## F4 - Implicit solutions for exact IVPs

#### Example 1 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( -1 )= 1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( 1 )= -1$$.

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 π

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 $$y( -1 )= 1$$.

$2 \, t^{2} y {y'} = -16 \, y^{3} {y'} + 12 \, t^{3} - 2 \, t y^{2}$
$3 \, t^{4} - t^{2} y^{2} - 4 \, y^{4} = \left(-2\right)$