y'= 2 \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{7}{5}} \hspace{2em} x( -4 )= 2" data-latex="y'= 2 \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{7}{5}} \hspace{2em} x( -4 )= 2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{7}{5}}" alt="F(t,y)= 2 \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{7}{5}}" title="F(t,y)= 2 \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{7}{5}}" data-latex="F(t,y)= 2 \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{7}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{42}{5} \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{2}{5}}" alt="F_y= \frac{42}{5} \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{2}{5}}" title="F_y= \frac{42}{5} \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{2}{5}}" data-latex="F_y= \frac{42}{5} \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{2}{5}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%2022%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B5%7D%7D" alt="F(t,y)= 2 \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{7}{5}}" title="F(t,y)= 2 \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{7}{5}}" data-latex="F(t,y)= 2 \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{7}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B42%7D%7B5%7D%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%2022%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D" alt="F_y= \frac{42}{5} \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{2}{5}}" title="F_y= \frac{42}{5} \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{2}{5}}" data-latex="F_y= \frac{42}{5} \, {\left(3 \, {y} - 4 \, t - 22\right)}^{\frac{2}{5}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3642" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3642"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 2" alt="y'= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 2" title="y'= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 2" data-latex="y'= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(5%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2019%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%203%20)=%202" alt="y'= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 2" title="y'= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 2" data-latex="y'= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}}" alt="F(t,y)= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}}" title="F(t,y)= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{8}{3 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{1}{3}}}" alt="F_y= \frac{8}{3 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{1}{3}}}" title="F_y= \frac{8}{3 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{8}{3 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{1}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(5%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2019%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D" alt="F(t,y)= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}}" title="F(t,y)= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 2 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{2}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B8%7D%7B3%20%5C,%20%7B%5Cleft(5%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20-%2019%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D" alt="F_y= \frac{8}{3 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{1}{3}}}" title="F_y= \frac{8}{3 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{8}{3 \, {\left(5 \, t + 2 \, {y} - 19\right)}^{\frac{1}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-8806" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 8806"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}} \hspace{2em} x( -5 )= -4" alt="y'= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}} \hspace{2em} x( -5 )= -4" title="y'= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}} \hspace{2em} x( -5 )= -4" data-latex="y'= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}} \hspace{2em} x( -5 )= -4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%2018%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%20-4" alt="y'= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}} \hspace{2em} x( -5 )= -4" title="y'= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}} \hspace{2em} x( -5 )= -4" data-latex="y'= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}} \hspace{2em} x( -5 )= -4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}}" alt="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}}" title="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{2}{{\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}}}" alt="F_y= \frac{2}{{\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}}}" title="F_y= \frac{2}{{\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{2}{{\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%2018%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D" alt="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}}" title="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{1}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B2%7D%7B%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%2018%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%7D" alt="F_y= \frac{2}{{\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}}}" title="F_y= \frac{2}{{\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{2}{{\left(2 \, {y} + 2 \, t + 18\right)}^{\frac{4}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-1088" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 1088"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -2" alt="y'= 3 \, {\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -2" title="y'= 3 \, {\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -2" data-latex="y'= 3 \, {\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%2014%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%20-2" alt="y'= 3 \, {\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -2" title="y'= 3 \, {\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -2" data-latex="y'= 3 \, {\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{1}{3}}" alt="F(t,y)= 3 \, {\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{1}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{1}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{2}{{\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{2}{3}}}" alt="F_y= \frac{2}{{\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{2}{3}}}" title="F_y= \frac{2}{{\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{2}{{\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{2}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%2014%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F(t,y)= 3 \, {\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{1}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{1}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B2%7D%7B%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20+%2014%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D" alt="F_y= \frac{2}{{\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{2}{3}}}" title="F_y= \frac{2}{{\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{2}{{\left(2 \, {y} + 2 \, t + 14\right)}^{\frac{2}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9786" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9786"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{2}{5}} \hspace{2em} x( -6 )= -2" alt="y'= 2 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{2}{5}} \hspace{2em} x( -6 )= -2" title="y'= 2 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{2}{5}} \hspace{2em} x( -6 )= -2" data-latex="y'= 2 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{2}{5}} \hspace{2em} x( -6 )= -2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20-%2032%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-6%20)=%20-2" alt="y'= 2 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{2}{5}} \hspace{2em} x( -6 )= -2" title="y'= 2 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{2}{5}} \hspace{2em} x( -6 )= -2" data-latex="y'= 2 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{2}{5}} \hspace{2em} x( -6 )= -2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{2}{5}}" alt="F(t,y)= 2 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{2}{5}}" title="F(t,y)= 2 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 2 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{2}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{8}{5 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{3}{5}}}" alt="F_y= \frac{8}{5 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{3}{5}}}" title="F_y= \frac{8}{5 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{8}{5 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{3}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20-%2032%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D" alt="F(t,y)= 2 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{2}{5}}" title="F(t,y)= 2 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 2 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{2}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B8%7D%7B5%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20-%2032%5Cright)%7D%5E%7B%5Cfrac%7B3%7D%7B5%7D%7D%7D" alt="F_y= \frac{8}{5 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{3}{5}}}" title="F_y= \frac{8}{5 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{8}{5 \, {\left(2 \, {y} - 6 \, t - 32\right)}^{\frac{3}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9111" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9111"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}} \hspace{2em} x( 4 )= 3" alt="y'= 6 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}} \hspace{2em} x( 4 )= 3" title="y'= 6 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}} \hspace{2em} x( 4 )= 3" data-latex="y'= 6 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}} \hspace{2em} x( 4 )= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%203" alt="y'= 6 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}} \hspace{2em} x( 4 )= 3" title="y'= 6 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}} \hspace{2em} x( 4 )= 3" data-latex="y'= 6 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}} \hspace{2em} x( 4 )= 3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}}" alt="F(t,y)= 6 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}}" title="F(t,y)= 6 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 6 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{48}{5 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{5}}}" alt="F_y= \frac{48}{5 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{5}}}" title="F_y= \frac{48}{5 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{48}{5 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D" alt="F(t,y)= 6 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}}" title="F(t,y)= 6 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 6 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{4}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B48%7D%7B5%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%7D" alt="F_y= \frac{48}{5 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{5}}}" title="F_y= \frac{48}{5 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{48}{5 \, {\left(2 \, {y} - 2 \, t + 2\right)}^{\frac{1}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3623" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3623"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= -3" alt="y'= 2 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= -3" title="y'= 2 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= -3" data-latex="y'= 2 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%2019%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%20-3" alt="y'= 2 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= -3" title="y'= 2 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= -3" data-latex="y'= 2 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= -3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{8}{3}}" alt="F(t,y)= 2 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{8}{3}}" title="F(t,y)= 2 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 2 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{8}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 16 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{5}{3}}" alt="F_y= 16 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{5}{3}}" title="F_y= 16 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{5}{3}}" data-latex="F_y= 16 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{5}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%2019%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D" alt="F(t,y)= 2 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{8}{3}}" title="F(t,y)= 2 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 2 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{8}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2016%20%5C,%20%7B%5Cleft(2%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20+%2019%5Cright)%7D%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D" alt="F_y= 16 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{5}{3}}" title="F_y= 16 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{5}{3}}" data-latex="F_y= 16 \, {\left(2 \, t + 3 \, {y} + 19\right)}^{\frac{5}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3892" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3892"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -5" alt="y'= 5 \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -5" title="y'= 5 \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -5" data-latex="y'= 5 \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20+%2028%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-6%20)=%20-5" alt="y'= 5 \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -5" title="y'= 5 \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -5" data-latex="y'= 5 \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{8}{3}}" alt="F(t,y)= 5 \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{8}{3}}" title="F(t,y)= 5 \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{8}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{80}{3} \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{5}{3}}" alt="F_y= \frac{80}{3} \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{5}{3}}" title="F_y= \frac{80}{3} \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{5}{3}}" data-latex="F_y= \frac{80}{3} \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{5}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20+%2028%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{8}{3}}" title="F(t,y)= 5 \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{8}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B80%7D%7B3%7D%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20+%2028%5Cright)%7D%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D" alt="F_y= \frac{80}{3} \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{5}{3}}" title="F_y= \frac{80}{3} \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{5}{3}}" data-latex="F_y= \frac{80}{3} \, {\left(2 \, {y} + 3 \, t + 