Determine which of the following ODEs is exact.
\[ -16 \, t^{3} - 8 \, t y {y'} - t {y'} = 6 \, t y^{2} + 4 \, t^{2} {y'} \]
\[ 4 \, t^{2} {y'} = -6 \, t^{2} y {y'} - 6 \, t y^{2} - 15 \, y^{2} {y'} - 8 \, t y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ 4 \, t^{2} {y'} = -6 \, t^{2} y {y'} - 6 \, t y^{2} - 15 \, y^{2} {y'} - 8 \, t y \]
Its implicit solution satisfying the initial value is:
\[ 3 \, t^{2} y^{2} + 4 \, t^{2} y + 5 \, y^{3} = \left(-6\right) \]
Determine which of the following ODEs is exact.
\[ -3 \, t^{2} {y'} = -8 \, t y {y'} + 6 \, y^{2} {y'} + 6 \, t y - 4 \, y^{2} \]
\[ -6 \, t y^{2} + 6 \, y^{2} {y'} + 6 \, t y - 4 \, y^{2} - 5 \, t {y'} = 0 \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ -3 \, t^{2} {y'} = -8 \, t y {y'} + 6 \, y^{2} {y'} + 6 \, t y - 4 \, y^{2} \]
Its implicit solution satisfying the initial value is:
\[ -3 \, t^{2} y + 4 \, t y^{2} - 2 \, y^{3} = \left(-9\right) \]
Determine which of the following ODEs is exact.
\[ 10 \, t^{2} y {y'} - 20 \, t^{3} + 6 \, t y = -4 \, t y {y'} + t {y'} \]
\[ 9 \, y^{2} {y'} - 2 \, y^{2} = 3 \, t^{2} {y'} + 4 \, t y {y'} + 6 \, t y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 9 \, y^{2} {y'} - 2 \, y^{2} = 3 \, t^{2} {y'} + 4 \, t y {y'} + 6 \, t y \]
Its implicit solution satisfying the initial value is:
\[ 3 \, t^{2} y + 2 \, t y^{2} - 3 \, y^{3} = \left(-2\right) \]
Determine which of the following ODEs is exact.
\[ -12 \, y^{3} {y'} + 4 \, t y^{2} + 5 \, t^{2} {y'} + 5 \, y = y^{2} \]
\[ -2 \, t y {y'} + 5 \, t {y'} = y^{2} + 4 \, t - 5 \, y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -2 \, t y {y'} + 5 \, t {y'} = y^{2} + 4 \, t - 5 \, y \]
Its implicit solution satisfying the initial value is:
\[ t y^{2} + 2 \, t^{2} - 5 \, t y = 8 \]
Determine which of the following ODEs is exact.
\[ -6 \, t y = -4 \, t y^{2} - 2 \, t y {y'} + 4 \, y {y'} - 2 \, y \]
\[ 6 \, t y - y^{2} - 2 \, t {y'} - 2 \, y = -3 \, t^{2} {y'} + 2 \, t y {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ 6 \, t y - y^{2} - 2 \, t {y'} - 2 \, y = -3 \, t^{2} {y'} + 2 \, t y {y'} \]
Its implicit solution satisfying the initial value is:
\[ -3 \, t^{2} y + t y^{2} + 2 \, t y = 2 \]
Determine which of the following ODEs is exact.
\[ 2 \, t^{2} {y'} + 2 \, t y {y'} - 2 \, y = 6 \, t y^{2} \]
\[ 2 \, y {y'} - 10 \, t - 2 \, y = 2 \, t {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 2 \, y {y'} - 10 \, t - 2 \, y = 2 \, t {y'} \]
Its implicit solution satisfying the initial value is:
\[ 5 \, t^{2} + 2 \, t y - y^{2} = 2 \]
Determine which of the following ODEs is exact.
