Determine which of the following ODEs is exact.
\[ -9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y \]
\[ -9 \, y^{2} {y'} = 4 \, t y^{2} + t^{2} {y'} + 6 \, t y {y'} + 12 \, t^{2} - y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -9 \, y^{2} {y'} - 12 \, t^{2} = -t {y'} - y \]
Its implicit solution satisfying the initial value is:
\[ -4 \, t^{3} - 3 \, y^{3} + t y = \left(-2\right) \]