28\right)}^{\frac{5}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-6516" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 6516"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{2}{5}} \hspace{2em} x( 2 )= 5" alt="y'= 3 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{2}{5}} \hspace{2em} x( 2 )= 5" title="y'= 3 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{2}{5}} \hspace{2em} x( 2 )= 5" data-latex="y'= 3 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{2}{5}} \hspace{2em} x( 2 )= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%202%20)=%205" alt="y'= 3 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{2}{5}} \hspace{2em} x( 2 )= 5" title="y'= 3 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{2}{5}} \hspace{2em} x( 2 )= 5" data-latex="y'= 3 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{2}{5}} \hspace{2em} x( 2 )= 5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{2}{5}}" alt="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{2}{5}}" title="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{2}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{12}{5 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{3}{5}}}" alt="F_y= \frac{12}{5 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{3}{5}}}" title="F_y= \frac{12}{5 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{12}{5 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{3}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D" alt="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{2}{5}}" title="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{2}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B12%7D%7B5%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%202%5Cright)%7D%5E%7B%5Cfrac%7B3%7D%7B5%7D%7D%7D" alt="F_y= \frac{12}{5 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{3}{5}}}" title="F_y= \frac{12}{5 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{12}{5 \, {\left(2 \, {y} - 6 \, t + 2\right)}^{\frac{3}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-8631" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 8631"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{2}{5}} \hspace{2em} x( 5 )= -4" alt="y'= 2 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{2}{5}} \hspace{2em} x( 5 )= -4" title="y'= 2 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{2}{5}} \hspace{2em} x( 5 )= -4" data-latex="y'= 2 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{2}{5}} \hspace{2em} x( 5 )= -4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2038%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%205%20)=%20-4" alt="y'= 2 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{2}{5}} \hspace{2em} x( 5 )= -4" title="y'= 2 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{2}{5}} \hspace{2em} x( 5 )= -4" data-latex="y'= 2 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{2}{5}} \hspace{2em} x( 5 )= -4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{2}{5}}" alt="F(t,y)= 2 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{2}{5}}" title="F(t,y)= 2 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 2 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{2}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{8}{5 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{3}{5}}}" alt="F_y= \frac{8}{5 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{3}{5}}}" title="F_y= \frac{8}{5 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{8}{5 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{3}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2038%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D" alt="F(t,y)= 2 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{2}{5}}" title="F(t,y)= 2 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 2 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{2}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B8%7D%7B5%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2038%5Cright)%7D%5E%7B%5Cfrac%7B3%7D%7B5%7D%7D%7D" alt="F_y= \frac{8}{5 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{3}{5}}}" title="F_y= \frac{8}{5 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{8}{5 \, {\left(2 \, {y} - 6 \, t + 38\right)}^{\frac{3}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-7910" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 7910"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= 4" alt="y'= 5 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= 4" title="y'= 5 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= 4" data-latex="y'= 5 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-6%20)=%204" alt="y'= 5 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= 4" title="y'= 5 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= 4" data-latex="y'= 5 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= 4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{8}{3}}" alt="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{8}{3}}" title="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{8}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 40 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{5}{3}}" alt="F_y= 40 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{5}{3}}" title="F_y= 40 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{5}{3}}" data-latex="F_y= 40 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{5}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{8}{3}}" title="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{8}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2040%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20+%206%5Cright)%7D%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D" alt="F_y= 40 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{5}{3}}" title="F_y= 40 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{5}{3}}" data-latex="F_y= 40 \, {\left(3 \, {y} + 3 \, t + 6\right)}^{\frac{5}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3527" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3527"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{7}{5}} \hspace{2em} x( -5 )= -6" alt="y'= 5 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{7}{5}} \hspace{2em} x( -5 )= -6" title="y'= 5 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{7}{5}} \hspace{2em} x( -5 )= -6" data-latex="y'= 5 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{7}{5}} \hspace{2em} x( -5 )= -6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%208%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%20-6" alt="y'= 5 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{7}{5}} \hspace{2em} x( -5 )= -6" title="y'= 5 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{7}{5}} \hspace{2em} x( -5 )= -6" data-latex="y'= 5 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{7}{5}} \hspace{2em} x( -5 )= -6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{7}{5}}" alt="F(t,y)= 5 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{7}{5}}" title="F(t,y)= 5 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{7}{5}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{7}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 14 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{2}{5}}" alt="F_y= 14 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{2}{5}}" title="F_y= 14 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{2}{5}}" data-latex="F_y= 14 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{2}{5}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%208%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B5%7D%7D" alt="F(t,y)= 5 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{7}{5}}" title="F(t,y)= 5 