\[ 3 \, y^{2} = 20 \, t^{3} - 6 \, t y {y'} - 5 \, t {y'} - 5 \, y \]
\[ 20 \, t^{3} - 9 \, y^{2} {y'} - 3 \, y^{2} = -10 \, t y^{2} - 3 \, t^{2} {y'} + 5 \, t {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 3 \, y^{2} = 20 \, t^{3} - 6 \, t y {y'} - 5 \, t {y'} - 5 \, y \]
Its implicit solution satisfying the initial value is:
\[ 5 \, t^{4} - 3 \, t y^{2} - 5 \, t y = 13 \]
Determine which of the following ODEs is exact.
\[ 10 \, t y {y'} + 2 \, t y = -8 \, t^{2} y {y'} - 2 \, y \]
\[ -2 \, t {y'} - 6 \, t = 3 \, y^{2} {y'} + 2 \, y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -2 \, t {y'} - 6 \, t = 3 \, y^{2} {y'} + 2 \, y \]
Its implicit solution satisfying the initial value is:
\[ -y^{3} - 3 \, t^{2} - 2 \, t y = 0 \]
Determine which of the following ODEs is exact.
\[ 0 = -4 \, t^{2} {y'} - 8 \, t y + 2 \, t {y'} - 8 \, y {y'} + 2 \, y \]
\[ 2 \, t^{2} y {y'} - y^{2} = 8 \, t y - 2 \, t {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 0 = -4 \, t^{2} {y'} - 8 \, t y + 2 \, t {y'} - 8 \, y {y'} + 2 \, y \]
Its implicit solution satisfying the initial value is:
\[ 4 \, t^{2} y - 2 \, t y + 4 \, y^{2} = 10 \]
Determine which of the following ODEs is exact.
\[ -4 \, t y {y'} = -3 \, t^{2} {y'} - 6 \, t y + 2 \, y^{2} + 6 \, y {y'} \]
\[ -6 \, t^{2} y {y'} - 4 \, t y {y'} = -6 \, t y + 3 \, y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ -4 \, t y {y'} = -3 \, t^{2} {y'} - 6 \, t y + 2 \, y^{2} + 6 \, y {y'} \]
Its implicit solution satisfying the initial value is:
\[ -3 \, t^{2} y + 2 \, t y^{2} + 3 \, y^{2} = \left(-2\right) \]
Determine which of the following ODEs is exact.
\[ 10 \, t^{2} y {y'} + 4 \, t^{2} {y'} + 5 \, y^{2} = -3 \, y \]
\[ -10 \, t y {y'} = 10 \, t^{2} y {y'} + 10 \, t y^{2} + 5 \, y^{2} + 6 \, y {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ -10 \, t y {y'} = 10 \, t^{2} y {y'} + 10 \, t y^{2} + 5 \, y^{2} + 6 \, y {y'} \]
Its implicit solution satisfying the initial value is:
\[ -5 \, t^{2} y^{2} - 5 \, t y^{2} - 3 \, y^{2} = \left(-3\right) \]
Determine which of the following ODEs is exact.
\[ 6 \, t^{2} y {y'} + 2 \, t y {y'} + y^{2} = -6 \, t y^{2} - 4 \, t^{2} {y'} - 8 \, t y \]
\[ 8 \, t y - 5 \, t {y'} = -6 \, t^{2} y {y'} - y^{2} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ 6 \, t^{2} y {y'} + 2 \, t y {y'} + y^{2} = -6 \, t y^{2} - 4 \, t^{2} {y'} - 8 \, t y \]
Its implicit solution satisfying the initial value is:
\[ -3 \, t^{2} y^{2} - 4 \, t^{2} y - t y^{2} = 0 \]
Determine which of the following ODEs is exact.
\[ 3 \, t^{2} + 5 \, t {y'} - 4 \, y {y'} = -5 \, y \]
\[ -10 \, t^{2} y {y'} + 2 \, t^{2} {y'} - 5 \, y = 4 \, y^{2} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 3 \, t^{2} + 5 \, t {y'} - 4 \, y {y'} = -5 \, y \]
Its implicit solution satisfying the initial value is:
\[ -t^{3} - 5 \, t y + 2 \, y^{2} = 8 \]
Determine which of the following ODEs is exact.
