Determine which of the following ODEs is exact.
\[ -4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t \]
\[ -4 \, t y^{2} + 3 \, y^{2} + 6 \, y {y'} - 4 \, y = 2 \, t^{2} {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -4 \, t {y'} - 4 \, y = -6 \, t y {y'} - 3 \, y^{2} - 10 \, t \]
Its implicit solution satisfying the initial value is:
\[ 3 \, t y^{2} + 5 \, t^{2} - 4 \, t y = 12 \]