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{7}{5}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{7}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2014%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%208%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D" alt="F_y= 14 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{2}{5}}" title="F_y= 14 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{2}{5}}" data-latex="F_y= 14 \, {\left(2 \, {y} - 4 \, t - 8\right)}^{\frac{2}{5}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9206" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9206"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= 3" alt="y'= 4 \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= 3" title="y'= 4 \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= 3" data-latex="y'= 4 \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= 3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(6%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%2024%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%203" alt="y'= 4 \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= 3" title="y'= 4 \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= 3" data-latex="y'= 4 \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{8}{3}} \hspace{2em} x( -5 )= 3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{8}{3}}" alt="F(t,y)= 4 \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{8}{3}}" title="F(t,y)= 4 \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 4 \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{8}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{64}{3} \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{5}{3}}" alt="F_y= \frac{64}{3} \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{5}{3}}" title="F_y= \frac{64}{3} \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{5}{3}}" data-latex="F_y= \frac{64}{3} \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{5}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(6%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%2024%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D" alt="F(t,y)= 4 \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{8}{3}}" title="F(t,y)= 4 \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 4 \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{8}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B64%7D%7B3%7D%20%5C,%20%7B%5Cleft(6%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%2024%5Cright)%7D%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D" alt="F_y= \frac{64}{3} \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{5}{3}}" title="F_y= \frac{64}{3} \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{5}{3}}" data-latex="F_y= \frac{64}{3} \, {\left(6 \, t + 2 \, {y} + 24\right)}^{\frac{5}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-2723" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 2723"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{8}{3}} \hspace{2em} x( -3 )= 5" alt="y'= 6 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{8}{3}} \hspace{2em} x( -3 )= 5" title="y'= 6 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{8}{3}} \hspace{2em} x( -3 )= 5" data-latex="y'= 6 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{8}{3}} \hspace{2em} x( -3 )= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(5%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%205%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-3%20)=%205" alt="y'= 6 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{8}{3}} \hspace{2em} x( -3 )= 5" title="y'= 6 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{8}{3}} \hspace{2em} x( -3 )= 5" data-latex="y'= 6 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{8}{3}} \hspace{2em} x( -3 )= 5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{8}{3}}" alt="F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{8}{3}}" title="F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{8}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 32 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{5}{3}}" alt="F_y= 32 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{5}{3}}" title="F_y= 32 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{5}{3}}" data-latex="F_y= 32 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{5}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(5%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%205%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D" alt="F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{8}{3}}" title="F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 6 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{8}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2032%20%5C,%20%7B%5Cleft(5%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%205%5Cright)%7D%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D" alt="F_y= 32 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{5}{3}}" title="F_y= 32 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{5}{3}}" data-latex="F_y= 32 \, {\left(5 \, t + 2 \, {y} + 5\right)}^{\frac{5}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-7933" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 7933"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{4}{5}} \hspace{2em} x( -6 )= -6" alt="y'= 6 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{4}{5}} \hspace{2em} x( -6 )= -6" title="y'= 6 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{4}{5}} \hspace{2em} x( -6 )= -6" data-latex="y'= 6 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{4}{5}} \hspace{2em} x( -6 )= -6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%20-6%20)=%20-6" alt="y'= 6 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{4}{5}} \hspace{2em} x( -6 )= -6" title="y'= 6 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{4}{5}} \hspace{2em} x( -6 )= -6" data-latex="y'= 6 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{4}{5}} \hspace{2em} x( -6 )= -6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{4}{5}}" alt="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{4}{5}}" title="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{4}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{72}{5 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{1}{5}}}" alt="F_y= \frac{72}{5 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{1}{5}}}" title="F_y= \frac{72}{5 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{72}{5 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{1}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D" alt="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{4}{5}}" title="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{4}{5}}" data-latex="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{4}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B72%7D%7B5%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%7D" alt="F_y= \frac{72}{5 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{1}{5}}}" title="F_y= \frac{72}{5 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{1}{5}}}" data-latex="F_y= \frac{72}{5 \, {\left(3 \, {y} - 5 \, t - 12\right)}^{\frac{1}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9893" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9893"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -4" alt="y'= 2 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -4" title="y'= 2 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -4" data-latex="y'= 2 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-5%20)=%20-4" alt="y'= 2 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -4" title="y'= 2 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -4" data-latex="y'= 2 