\[ 0 = -4 \, t^{2} y {y'} - 6 \, t y + 5 \, y^{2} + 10 \, y {y'} - y \]
\[ 3 \, t^{2} {y'} - 10 \, t y {y'} - 5 \, y^{2} - 10 \, y {y'} = -6 \, t y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ 3 \, t^{2} {y'} - 10 \, t y {y'} - 5 \, y^{2} - 10 \, y {y'} = -6 \, t y \]
Its implicit solution satisfying the initial value is:
\[ -3 \, t^{2} y + 5 \, t y^{2} + 5 \, y^{2} = 13 \]
Determine which of the following ODEs is exact.
\[ 5 \, y = 12 \, t^{3} - 8 \, t y^{2} - t^{2} {y'} - 4 \, t y {y'} - 4 \, y {y'} \]
\[ 8 \, t^{2} y {y'} - 12 \, t^{3} = -8 \, t y^{2} - 4 \, y {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 8 \, t^{2} y {y'} - 12 \, t^{3} = -8 \, t y^{2} - 4 \, y {y'} \]
Its implicit solution satisfying the initial value is:
\[ 3 \, t^{4} - 4 \, t^{2} y^{2} - 2 \, y^{2} = \left(-3\right) \]
Determine which of the following ODEs is exact.
\[ -10 \, t + 3 \, y = -6 \, t^{2} y {y'} - t^{2} {y'} + 8 \, t y {y'} \]
\[ 8 \, t y {y'} + 4 \, y^{2} + 10 \, t = -8 \, y^{3} {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 8 \, t y {y'} + 4 \, y^{2} + 10 \, t = -8 \, y^{3} {y'} \]
Its implicit solution satisfying the initial value is:
\[ -2 \, y^{4} - 4 \, t y^{2} - 5 \, t^{2} = \left(-3\right) \]
Determine which of the following ODEs is exact.
\[ -10 \, t^{2} y {y'} - 10 \, t y^{2} + 2 \, t {y'} + 2 \, y = 6 \, t y {y'} + 3 \, y^{2} \]
\[ -2 \, t^{2} {y'} = 10 \, t y^{2} + 3 \, y^{2} - 2 \, t {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -10 \, t^{2} y {y'} - 10 \, t y^{2} + 2 \, t {y'} + 2 \, y = 6 \, t y {y'} + 3 \, y^{2} \]
Its implicit solution satisfying the initial value is:
\[ 5 \, t^{2} y^{2} + 3 \, t y^{2} - 2 \, t y = 10 \]
Determine which of the following ODEs is exact.
\[ -4 \, t^{3} - 2 \, t y^{2} - 4 \, t y {y'} = -2 \, t^{2} {y'} + 5 \, t {y'} \]
\[ 2 \, t^{2} y {y'} + 2 \, y^{2} = 12 \, y^{3} {y'} - 2 \, t y^{2} - 4 \, t y {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ 2 \, t^{2} y {y'} + 2 \, y^{2} = 12 \, y^{3} {y'} - 2 \, t y^{2} - 4 \, t y {y'} \]
Its implicit solution satisfying the initial value is:
\[ t^{2} y^{2} - 3 \, y^{4} + 2 \, t y^{2} = 0 \]
Determine which of the following ODEs is exact.
\[ -4 \, t^{2} y {y'} - 4 \, t y^{2} - t^{2} {y'} - 2 \, t y - 3 \, y^{2} = 6 \, t y {y'} \]
\[ -4 \, t y^{2} - t^{2} {y'} = 3 \, y^{2} + 6 \, y {y'} - 4 \, y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ -4 \, t^{2} y {y'} - 4 \, t y^{2} - t^{2} {y'} - 2 \, t y - 3 \, y^{2} = 6 \, t y {y'} \]
Its implicit solution satisfying the initial value is:
\[ 2 \, t^{2} y^{2} + t^{2} y + 3 \, t y^{2} = 0 \]
Determine which of the following ODEs is exact.
