0001.qti

<item ident="X3-2079" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 2079">
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          <p>
            <strong>X3.</strong>
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          <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"/>
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      <mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;X3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?y'=%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"&gt;
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            <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>
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        <mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p&gt;&lt;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}}"&gt; is continuous at and nearby the initial value so a solution exists for a nearby interval. &lt;/p&gt;
  &lt;p&gt;&lt;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}}"&gt; is continous at and nearby the initial value so the solution is unique for a nearby interval. &lt;/p&gt;
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