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{1}{3}} \hspace{2em} x( -5 )= -4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{1}{3}}" alt="F(t,y)= 2 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{1}{3}}" title="F(t,y)= 2 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 2 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{1}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{4}{3 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{2}{3}}}" alt="F_y= \frac{4}{3 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{2}{3}}}" title="F_y= \frac{4}{3 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{4}{3 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{2}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F(t,y)= 2 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{1}{3}}" title="F(t,y)= 2 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{1}{3}}" data-latex="F(t,y)= 2 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{1}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B4%7D%7B3%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%204%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D" alt="F_y= \frac{4}{3 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{2}{3}}}" title="F_y= \frac{4}{3 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{2}{3}}}" data-latex="F_y= \frac{4}{3 \, {\left(2 \, {y} - 4 \, t - 12\right)}^{\frac{2}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-2787" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 2787"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{7}{3}} \hspace{2em} x( -6 )= 4" alt="y'= 3 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{7}{3}} \hspace{2em} x( -6 )= 4" title="y'= 3 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{7}{3}} \hspace{2em} x( -6 )= 4" data-latex="y'= 3 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{7}{3}} \hspace{2em} x( -6 )= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%2026%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-6%20)=%204" alt="y'= 3 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{7}{3}} \hspace{2em} x( -6 )= 4" title="y'= 3 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{7}{3}} \hspace{2em} x( -6 )= 4" data-latex="y'= 3 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{7}{3}} \hspace{2em} x( -6 )= 4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{7}{3}}" alt="F(t,y)= 3 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{7}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{7}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 14 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{4}{3}}" alt="F_y= 14 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{4}{3}}" title="F_y= 14 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{4}{3}}" data-latex="F_y= 14 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{4}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%2026%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B3%7D%7D" alt="F(t,y)= 3 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{7}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{7}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{7}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2014%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%2026%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F_y= 14 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{4}{3}}" title="F_y= 14 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{4}{3}}" data-latex="F_y= 14 \, {\left(2 \, {y} - 3 \, t - 26\right)}^{\frac{4}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-8446" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 8446"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -6" alt="y'= 3 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -6" title="y'= 3 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -6" data-latex="y'= 3 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%206%20)=%20-6" alt="y'= 3 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -6" title="y'= 3 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -6" data-latex="y'= 3 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{4}{3}} \hspace{2em} x( 6 )= -6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{4}{3}}" alt="F(t,y)= 3 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{4}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{4}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 8 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{1}{3}}" alt="F_y= 8 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{1}{3}}" title="F_y= 8 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{1}{3}}" data-latex="F_y= 8 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{1}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F(t,y)= 3 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{4}{3}}" title="F(t,y)= 3 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{4}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%208%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20-%2018%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F_y= 8 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{1}{3}}" title="F_y= 8 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{1}{3}}" data-latex="F_y= 8 \, {\left(2 \, {y} + 5 \, t - 18\right)}^{\frac{1}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-6130" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 6130"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 6" alt="y'= 2 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 6" title="y'= 2 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 6" data-latex="y'= 2 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%203%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%203%20)=%206" alt="y'= 2 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 6" title="y'= 2 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 6" data-latex="y'= 2 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}} \hspace{2em} x( 3 )= 6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}}" alt="F(t,y)= 2 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}}" title="F(t,y)= 2 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 2 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{4}{{\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}}}" alt="F_y= \frac{4}{{\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}}}" title="F_y= \frac{4}{{\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{4}{{\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%203%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D" alt="F(t,y)= 2 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}}" title="F(t,y)= 2 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 2 \, {\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{2}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B4%7D%7B%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20-%203%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D" alt="F_y= \frac{4}{{\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}}}" title="F_y= \frac{4}{{\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{4}{{\left(3 \, {y} - 5 \, t - 3\right)}^{\frac{1}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3496" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3496"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 6 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= -4" alt="y'= 6 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= -4" title="y'= 6 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= -4" data-latex="y'= 6 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= -4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%206%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%2032%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%204%20)=%20-4" alt="y'= 