\begin{exerciseStatement}
Determine which of the following ODEs is exact.
\[ -4 \, t y^{2} - 5 \, y^{2} = -3 \, t^{2} {y'} + 4 \, t {y'} \]\[ -8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y \]
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.
\[ -8 \, y^{3} {y'} - 3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 4 \, t y^{2} + 6 \, t y \]
Its implicit solution satisfying the initial value is:
\[ 2 \, t^{2} y^{2} - 2 \, y^{4} - 3 \, t^{2} y = \left(-3\right) \]
\end{exerciseAnswer}
