\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.
\[
y'= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}} \hspace{2em}
x( 3 )= -5 \]
\end{exerciseStatement}
\begin{exerciseAnswer}
\(F(t,y)= 6 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{8}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.
\(F_y= 48 \, {\left(3 \, t + 3 \, {y} + 6\right)}^{\frac{5}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval.
\end{exerciseAnswer}