6 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= -4" title="y'= 6 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= -4" data-latex="y'= 6 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{4}{3}} \hspace{2em} x( 4 )= -4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{4}{3}}" alt="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{4}{3}}" title="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{4}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 24 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{1}{3}}" alt="F_y= 24 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{1}{3}}" title="F_y= 24 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{1}{3}}" data-latex="F_y= 24 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{1}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%206%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%2032%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D" alt="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{4}{3}}" title="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{4}{3}}" data-latex="F(t,y)= 6 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{4}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2024%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%2032%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D" alt="F_y= 24 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{1}{3}}" title="F_y= 24 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{1}{3}}" data-latex="F_y= 24 \, {\left(3 \, {y} - 5 \, t + 32\right)}^{\frac{1}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-2248" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 2248"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{8}{5}} \hspace{2em} x( 3 )= 5" alt="y'= 5 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{8}{5}} \hspace{2em} x( 3 )= 5" title="y'= 5 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{8}{5}} \hspace{2em} x( 3 )= 5" data-latex="y'= 5 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{8}{5}} \hspace{2em} x( 3 )= 5"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20-%2025%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%203%20)=%205" alt="y'= 5 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{8}{5}} \hspace{2em} x( 3 )= 5" title="y'= 5 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{8}{5}} \hspace{2em} x( 3 )= 5" data-latex="y'= 5 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{8}{5}} \hspace{2em} x( 3 )= 5">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{8}{5}}" alt="F(t,y)= 5 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{8}{5}}" title="F(t,y)= 5 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{8}{5}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{8}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 16 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{3}{5}}" alt="F_y= 16 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{3}{5}}" title="F_y= 16 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{3}{5}}" data-latex="F_y= 16 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{3}{5}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20-%2025%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B5%7D%7D" alt="F(t,y)= 5 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{8}{5}}" title="F(t,y)= 5 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{8}{5}}" data-latex="F(t,y)= 5 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{8}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2016%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%205%20%5C,%20t%20-%2025%5Cright)%7D%5E%7B%5Cfrac%7B3%7D%7B5%7D%7D" alt="F_y= 16 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{3}{5}}" title="F_y= 16 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{3}{5}}" data-latex="F_y= 16 \, {\left(2 \, {y} + 5 \, t - 25\right)}^{\frac{3}{5}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-3661" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 3661"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 4 \, {\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -6" alt="y'= 4 \, {\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -6" title="y'= 4 \, {\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -6" data-latex="y'= 4 \, {\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%2028%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%205%20)=%20-6" alt="y'= 4 \, {\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -6" title="y'= 4 \, {\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -6" data-latex="y'= 4 \, {\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{2}{3}} \hspace{2em} x( 5 )= -6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 4 \, {\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{2}{3}}" alt="F(t,y)= 4 \, {\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{2}{3}}" title="F(t,y)= 4 \, {\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{2}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{8}{{\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{1}{3}}}" alt="F_y= \frac{8}{{\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{1}{3}}}" title="F_y= \frac{8}{{\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{8}{{\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{1}{3}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%204%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%2028%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D" alt="F(t,y)= 4 \, {\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{2}{3}}" title="F(t,y)= 4 \, {\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{2}{3}}" data-latex="F(t,y)= 4 \, {\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{2}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B8%7D%7B%7B%5Cleft(3%20%5C,%20%7By%7D%20-%202%20%5C,%20t%20+%2028%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D" alt="F_y= \frac{8}{{\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{1}{3}}}" title="F_y= \frac{8}{{\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{1}{3}}}" data-latex="F_y= \frac{8}{{\left(3 \, {y} - 2 \, t + 28\right)}^{\frac{1}{3}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-4923" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 4923"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{1}{5}} \hspace{2em} x( 2 )= 6" alt="y'= 3 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{1}{5}} \hspace{2em} x( 2 )= 6" title="y'= 3 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{1}{5}} \hspace{2em} x( 2 )= 6" data-latex="y'= 3 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{1}{5}} \hspace{2em} x( 2 )= 6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20-%2016%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%202%20)=%206" alt="y'= 3 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{1}{5}} \hspace{2em} x( 2 )= 6" title="y'= 3 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{1}{5}} \hspace{2em} x( 2 )= 6" data-latex="y'= 3 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{1}{5}} \hspace{2em} x( 2 )= 6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{1}{5}}" alt="F(t,y)= 3 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{1}{5}}" title="F(t,y)= 3 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{1}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{6}{5 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{4}{5}}}" alt="F_y= \frac{6}{5 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{4}{5}}}" title="F_y= \frac{6}{5 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{6}{5 