Determine which of the following ODEs is exact.
\[ -3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'} \]
\[ -6 \, t^{2} y {y'} + 2 \, t = -4 \, t y {y'} - 6 \, t y + 4 \, t {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -3 \, t^{2} {y'} - 6 \, t y - 2 \, y^{2} = 16 \, y^{3} {y'} + 4 \, t y {y'} \]
Its implicit solution satisfying the initial value is:
\[ -4 \, y^{4} - 3 \, t^{2} y - 2 \, t y^{2} = \left(-3\right) \]