\begin{exerciseStatement}
Determine which of the following ODEs is exact.
\[ 16 \, y^{3} {y'} + 2 \, t y {y'} - 8 \, t y - 5 \, t {y'} = 10 \, t y^{2} + 10 \, t \]\[ 16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y \]
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.
\[ 16 \, y^{3} {y'} - 4 \, t^{2} {y'} - 10 \, t = 8 \, t y \]
Its implicit solution satisfying the initial value is:
\[ 4 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = \left(-5\right) \]
\end{exerciseAnswer}