\[ -3 \, t^{2} {y'} = -4 \, t^{2} y {y'} - 8 \, t y {y'} + 6 \, t^{2} + 4 \, y \]
\[ -6 \, t y - 4 \, t {y'} - 4 \, y = 3 \, t^{2} {y'} + 15 \, y^{2} {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ -6 \, t y - 4 \, t {y'} - 4 \, y = 3 \, t^{2} {y'} + 15 \, y^{2} {y'} \]
Its implicit solution satisfying the initial value is:
\[ -3 \, t^{2} y - 5 \, y^{3} - 4 \, t y = \left(-4\right) \]
Determine which of the following ODEs is exact.
\[ -8 \, t y - y^{2} = -2 \, t^{2} y {y'} - 4 \, t {y'} \]
\[ 2 \, t y {y'} + 8 \, t y + y^{2} = -4 \, t^{2} {y'} - 15 \, t^{2} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 2 \, t y {y'} + 8 \, t y + y^{2} = -4 \, t^{2} {y'} - 15 \, t^{2} \]
Its implicit solution satisfying the initial value is:
\[ -5 \, t^{3} - 4 \, t^{2} y - t y^{2} = 2 \]
Determine which of the following ODEs is exact.
\[ -6 \, t^{2} y {y'} + 9 \, y^{2} {y'} + 3 \, y^{2} - 4 \, y = 20 \, t^{3} + 5 \, t^{2} {y'} \]
\[ 20 \, t^{3} - 6 \, t y {y'} = 9 \, y^{2} {y'} + 3 \, y^{2} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ 20 \, t^{3} - 6 \, t y {y'} = 9 \, y^{2} {y'} + 3 \, y^{2} \]
Its implicit solution satisfying the initial value is:
\[ -5 \, t^{4} + 3 \, t y^{2} + 3 \, y^{3} = \left(-5\right) \]
Determine which of the following ODEs is exact.
\[ -4 \, t y = -12 \, y^{3} {y'} + 2 \, t^{2} {y'} + t {y'} + y \]
\[ -2 \, t^{2} y {y'} - 12 \, y^{3} {y'} - 16 \, t^{3} + 6 \, t y {y'} = -4 \, t y - y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -4 \, t y = -12 \, y^{3} {y'} + 2 \, t^{2} {y'} + t {y'} + y \]
Its implicit solution satisfying the initial value is:
\[ -3 \, y^{4} + 2 \, t^{2} y + t y = \left(-6\right) \]
Determine which of the following ODEs is exact.
\[ -9 \, y^{2} {y'} + 3 \, y^{2} = 4 \, t^{2} y {y'} + 4 \, t y^{2} - 6 \, t y {y'} \]
\[ -4 \, t y^{2} = 3 \, t^{2} {y'} + 9 \, y^{2} {y'} - 15 \, t^{2} - 3 \, y^{2} + t {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ -9 \, y^{2} {y'} + 3 \, y^{2} = 4 \, t^{2} y {y'} + 4 \, t y^{2} - 6 \, t y {y'} \]
Its implicit solution satisfying the initial value is:
\[ 2 \, t^{2} y^{2} - 3 \, t y^{2} + 3 \, y^{3} = 8 \]
Determine which of the following ODEs is exact.
\[ -6 \, t y^{2} + 10 \, t y {y'} - 2 \, t {y'} = -8 \, t y \]
\[ -6 \, t^{2} y {y'} - 2 \, t {y'} - 2 \, y = 6 \, t y^{2} + 4 \, y {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ -6 \, t^{2} y {y'} - 2 \, t {y'} - 2 \, y = 6 \, t y^{2} + 4 \, y {y'} \]
Its implicit solution satisfying the initial value is:
\[ 3 \, t^{2} y^{2} + 2 \, t y + 2 \, y^{2} = 3 \]
Determine which of the following ODEs is exact.
