0011.qti

<item ident="X3-7273" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 7273">
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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(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"/>
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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(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"&gt;
  &lt;/p&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(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>
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  &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(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}}"&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%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}}}"&gt; is not continous (or even defined) at the initial value so the guaranteed solution may not be unique. &lt;/p&gt;
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