Determine which of the following ODEs is exact.
\[ 0 = -8 \, t^{2} y {y'} + 2 \, t^{2} {y'} + 3 \, y^{2} + 5 \, y \]
\[ 6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ 6 \, t y {y'} + 3 \, y^{2} = -2 \, t^{2} {y'} - 4 \, t y - 5 \, t {y'} - 5 \, y \]
Its implicit solution satisfying the initial value is:
\[ -2 \, t^{2} y - 3 \, t y^{2} - 5 \, t y = 4 \]