## F6: Exact ODEs (ver. 1)

Determine which of the following ODEs is exact.

$( -6 \, t y - y )+( -4 \, t^{2} y - 2 \, t y )y'=0$

$( -6 \, t y - y^{2} )+( 4 \, y^{3} - 3 \, t^{2} - 2 \, t y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= 1$$.

The following ODE is exact.

$( -6 \, t y - y^{2} )+( 4 \, y^{3} - 3 \, t^{2} - 2 \, t y )y'=0$

Its implicit solution satisfying $$y( 0 )= 1$$ is:

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

## F6: Exact ODEs (ver. 2)

Determine which of the following ODEs is exact.

$( 6 \, t y + 2 \, y^{2} )+( -10 \, t^{2} y + 5 \, t )y'=0$

$( 2 \, y^{2} + 5 \, y )+( -8 \, y^{3} + 4 \, t y + 5 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= 0$$.

The following ODE is exact.

$( 2 \, y^{2} + 5 \, y )+( -8 \, y^{3} + 4 \, t y + 5 \, t )y'=0$

Its implicit solution satisfying $$y( 1 )= 0$$ is:

$-2 \, y^{4} + 2 \, t y^{2} + 5 \, t y = 0$

## F6: Exact ODEs (ver. 3)

Determine which of the following ODEs is exact.

$( 8 \, t y + 5 \, y^{2} )+( 4 \, t^{2} y - 4 \, t )y'=0$

$( 8 \, t y + 5 \, y^{2} - 4 \, y )+( 4 \, t^{2} + 10 \, t y - 4 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( -1 )= 1$$.

The following ODE is exact.

$( 8 \, t y + 5 \, y^{2} - 4 \, y )+( 4 \, t^{2} + 10 \, t y - 4 \, t )y'=0$

Its implicit solution satisfying $$y( -1 )= 1$$ is:

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

## F6: Exact ODEs (ver. 4)

Determine which of the following ODEs is exact.

$( 8 \, t y - y^{2} - 3 \, y )+( 4 \, t^{2} - 2 \, t y - 3 \, t )y'=0$

$( -y^{2} - 8 \, t - 3 \, y )+( 6 \, t^{2} y + 4 \, t^{2} + 6 \, y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= 0$$.

The following ODE is exact.

$( 8 \, t y - y^{2} - 3 \, y )+( 4 \, t^{2} - 2 \, t y - 3 \, t )y'=0$

Its implicit solution satisfying $$y( 1 )= 0$$ is:

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

## F6: Exact ODEs (ver. 5)

Determine which of the following ODEs is exact.

$( -2 \, t y^{2} + 2 \, t - 3 \, y )+( -2 \, t^{2} y - 3 \, t )y'=0$

$( -8 \, t y + 2 \, t - 3 \, y )+( -2 \, t^{2} y + 4 \, t y + 4 \, y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= 0$$.

The following ODE is exact.

$( -2 \, t y^{2} + 2 \, t - 3 \, y )+( -2 \, t^{2} y - 3 \, t )y'=0$

Its implicit solution satisfying $$y( 1 )= 0$$ is:

$-t^{2} y^{2} + t^{2} - 3 \, t y = 1$

## F6: Exact ODEs (ver. 6)

Determine which of the following ODEs is exact.

$( 15 \, t^{2} + 4 \, y )+( 6 \, y^{2} + 4 \, t )y'=0$

$( 15 \, t^{2} + y^{2} + 4 \, y )+( 8 \, t^{2} y - t^{2} + 6 \, y^{2} )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= 0$$.

The following ODE is exact.

$( 15 \, t^{2} + 4 \, y )+( 6 \, y^{2} + 4 \, t )y'=0$

Its implicit solution satisfying $$y( 1 )= 0$$ is:

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

## F6: Exact ODEs (ver. 7)

Determine which of the following ODEs is exact.

