\begin{exerciseStatement}
Determine which of the following ODEs is exact.
\[ -6 \, t y^{2} + 10 \, t y {y'} + 6 \, t y - 5 \, y = 9 \, y^{2} {y'} \]\[ -6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
\end{exerciseStatement}
\begin{exerciseAnswer}
The following ODE is exact.
\[ -6 \, t^{2} y {y'} - 6 \, t y^{2} + 3 \, t^{2} {y'} + 10 \, t y {y'} + 6 \, t y = -5 \, y^{2} \]
Its implicit solution satisfying the initial value is:
\[ 3 \, t^{2} y^{2} - 3 \, t^{2} y - 5 \, t y^{2} = 1 \]
\end{exerciseAnswer}
