Determine which of the following ODEs is exact.
\[ -4 \, t^{2} y {y'} - 4 \, y^{3} {y'} - 3 \, y^{2} = t^{2} {y'} - 15 \, t^{2} + y \]
\[ -4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -4 \, y^{3} {y'} - 6 \, t y {y'} + 15 \, t^{2} = 3 \, y^{2} \]
Its implicit solution satisfying the initial value is:
\[ y^{4} - 5 \, t^{3} + 3 \, t y^{2} = \left(-1\right) \]