Determine which of the following ODEs is exact.
\[ 6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3} \]
\[ 6 \, t y - 2 \, t {y'} = 6 \, t y^{2} + 12 \, y^{2} {y'} - 4 \, y^{2} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ 6 \, t y^{2} - 3 \, t^{2} {y'} - 6 \, t y = -6 \, t^{2} y {y'} + 4 \, t^{3} \]
Its implicit solution satisfying the initial value is:
\[ -t^{4} + 3 \, t^{2} y^{2} - 3 \, t^{2} y = 5 \]