\[ -20 \, y^{3} {y'} + 6 \, t^{2} - 5 \, y^{2} = 10 \, t y {y'} \]
\[ -8 \, t^{2} y {y'} + 20 \, y^{3} {y'} + 5 \, t {y'} = 6 \, t^{2} - 2 \, t y - 5 \, y^{2} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ -20 \, y^{3} {y'} + 6 \, t^{2} - 5 \, y^{2} = 10 \, t y {y'} \]
Its implicit solution satisfying the initial value is:
\[ -5 \, y^{4} + 2 \, t^{3} - 5 \, t y^{2} = \left(-2\right) \]
Determine which of the following ODEs is exact.
\[ -2 \, t y = 6 \, t^{2} y {y'} - 3 \, y^{2} - 5 \, t {y'} \]
\[ -6 \, t y^{2} = 6 \, t^{2} y {y'} + 12 \, y^{3} {y'} + t^{2} {y'} + 2 \, t y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -6 \, t y^{2} = 6 \, t^{2} y {y'} + 12 \, y^{3} {y'} + t^{2} {y'} + 2 \, t y \]
Its implicit solution satisfying the initial value is:
\[ 3 \, t^{2} y^{2} + 3 \, y^{4} + t^{2} y = 5 \]
Determine which of the following ODEs is exact.
\[ -4 \, t y {y'} - 2 \, t y + 3 \, t {y'} = -4 \, t y^{2} \]
\[ 12 \, y^{3} {y'} - 4 \, t y^{2} = 4 \, t^{2} y {y'} - 4 \, t y {y'} - 2 \, y^{2} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 12 \, y^{3} {y'} - 4 \, t y^{2} = 4 \, t^{2} y {y'} - 4 \, t y {y'} - 2 \, y^{2} \]
Its implicit solution satisfying the initial value is:
\[ -2 \, t^{2} y^{2} + 3 \, y^{4} + 2 \, t y^{2} = \left(-1\right) \]
Determine which of the following ODEs is exact.
\[ 20 \, y^{3} {y'} + t^{2} {y'} - 2 \, t = -2 \, t y \]
\[ 20 \, y^{3} {y'} - y^{2} - 2 \, t = 4 \, t^{2} y {y'} - 2 \, t y + 2 \, t {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 20 \, y^{3} {y'} + t^{2} {y'} - 2 \, t = -2 \, t y \]
Its implicit solution satisfying the initial value is:
\[ -5 \, y^{4} - t^{2} y + t^{2} = \left(-5\right) \]
Determine which of the following ODEs is exact.
\[ -6 \, t y {y'} - 3 \, y^{2} + 4 \, y = 10 \, t^{2} y {y'} + 10 \, t y^{2} - 4 \, t {y'} \]
\[ 10 \, t y^{2} - 5 \, t^{2} {y'} = -3 \, y^{2} + 4 \, t {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ -6 \, t y {y'} - 3 \, y^{2} + 4 \, y = 10 \, t^{2} y {y'} + 10 \, t y^{2} - 4 \, t {y'} \]
Its implicit solution satisfying the initial value is:
\[ -5 \, t^{2} y^{2} - 3 \, t y^{2} + 4 \, t y = \left(-6\right) \]
Determine which of the following ODEs is exact.
\[ -10 \, t^{2} y {y'} - 10 \, t y^{2} + 10 \, t = -4 \, t {y'} - 4 \, y \]
\[ 0 = 10 \, t^{2} y {y'} + 2 \, t^{2} {y'} + 4 \, t y {y'} - 10 \, t - 4 \, y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ -10 \, t^{2} y {y'} - 10 \, t y^{2} + 10 \, t = -4 \, t {y'} - 4 \, y \]
Its implicit solution satisfying the initial value is:
\[ -5 \, t^{2} y^{2} + 5 \, t^{2} + 4 \, t y = \left(-4\right) \]
Determine which of the following ODEs is exact.
\[ -t^{2} {y'} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 2 \, t y + 5 \, t {y'} + 5 \, y \]
\[ 4 \, t y {y'} + 2 \, t y + 5 \, y = 8 \, t^{2} y {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ -t^{2} {y'} = -8 \, t^{2} y {y'} - 8 \, t y^{2} + 2 \, t y + 5 \, t {y'} + 5 \, y \]
Its implicit solution satisfying the initial value is:
\[ 4 \, t^{2} y^{2} - t^{2} y - 5 \, t y = 8 \]
Determine which of the following ODEs is exact.
