\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.
\[
y'= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}} \hspace{2em}
x( 4 )= 3 \]
\end{exerciseStatement}
\begin{exerciseAnswer}
\(F(t,y)= 4 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{2}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval.
\(F_y= \frac{16}{3 \, {\left(2 \, {y} - 3 \, t + 6\right)}^{\frac{1}{3}}} \) is not continous (or even defined) at the initial value so the guaranteed solution may not be unique.
\end{exerciseAnswer}