$( 15 \, t^{2} - 4 \, t y - 5 \, y^{2} )+( -2 \, t^{2} - 10 \, t y )y'=0$

$( -4 \, t y - 5 \, y^{2} + 4 \, y )+( -10 \, t^{2} y + 15 \, y^{2} )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( -1 )= 0$$.

The following ODE is exact.

$( 15 \, t^{2} - 4 \, t y - 5 \, y^{2} )+( -2 \, t^{2} - 10 \, t y )y'=0$

Its implicit solution satisfying $$y( -1 )= 0$$ is:

$5 \, t^{3} - 2 \, t^{2} y - 5 \, t y^{2} = -5$

## F6: Exact ODEs (ver. 8)

Determine which of the following ODEs is exact.

$( -2 \, t y^{2} - 8 \, t y + 3 \, y )+( -2 \, t^{2} y - 4 \, t^{2} + 3 \, t )y'=0$

$( -2 \, t y^{2} - 8 \, t y + y^{2} )+( 4 \, y^{3} + 3 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= 0$$.

The following ODE is exact.

$( -2 \, t y^{2} - 8 \, t y + 3 \, y )+( -2 \, t^{2} y - 4 \, t^{2} + 3 \, t )y'=0$

Its implicit solution satisfying $$y( 1 )= 0$$ is:

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

## F6: Exact ODEs (ver. 9)

Determine which of the following ODEs is exact.

$( -6 \, t y^{2} - 12 \, t^{2} )+( -6 \, t^{2} y - 6 \, y^{2} )y'=0$

$( -6 \, t y^{2} - 12 \, t^{2} + 6 \, t y )+( 8 \, t y - 6 \, y^{2} + 3 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= -1$$.

The following ODE is exact.

$( -6 \, t y^{2} - 12 \, t^{2} )+( -6 \, t^{2} y - 6 \, y^{2} )y'=0$

Its implicit solution satisfying $$y( 1 )= -1$$ is:

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

## F6: Exact ODEs (ver. 10)

Determine which of the following ODEs is exact.

$( -15 \, t^{2} - 2 \, y )+( -2 \, t + 2 \, y )y'=0$

$( -15 \, t^{2} - 4 \, y^{2} - 2 \, y )+( 6 \, t^{2} y + t^{2} + 2 \, y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( -1 )= 0$$.

The following ODE is exact.

$( -15 \, t^{2} - 2 \, y )+( -2 \, t + 2 \, y )y'=0$

Its implicit solution satisfying $$y( -1 )= 0$$ is:

$-5 \, t^{3} - 2 \, t y + y^{2} = 5$

## F6: Exact ODEs (ver. 11)

Determine which of the following ODEs is exact.

$( -8 \, t y - 2 \, y^{2} )+( -4 \, t^{2} y - 2 \, t )y'=0$

$( -8 \, t y - 2 \, y^{2} - 6 \, t )+( -4 \, t^{2} - 4 \, t y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= -1$$.

The following ODE is exact.

$( -8 \, t y - 2 \, y^{2} - 6 \, t )+( -4 \, t^{2} - 4 \, t y )y'=0$

Its implicit solution satisfying $$y( 0 )= -1$$ is:

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

## F6: Exact ODEs (ver. 12)

Determine which of the following ODEs is exact.

$( 9 \, t^{2} + 8 \, t y + 4 \, y^{2} )+( 2 \, t^{2} y - 20 \, y^{3} + 2 \, t )y'=0$

$( 2 \, t y^{2} + 9 \, t^{2} + 4 \, y^{2} )+( 2 \, t^{2} y + 8 \, t y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= 1$$.

The following ODE is exact.

$( 2 \, t y^{2} + 9 \, t^{2} + 4 \, y^{2} )+( 2 \, t^{2} y + 8 \, t y )y'=0$

Its implicit solution satisfying $$y( 0 )= 1$$ is:

$t^{2} y^{2} + 3 \, t^{3} + 4 \, t y^{2} = 0$

## F6: Exact ODEs (ver. 13)

Determine which of the following ODEs is exact.