\[ -6 \, t y^{2} - 4 \, t y {y'} = 9 \, t^{2} - 8 \, t y + t {y'} - 8 \, y {y'} \]
\[ 4 \, t^{2} {y'} - t {y'} - y = 9 \, t^{2} - 8 \, t y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 4 \, t^{2} {y'} - t {y'} - y = 9 \, t^{2} - 8 \, t y \]
Its implicit solution satisfying the initial value is:
\[ -3 \, t^{3} + 4 \, t^{2} y - t y = 8 \]
Determine which of the following ODEs is exact.
\[ 15 \, y^{2} {y'} = -8 \, t^{2} y {y'} + 20 \, t^{3} + 3 \, t^{2} {y'} - 3 \, y^{2} - 2 \, y \]
\[ 8 \, t^{2} y {y'} + 3 \, y^{2} + 2 \, y = -8 \, t y^{2} - 6 \, t y {y'} - 2 \, t {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 8 \, t^{2} y {y'} + 3 \, y^{2} + 2 \, y = -8 \, t y^{2} - 6 \, t y {y'} - 2 \, t {y'} \]
Its implicit solution satisfying the initial value is:
\[ -4 \, t^{2} y^{2} - 3 \, t y^{2} - 2 \, t y = 1 \]
Determine which of the following ODEs is exact.
\[ -12 \, t^{2} - 10 \, t y - t {y'} = 8 \, t y^{2} + 6 \, t y {y'} - 6 \, y {y'} \]
\[ -6 \, t y {y'} - 12 \, t^{2} - 3 \, y^{2} = 8 \, t^{2} y {y'} + 8 \, t y^{2} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ -6 \, t y {y'} - 12 \, t^{2} - 3 \, y^{2} = 8 \, t^{2} y {y'} + 8 \, t y^{2} \]
Its implicit solution satisfying the initial value is:
\[ 4 \, t^{2} y^{2} + 4 \, t^{3} + 3 \, t y^{2} = \left(-3\right) \]
Determine which of the following ODEs is exact.
\[ -4 \, y = 20 \, y^{3} {y'} + 2 \, t y^{2} + 2 \, t^{2} {y'} + 8 \, t y {y'} + 6 \, t^{2} \]
\[ 2 \, t^{2} y {y'} + 2 \, t^{2} {y'} + 4 \, t y = -2 \, t y^{2} - 4 \, t {y'} - 4 \, y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 2 \, t^{2} y {y'} + 2 \, t^{2} {y'} + 4 \, t y = -2 \, t y^{2} - 4 \, t {y'} - 4 \, y \]
Its implicit solution satisfying the initial value is:
\[ t^{2} y^{2} + 2 \, t^{2} y + 4 \, t y = \left(-1\right) \]
Determine which of the following ODEs is exact.
\[ -6 \, t^{2} y {y'} + 16 \, t^{3} = -16 \, y^{3} {y'} + 6 \, t y^{2} \]
\[ -16 \, t^{3} - 6 \, t y {y'} + 4 \, y = -6 \, t^{2} y {y'} + t^{2} {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -6 \, t^{2} y {y'} + 16 \, t^{3} = -16 \, y^{3} {y'} + 6 \, t y^{2} \]
Its implicit solution satisfying the initial value is:
\[ 4 \, t^{4} - 3 \, t^{2} y^{2} + 4 \, y^{4} = 5 \]
Determine which of the following ODEs is exact.
\[ -2 \, t^{2} {y'} = -6 \, t^{2} y {y'} - 6 \, t y^{2} + 4 \, t y + 4 \, t {y'} + 4 \, y \]
\[ 4 \, t y = 6 \, t^{2} y {y'} - 6 \, y^{2} {y'} + 5 \, y^{2} - 4 \, y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -2 \, t^{2} {y'} = -6 \, t^{2} y {y'} - 6 \, t y^{2} + 4 \, t y + 4 \, t {y'} + 4 \, y \]
Its implicit solution satisfying the initial value is:
\[ -3 \, t^{2} y^{2} + 2 \, t^{2} y + 4 \, t y = \left(-9\right) \]
Determine which of the following ODEs is exact.
