0003.tex

\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. 

\[
      y'= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}} \hspace{2em}
      x( 2 )= 3 \]
\end{exerciseStatement}
    
\begin{exerciseAnswer} 


\(F(t,y)= 5 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{8}{3}} \) is continuous at and nearby the initial value so a solution exists for a nearby interval. 



\(F_y= 40 \, {\left(4 \, t + 3 \, {y} - 17\right)}^{\frac{5}{3}} \) is continous at and nearby the initial value so the solution is unique for a nearby interval. 


\end{exerciseAnswer}