<item ident="X3-7896" title="X3 | Existence/uniqueness theorem for first-order IVPs | ver. 7896">
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<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>
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<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">
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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)= 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>
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<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>
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