$( 4 \, t - y )+( 2 \, t^{2} y + t^{2} + 8 \, t y )y'=0$

$( 4 \, y^{2} - y )+( 8 \, t y - 12 \, y^{2} - t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= -1$$.

The following ODE is exact.

$( 4 \, y^{2} - y )+( 8 \, t y - 12 \, y^{2} - t )y'=0$

Its implicit solution satisfying $$y( 1 )= -1$$ is:

$4 \, t y^{2} - 4 \, y^{3} - t y = 9$

## F6: Exact ODEs (ver. 14)

Determine which of the following ODEs is exact.

$( -4 \, t^{3} + 6 \, t y + 2 \, y )+( -4 \, t^{2} y - 8 \, t y - 6 \, y^{2} )y'=0$

$( -4 \, t^{3} - 4 \, t y^{2} + 2 \, y )+( -4 \, t^{2} y + 2 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( -1 )= 0$$.

The following ODE is exact.

$( -4 \, t^{3} - 4 \, t y^{2} + 2 \, y )+( -4 \, t^{2} y + 2 \, t )y'=0$

Its implicit solution satisfying $$y( -1 )= 0$$ is:

$-t^{4} - 2 \, t^{2} y^{2} + 2 \, t y = -1$

## F6: Exact ODEs (ver. 15)

Determine which of the following ODEs is exact.

$( -4 \, t y^{2} + 8 \, t )+( t^{2} - 8 \, t y - 4 \, t )y'=0$

$( -4 \, y^{2} + 8 \, t )+( -8 \, t y + 8 \, y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= -1$$.

The following ODE is exact.

$( -4 \, y^{2} + 8 \, t )+( -8 \, t y + 8 \, y )y'=0$

Its implicit solution satisfying $$y( 0 )= -1$$ is:

$-4 \, t y^{2} + 4 \, t^{2} + 4 \, y^{2} = 4$

## F6: Exact ODEs (ver. 16)

Determine which of the following ODEs is exact.

$( 2 \, t y - 3 \, y^{2} - y )+( 6 \, t^{2} y - 16 \, y^{3} )y'=0$

$( 2 \, t y + 6 \, t - y )+( t^{2} - t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= -1$$.

The following ODE is exact.

$( 2 \, t y + 6 \, t - y )+( t^{2} - t )y'=0$

Its implicit solution satisfying $$y( 0 )= -1$$ is:

$t^{2} y + 3 \, t^{2} - t y = 0$

## F6: Exact ODEs (ver. 17)

Determine which of the following ODEs is exact.

$( -4 \, t y^{2} + 10 \, t y - y^{2} )+( -20 \, y^{3} + 2 \, t )y'=0$

$( -4 \, t y^{2} + 10 \, t y )+( -4 \, t^{2} y - 20 \, y^{3} + 5 \, t^{2} )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= -1$$.

The following ODE is exact.

$( -4 \, t y^{2} + 10 \, t y )+( -4 \, t^{2} y - 20 \, y^{3} + 5 \, t^{2} )y'=0$

Its implicit solution satisfying $$y( 1 )= -1$$ is:

$-2 \, t^{2} y^{2} - 5 \, y^{4} + 5 \, t^{2} y = -12$

## F6: Exact ODEs (ver. 18)

Determine which of the following ODEs is exact.

$( -15 \, t^{2} + 6 \, t y - 4 \, y )+( 6 \, t^{2} y + 2 \, t y + 6 \, y^{2} )y'=0$

$( -15 \, t^{2} + y^{2} - 4 \, y )+( 2 \, t y - 4 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= 0$$.

The following ODE is exact.