\[ -8 \, y^{3} {y'} + 10 \, t y^{2} + 2 \, t y {y'} + 2 \, t y = 2 \, y \]
\[ 12 \, t^{3} + 10 \, t y^{2} + t^{2} {y'} + 2 \, t y = -10 \, t^{2} y {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 12 \, t^{3} + 10 \, t y^{2} + t^{2} {y'} + 2 \, t y = -10 \, t^{2} y {y'} \]
Its implicit solution satisfying the initial value is:
\[ -3 \, t^{4} - 5 \, t^{2} y^{2} - t^{2} y = \left(-9\right) \]
Determine which of the following ODEs is exact.
\[ 2 \, t y^{2} - 4 \, y^{2} + 3 \, t {y'} + 6 \, t = 2 \, t^{2} {y'} + 4 \, y {y'} \]
\[ 2 \, t^{2} y {y'} + 2 \, t y^{2} = 4 \, y {y'} - 6 \, t \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 2 \, t^{2} y {y'} + 2 \, t y^{2} = 4 \, y {y'} - 6 \, t \]
Its implicit solution satisfying the initial value is:
\[ -t^{2} y^{2} - 3 \, t^{2} + 2 \, y^{2} = \left(-2\right) \]
Determine which of the following ODEs is exact.
\[ 8 \, t^{2} y {y'} + 9 \, y^{2} {y'} - y^{2} = t^{2} {y'} + 2 \, t + 2 \, y \]
\[ 9 \, y^{2} {y'} - 2 \, y = 2 \, t {y'} + 2 \, t \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ 9 \, y^{2} {y'} - 2 \, y = 2 \, t {y'} + 2 \, t \]
Its implicit solution satisfying the initial value is:
\[ 3 \, y^{3} - t^{2} - 2 \, t y = \left(-2\right) \]
Determine which of the following ODEs is exact.
\[ 6 \, t^{2} y {y'} + 12 \, t^{2} - t {y'} = 8 \, t y {y'} + 2 \, t y \]
\[ -6 \, t y^{2} + t^{2} {y'} - 12 \, t^{2} = 6 \, t^{2} y {y'} - 2 \, t y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -6 \, t y^{2} + t^{2} {y'} - 12 \, t^{2} = 6 \, t^{2} y {y'} - 2 \, t y \]
Its implicit solution satisfying the initial value is:
\[ -3 \, t^{2} y^{2} - 4 \, t^{3} + t^{2} y = \left(-8\right) \]
Determine which of the following ODEs is exact.
\[ 4 \, y^{3} {y'} = 4 \, t {y'} - 8 \, t + 4 \, y \]
\[ -2 \, t y {y'} + 2 \, t y + 4 \, t {y'} = 4 \, t^{2} y {y'} + 8 \, t \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ 4 \, y^{3} {y'} = 4 \, t {y'} - 8 \, t + 4 \, y \]
Its implicit solution satisfying the initial value is:
\[ y^{4} + 4 \, t^{2} - 4 \, t y = 9 \]
Determine which of the following ODEs is exact.
\[ 4 \, t^{2} {y'} + 2 \, y^{2} - 8 \, y {y'} = -10 \, t^{2} y {y'} + 6 \, t + 2 \, y \]
\[ -4 \, t y {y'} - 2 \, y^{2} + 6 \, t = 10 \, t^{2} y {y'} + 10 \, t y^{2} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -4 \, t y {y'} - 2 \, y^{2} + 6 \, t = 10 \, t^{2} y {y'} + 10 \, t y^{2} \]
Its implicit solution satisfying the initial value is:
\[ -5 \, t^{2} y^{2} - 2 \, t y^{2} + 3 \, t^{2} = \left(-4\right) \]
Determine which of the following ODEs is exact.
