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