F4 - Implicit solutions for exact IVPs


Example 1 πŸ”—

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:

\[ 3 \, t^{2} y^{2} + 4 \, t^{2} y + 5 \, y^{3} = \left(-6\right) \]


Example 2 πŸ”—

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:

\[ -3 \, t^{2} y + 4 \, t y^{2} - 2 \, y^{3} = \left(-9\right) \]


Example 3 πŸ”—

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:

\[ 3 \, t^{2} y + 2 \, t y^{2} - 3 \, y^{3} = \left(-2\right) \]


Example 4 πŸ”—

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:

\[ t y^{2} + 2 \, t^{2} - 5 \, t y = 8 \]


Example 5 πŸ”—

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:

\[ -3 \, t^{2} y + t y^{2} + 2 \, t y = 2 \]


Example 6 πŸ”—

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:

\[ 5 \, t^{2} + 2 \, t y - y^{2} = 2 \]


Example 7 πŸ”—

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:

\[ 5 \, t^{4} - 3 \, t y^{2} - 5 \, t y = 13 \]


Example 8 πŸ”—

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:

\[ -y^{3} - 3 \, t^{2} - 2 \, t y = 0 \]


Example 9 πŸ”—

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:

\[ 4 \, t^{2} y - 2 \, t y + 4 \, y^{2} = 10 \]


Example 10 πŸ”—

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:

\[ -3 \, t^{2} y + 2 \, t y^{2} + 3 \, y^{2} = \left(-2\right) \]


Example 11 πŸ”—

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:

\[ -5 \, t^{2} y^{2} - 5 \, t y^{2} - 3 \, y^{2} = \left(-3\right) \]


Example 12 πŸ”—

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:

\[ -3 \, t^{2} y^{2} - 4 \, t^{2} y - t y^{2} = 0 \]


Example 13 πŸ”—

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:

\[ -t^{3} - 5 \, t y + 2 \, y^{2} = 8 \]


Example 14 πŸ”—

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:

\[ -3 \, t^{2} y + 5 \, t y^{2} + 5 \, y^{2} = 13 \]


Example 15 πŸ”—

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:

\[ 3 \, t^{4} - 4 \, t^{2} y^{2} - 2 \, y^{2} = \left(-3\right) \]


Example 16 πŸ”—

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:

\[ -2 \, y^{4} - 4 \, t y^{2} - 5 \, t^{2} = \left(-3\right) \]


Example 17 πŸ”—

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:

\[ 5 \, t^{2} y^{2} + 3 \, t y^{2} - 2 \, t y = 10 \]


Example 18 πŸ”—

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:

\[ t^{2} y^{2} - 3 \, y^{4} + 2 \, t y^{2} = 0 \]


Example 19 πŸ”—

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:

\[ 2 \, t^{2} y^{2} + t^{2} y + 3 \, t y^{2} = 0 \]


Example 20 πŸ”—

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:

\[ -3 \, t^{2} y - 5 \, y^{3} - 4 \, t y = \left(-4\right) \]


Example 21 πŸ”—

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:

\[ -5 \, t^{3} - 4 \, t^{2} y - t y^{2} = 2 \]


Example 22 πŸ”—

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:

\[ -5 \, t^{4} + 3 \, t y^{2} + 3 \, y^{3} = \left(-5\right) \]


Example 23 πŸ”—

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:

\[ -3 \, y^{4} + 2 \, t^{2} y + t y = \left(-6\right) \]


Example 24 πŸ”—

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:

\[ 2 \, t^{2} y^{2} - 3 \, t y^{2} + 3 \, y^{3} = 8 \]


Example 25 πŸ”—

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:

\[ 3 \, t^{2} y^{2} + 2 \, t y + 2 \, y^{2} = 3 \]


Example 26 πŸ”—

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:

\[ -5 \, y^{4} + 2 \, t^{3} - 5 \, t y^{2} = \left(-2\right) \]


Example 27 πŸ”—

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:

\[ 3 \, t^{2} y^{2} + 3 \, y^{4} + t^{2} y = 5 \]


Example 28 πŸ”—

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:

\[ -2 \, t^{2} y^{2} + 3 \, y^{4} + 2 \, t y^{2} = \left(-1\right) \]


Example 29 πŸ”—

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:

\[ -5 \, y^{4} - t^{2} y + t^{2} = \left(-5\right) \]


Example 30 πŸ”—

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:

\[ -5 \, t^{2} y^{2} - 3 \, t y^{2} + 4 \, t y = \left(-6\right) \]


Example 31 πŸ”—

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:

\[ -5 \, t^{2} y^{2} + 5 \, t^{2} + 4 \, t y = \left(-4\right) \]


Example 32 πŸ”—

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:

\[ 4 \, t^{2} y^{2} - t^{2} y - 5 \, t y = 8 \]


Example 33 πŸ”—

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:

\[ -3 \, t^{3} + 4 \, t^{2} y - t y = 8 \]


Example 34 πŸ”—

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:

\[ -4 \, t^{2} y^{2} - 3 \, t y^{2} - 2 \, t y = 1 \]


Example 35 πŸ”—

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:

\[ 4 \, t^{2} y^{2} + 4 \, t^{3} + 3 \, t y^{2} = \left(-3\right) \]


Example 36 πŸ”—

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:

\[ t^{2} y^{2} + 2 \, t^{2} y + 4 \, t y = \left(-1\right) \]


Example 37 πŸ”—

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:

\[ 4 \, t^{4} - 3 \, t^{2} y^{2} + 4 \, y^{4} = 5 \]


Example 38 πŸ”—

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:

\[ -3 \, t^{2} y^{2} + 2 \, t^{2} y + 4 \, t y = \left(-9\right) \]


Example 39 πŸ”—

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:

\[ -3 \, t^{4} - 5 \, t^{2} y^{2} - t^{2} y = \left(-9\right) \]


Example 40 πŸ”—

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:

\[ -t^{2} y^{2} - 3 \, t^{2} + 2 \, y^{2} = \left(-2\right) \]


Example 41 πŸ”—

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:

\[ 3 \, y^{3} - t^{2} - 2 \, t y = \left(-2\right) \]


Example 42 πŸ”—

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:

\[ -3 \, t^{2} y^{2} - 4 \, t^{3} + t^{2} y = \left(-8\right) \]


Example 43 πŸ”—

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:

\[ y^{4} + 4 \, t^{2} - 4 \, t y = 9 \]


Example 44 πŸ”—

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:

\[ -5 \, t^{2} y^{2} - 2 \, t y^{2} + 3 \, t^{2} = \left(-4\right) \]


Example 45 πŸ”—

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:

\[ -2 \, t^{2} y^{2} - 4 \, t^{2} y + 4 \, t y = \left(-10\right) \]


Example 46 πŸ”—

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:

\[ 2 \, t^{2} y^{2} + 3 \, t^{3} - 2 \, t^{2} y = 7 \]


Example 47 πŸ”—

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:

\[ -3 \, t^{2} y^{2} - 4 \, t^{2} y + 5 \, t y = \left(-12\right) \]


Example 48 πŸ”—

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:

\[ -y^{4} - 4 \, t y^{2} + 2 \, t y = \left(-7\right) \]


Example 49 πŸ”—

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:

\[ -3 \, t^{2} y + t y - 4 \, y^{2} = \left(-2\right) \]


Example 50 πŸ”—

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:

\[ 3 \, t^{4} - t^{2} y^{2} - 4 \, y^{4} = \left(-2\right) \]