\[ 4 \, t y^{2} + 6 \, t y {y'} - 4 \, t {y'} = -8 \, t y \]
\[ 8 \, t y = -4 \, t^{2} y {y'} - 4 \, t y^{2} - 4 \, t^{2} {y'} + 4 \, t {y'} + 4 \, y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 8 \, t y = -4 \, t^{2} y {y'} - 4 \, t y^{2} - 4 \, t^{2} {y'} + 4 \, t {y'} + 4 \, y \]
Its implicit solution satisfying the initial value is:
\[ -2 \, t^{2} y^{2} - 4 \, t^{2} y + 4 \, t y = \left(-10\right) \]
Determine which of the following ODEs is exact.
\[ -4 \, t^{2} y {y'} + 16 \, y^{3} {y'} = 9 \, t^{2} - 4 \, t y - 4 \, y^{2} + 2 \, t {y'} \]
\[ -4 \, t^{2} y {y'} - 4 \, t y^{2} - 9 \, t^{2} + 4 \, t y = -2 \, t^{2} {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -4 \, t^{2} y {y'} - 4 \, t y^{2} - 9 \, t^{2} + 4 \, t y = -2 \, t^{2} {y'} \]
Its implicit solution satisfying the initial value is:
\[ 2 \, t^{2} y^{2} + 3 \, t^{3} - 2 \, t^{2} y = 7 \]
Determine which of the following ODEs is exact.
\[ 6 \, t^{2} y {y'} + 8 \, t y = 16 \, y^{3} {y'} + 2 \, y^{2} + 5 \, y \]
\[ -6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ -6 \, t y^{2} - 4 \, t^{2} {y'} + 5 \, t {y'} + 5 \, y = 6 \, t^{2} y {y'} + 8 \, t y \]
Its implicit solution satisfying the initial value is:
\[ -3 \, t^{2} y^{2} - 4 \, t^{2} y + 5 \, t y = \left(-12\right) \]
Determine which of the following ODEs is exact.
\[ -10 \, t y^{2} + 3 \, t^{2} {y'} - 8 \, t y {y'} + 2 \, y = 0 \]
\[ -4 \, y^{3} {y'} - 4 \, y^{2} + 2 \, y = 8 \, t y {y'} - 2 \, t {y'} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ -4 \, y^{3} {y'} - 4 \, y^{2} + 2 \, y = 8 \, t y {y'} - 2 \, t {y'} \]
Its implicit solution satisfying the initial value is:
\[ -y^{4} - 4 \, t y^{2} + 2 \, t y = \left(-7\right) \]
Determine which of the following ODEs is exact.
\[ 3 \, t^{2} {y'} - t {y'} + 8 \, y {y'} = -6 \, t y + y \]
\[ -3 \, t^{2} {y'} + y = 10 \, t^{2} y {y'} - 8 \, t y {y'} + 4 \, t \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( 1 )= -1 \).
Answer:
The following ODE is exact.
\[ 3 \, t^{2} {y'} - t {y'} + 8 \, y {y'} = -6 \, t y + y \]
Its implicit solution satisfying the initial value is:
\[ -3 \, t^{2} y + t y - 4 \, y^{2} = \left(-2\right) \]
Determine which of the following ODEs is exact.
\[ -12 \, t^{3} + 2 \, t y^{2} + 5 \, y^{2} - 3 \, t {y'} = -16 \, y^{3} {y'} - 2 \, t^{2} {y'} \]
\[ 2 \, t^{2} y {y'} = -16 \, y^{3} {y'} + 12 \, t^{3} - 2 \, t y^{2} \]
Then find an implicit solution for this exact ODE satisfying the initial value \(y( -1 )= 1 \).
Answer:
The following ODE is exact.
\[ 2 \, t^{2} y {y'} = -16 \, y^{3} {y'} + 12 \, t^{3} - 2 \, t y^{2} \]
Its implicit solution satisfying the initial value is:
\[ 3 \, t^{4} - t^{2} y^{2} - 4 \, y^{4} = \left(-2\right) \]