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