0001.tex

\begin{exerciseStatement}


 Apply each of the substitutions \(u=\frac{y}{t}\) and \(u=at+by+c\) to one of the following ODEs, and show that each result is separable. (Do not fully solve either ODE.) 


\begin{enumerate}[(a)]
\item \( {y'} + \frac{3}{3 \, t - y + 2} + 5 = -3 \, t + y \)
\item \( -2 \, t^{2} = -t y {y'} - y^{2} \)
\end{enumerate}
    
\end{exerciseStatement}
    
\begin{exerciseAnswer} 

\begin{enumerate}[(a)]
\item \( \frac{u {u'}}{u^{2} + 3} = 1 \)
\item \( \frac{u {u'}}{u^{2} - 2} = \frac{1}{t} \)
\end{enumerate}
    
\end{exerciseAnswer}