\begin{exerciseStatement}
Determine which of the following ODEs is exact.
\[ 3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'} \]\[ 12 \, t^{3} - 2 \, y = -8 \, t^{2} y {y'} - 4 \, y^{3} {y'} + 8 \, t y {y'} + 6 \, t y \]
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.
\[ 3 \, t^{2} {y'} + 6 \, t y + 2 \, t {y'} + 2 \, y = 4 \, y^{3} {y'} \]
Its implicit solution satisfying the initial value is:
\[ -y^{4} + 3 \, t^{2} y + 2 \, t y = 0 \]
\end{exerciseAnswer}
