Determine which of the following ODEs is exact.
\[ 0 = 20 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, t \]
\[ 6 \, t^{2} y {y'} - 4 \, t y {y'} - y = -8 \, t y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ 0 = 20 \, y^{3} {y'} - 4 \, t^{2} {y'} - 8 \, t y - 10 \, t \]
Its implicit solution satisfying the initial value is:
\[ 5 \, y^{4} - 4 \, t^{2} y - 5 \, t^{2} = 4 \]