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{4}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20-%2016%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D" alt="F(t,y)= 3 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{1}{5}}" title="F(t,y)= 3 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 3 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{1}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B6%7D%7B5%20%5C,%20%7B%5Cleft(2%20%5C,%20%7By%7D%20+%202%20%5C,%20t%20-%2016%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%7D" alt="F_y= \frac{6}{5 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{4}{5}}}" title="F_y= \frac{6}{5 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{6}{5 \, {\left(2 \, {y} + 2 \, t - 16\right)}^{\frac{4}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9596" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9596"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{1}{5}} \hspace{2em} x( 2 )= 6" alt="y'= 5 \, {\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{1}{5}} \hspace{2em} x( 2 )= 6" title="y'= 5 \, {\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{1}{5}} \hspace{2em} x( 2 )= 6" data-latex="y'= 5 \, {\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{1}{5}} \hspace{2em} x( 2 )= 6"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%202%20)=%206" alt="y'= 5 \, {\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{1}{5}} \hspace{2em} x( 2 )= 6" title="y'= 5 \, {\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{1}{5}} \hspace{2em} x( 2 )= 6" data-latex="y'= 5 \, {\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{1}{5}} \hspace{2em} x( 2 )= 6">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{1}{5}}" alt="F(t,y)= 5 \, {\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{1}{5}}" title="F(t,y)= 5 \, {\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{1}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{3}{{\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{4}{5}}}" alt="F_y= \frac{3}{{\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{4}{5}}}" title="F_y= \frac{3}{{\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{3}{{\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{4}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D" alt="F(t,y)= 5 \, {\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{1}{5}}" title="F(t,y)= 5 \, {\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{1}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B3%7D%7B%7B%5Cleft(3%20%5C,%20%7By%7D%20-%203%20%5C,%20t%20-%2012%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%7D" alt="F_y= \frac{3}{{\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{4}{5}}}" title="F_y= \frac{3}{{\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{3}{{\left(3 \, {y} - 3 \, t - 12\right)}^{\frac{4}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-9330" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 9330"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 3 \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{7}{5}} \hspace{2em} x( 5 )= -3" alt="y'= 3 \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{7}{5}} \hspace{2em} x( 5 )= -3" title="y'= 3 \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{7}{5}} \hspace{2em} x( 5 )= -3" data-latex="y'= 3 \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{7}{5}} \hspace{2em} x( 5 )= -3"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%203%20%5C,%20%7B%5Cleft(-6%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%2036%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%205%20)=%20-3" alt="y'= 3 \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{7}{5}} \hspace{2em} x( 5 )= -3" title="y'= 3 \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{7}{5}} \hspace{2em} x( 5 )= -3" data-latex="y'= 3 \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{7}{5}} \hspace{2em} x( 5 )= -3">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 3 \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{7}{5}}" alt="F(t,y)= 3 \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{7}{5}}" title="F(t,y)= 3 \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{7}{5}}" data-latex="F(t,y)= 3 \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{7}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{42}{5} \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{2}{5}}" alt="F_y= \frac{42}{5} \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{2}{5}}" title="F_y= \frac{42}{5} \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{2}{5}}" data-latex="F_y= \frac{42}{5} \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{2}{5}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%203%20%5C,%20%7B%5Cleft(-6%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%2036%5Cright)%7D%5E%7B%5Cfrac%7B7%7D%7B5%7D%7D" alt="F(t,y)= 3 \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{7}{5}}" title="F(t,y)= 3 \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{7}{5}}" data-latex="F(t,y)= 3 \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{7}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B42%7D%7B5%7D%20%5C,%20%7B%5Cleft(-6%20%5C,%20t%20+%202%20%5C,%20%7By%7D%20+%2036%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D" alt="F_y= \frac{42}{5} \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{2}{5}}" title="F_y= \frac{42}{5} \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{2}{5}}" data-latex="F_y= \frac{42}{5} \, {\left(-6 \, t + 2 \, {y} + 36\right)}^{\frac{2}{5}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-7621" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 7621"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{1}{5}} \hspace{2em} x( 5 )= 4" alt="y'= 2 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{1}{5}} \hspace{2em} x( 5 )= 4" title="y'= 2 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{1}{5}} \hspace{2em} x( 5 )= 4" data-latex="y'= 2 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{1}{5}} \hspace{2em} x( 5 )= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%2013%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%205%20)=%204" alt="y'= 2 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{1}{5}} \hspace{2em} x( 5 )= 4" title="y'= 2 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{1}{5}} \hspace{2em} x( 5 )= 4" data-latex="y'= 2 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{1}{5}} \hspace{2em} x( 5 )= 4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{1}{5}}" alt="F(t,y)= 2 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{1}{5}}" title="F(t,y)= 2 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 2 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{1}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{6}{5 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{4}{5}}}" alt="F_y= \frac{6}{5 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{4}{5}}}" title="F_y= \frac{6}{5 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{6}{5 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{4}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%2013%5Cright)%7D%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D" alt="F(t,y)= 2 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{1}{5}}" title="F(t,y)= 2 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{1}{5}}" data-latex="F(t,y)= 2 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{1}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B6%7D%7B5%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%205%20%5C,%20t%20+%2013%5Cright)%7D%5E%7B%5Cfrac%7B4%7D%7B5%7D%7D%7D" alt="F_y= \frac{6}{5 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{4}{5}}}" title="F_y= \frac{6}{5 