0006.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 \( -3 \, t + {y'} = y - \frac{2}{3 \, t + y - 5} - 2 \)
\item \( -t y {y'} - 2 \, t^{2} - y^{2} = 0 \)
\end{enumerate}
    
\end{exerciseStatement}
    
\begin{exerciseAnswer} 

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