0007.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 \( -4 \, t - {y'} - 1 = -y + \frac{1}{4 \, t - y - 3} \)
\item \( t y {y'} + 5 \, t^{2} = -y^{2} \)
\end{enumerate}
    
\end{exerciseStatement}
    
\begin{exerciseAnswer} 

\begin{enumerate}[(a)]
\item \( \frac{u {u'}}{u^{2} + 1} = 1 \)
\item \( \frac{u {u'}}{u^{2} + 5} = \frac{1}{t} \)
\end{enumerate}
    
\end{exerciseAnswer}