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{4}{5}}}" data-latex="F_y= \frac{6}{5 \, {\left(3 \, {y} - 5 \, t + 13\right)}^{\frac{4}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-0946" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 0946"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -2" alt="y'= 5 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -2" title="y'= 5 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -2" data-latex="y'= 5 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -2"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(-5%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%2024%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-6%20)=%20-2" alt="y'= 5 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -2" title="y'= 5 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -2" data-latex="y'= 5 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{8}{3}} \hspace{2em} x( -6 )= -2">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{8}{3}}" alt="F(t,y)= 5 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{8}{3}}" title="F(t,y)= 5 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 5 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{8}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 40 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{5}{3}}" alt="F_y= 40 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{5}{3}}" title="F_y= 40 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{5}{3}}" data-latex="F_y= 40 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{5}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(-5%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%2024%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{8}{3}}" title="F(t,y)= 5 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 5 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{8}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2040%20%5C,%20%7B%5Cleft(-5%20%5C,%20t%20+%203%20%5C,%20%7By%7D%20-%2024%5Cright)%7D%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D" alt="F_y= 40 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{5}{3}}" title="F_y= 40 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{5}{3}}" data-latex="F_y= 40 \, {\left(-5 \, t + 3 \, {y} - 24\right)}^{\frac{5}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-6536" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 6536"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 2 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{2}{5}} \hspace{2em} x( 2 )= -4" alt="y'= 2 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{2}{5}} \hspace{2em} x( 2 )= -4" title="y'= 2 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{2}{5}} \hspace{2em} x( 2 )= -4" data-latex="y'= 2 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{2}{5}} \hspace{2em} x( 2 )= -4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%202%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2024%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D%20%5Chspace%7B2em%7D%20x(%202%20)=%20-4" alt="y'= 2 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{2}{5}} \hspace{2em} x( 2 )= -4" title="y'= 2 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{2}{5}} \hspace{2em} x( 2 )= -4" data-latex="y'= 2 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{2}{5}} \hspace{2em} x( 2 )= -4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{2}{5}}" alt="F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{2}{5}}" title="F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{2}{5}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= \frac{12}{5 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{3}{5}}}" alt="F_y= \frac{12}{5 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{3}{5}}}" title="F_y= \frac{12}{5 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{12}{5 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{3}{5}}}"/> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%202%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2024%5Cright)%7D%5E%7B%5Cfrac%7B2%7D%7B5%7D%7D" alt="F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{2}{5}}" title="F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{2}{5}}" data-latex="F(t,y)= 2 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{2}{5}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%20%5Cfrac%7B12%7D%7B5%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20-%206%20%5C,%20t%20+%2024%5Cright)%7D%5E%7B%5Cfrac%7B3%7D%7B5%7D%7D%7D" alt="F_y= \frac{12}{5 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{3}{5}}}" title="F_y= \frac{12}{5 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{3}{5}}}" data-latex="F_y= \frac{12}{5 \, {\left(3 \, {y} - 6 \, t + 24\right)}^{\frac{3}{5}}}"> is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item><item ident="X3-5789" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 5789"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>X3.</strong></p><p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'= 5 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -2 )= 4" alt="y'= 5 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -2 )= 4" title="y'= 5 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -2 )= 4" data-latex="y'= 5 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -2 )= 4"/></p></div></mattextxml><mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>X3.</strong>
</p>
<p> Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions 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'=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%20%5Chspace%7B2em%7D%20x(%20-2%20)=%204" alt="y'= 5 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -2 )= 4" title="y'= 5 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -2 )= 4" data-latex="y'= 5 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{8}{3}} \hspace{2em} x( -2 )= 4">
</p>
</div>
</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><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{8}{3}}" alt="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{8}{3}}" title="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{8}{3}}"/> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p><p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y= 40 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{5}{3}}" alt="F_y= 40 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{5}{3}}" title="F_y= 40 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{5}{3}}" data-latex="F_y= 40 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{5}{3}}"/> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p></div></mattextxml><mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F(t,y)=%205%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D" alt="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{8}{3}}" title="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{8}{3}}" data-latex="F(t,y)= 5 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{8}{3}}"> is continuous at and nearby the initial value so a solution exists for a nearby interval. </p>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?F_y=%2040%20%5C,%20%7B%5Cleft(3%20%5C,%20%7By%7D%20+%203%20%5C,%20t%20-%206%5Cright)%7D%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D" alt="F_y= 40 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{5}{3}}" title="F_y= 40 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{5}{3}}" data-latex="F_y= 40 \, {\left(3 \, {y} + 3 \, t - 6\right)}^{\frac{5}{3}}"> is continous at and nearby the initial value so the solution is unique for a nearby interval. </p>
</div>
</mattext></material></flow_mat></itemfeedback></item></objectbank>
</questestinterop>
