\begin{exerciseStatement}
Explain what the Existence and Uniqueness Theorem for First Order IVPs guarantees about the existence and uniqueness of solutions for the following IVP.
\[
y'= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}} \hspace{2em}
x( -4 )= -2 \]
\end{exerciseStatement}
\begin{exerciseAnswer}
\(F(t,y)= 6 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{4}{5}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.
\(F_y= \frac{72}{5 \, {\left(5 \, t + 3 \, {y} + 26\right)}^{\frac{1}{5}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.
\end{exerciseAnswer}