$( -15 \, t^{2} + y^{2} - 4 \, y )+( 2 \, t y - 4 \, t )y'=0$

Its implicit solution satisfying $$y( 1 )= 0$$ is:

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

## F6: Exact ODEs (ver. 19)

Determine which of the following ODEs is exact.

$( 8 \, t^{3} - 8 \, t y^{2} )+( -8 \, t^{2} y - 6 \, y )y'=0$

$( -8 \, t y^{2} - 3 \, y^{2} )+( 5 \, t^{2} - 4 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= 1$$.

The following ODE is exact.

$( 8 \, t^{3} - 8 \, t y^{2} )+( -8 \, t^{2} y - 6 \, y )y'=0$

Its implicit solution satisfying $$y( 0 )= 1$$ is:

$2 \, t^{4} - 4 \, t^{2} y^{2} - 3 \, y^{2} = -3$

## F6: Exact ODEs (ver. 20)

Determine which of the following ODEs is exact.

$( -8 \, t y^{2} + 3 \, y^{2} + 2 \, y )+( 4 \, t^{2} + 8 \, y )y'=0$

$( -8 \, t y^{2} + 8 \, t y + 3 \, y^{2} )+( -8 \, t^{2} y + 4 \, t^{2} + 6 \, t y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= -1$$.

The following ODE is exact.

$( -8 \, t y^{2} + 8 \, t y + 3 \, y^{2} )+( -8 \, t^{2} y + 4 \, t^{2} + 6 \, t y )y'=0$

Its implicit solution satisfying $$y( 0 )= -1$$ is:

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

## F6: Exact ODEs (ver. 21)

Determine which of the following ODEs is exact.

$( 10 \, t y^{2} - 4 \, t )+( 3 \, t^{2} - 4 \, t y - 4 \, t )y'=0$

$( 10 \, t y^{2} - 2 \, y^{2} )+( 10 \, t^{2} y - 4 \, t y + 8 \, y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= -1$$.

The following ODE is exact.

$( 10 \, t y^{2} - 2 \, y^{2} )+( 10 \, t^{2} y - 4 \, t y + 8 \, y )y'=0$

Its implicit solution satisfying $$y( 1 )= -1$$ is:

$5 \, t^{2} y^{2} - 2 \, t y^{2} + 4 \, y^{2} = 7$

## F6: Exact ODEs (ver. 22)

Determine which of the following ODEs is exact.

$( -4 \, t y - 4 \, t + 3 \, y )+( -2 \, t^{2} + 3 \, t )y'=0$

$( -4 \, t y + 4 \, y^{2} + 3 \, y )+( -8 \, t^{2} y + 4 \, y^{3} )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= 1$$.

The following ODE is exact.

$( -4 \, t y - 4 \, t + 3 \, y )+( -2 \, t^{2} + 3 \, t )y'=0$

Its implicit solution satisfying $$y( 0 )= 1$$ is:

$-2 \, t^{2} y - 2 \, t^{2} + 3 \, t y = 0$

## F6: Exact ODEs (ver. 23)

Determine which of the following ODEs is exact.

$( 6 \, t y^{2} - 2 \, t )+( -2 \, t^{2} + 8 \, t y - 3 \, t )y'=0$

$( 6 \, t y^{2} + 4 \, y^{2} - 2 \, t )+( 6 \, t^{2} y + 8 \, t y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= 0$$.

The following ODE is exact.

$( 6 \, t y^{2} + 4 \, y^{2} - 2 \, t )+( 6 \, t^{2} y + 8 \, t y )y'=0$

Its implicit solution satisfying $$y( 1 )= 0$$ is:

$3 \, t^{2} y^{2} + 4 \, t y^{2} - t^{2} = -1$

## F6: Exact ODEs (ver. 24)

Determine which of the following ODEs is exact.

$( 8 \, t y^{2} + y^{2} - 2 \, y )+( 8 \, t^{2} y + 2 \, t y - 2 \, t )y'=0$

$( 8 \, t y^{2} + 6 \, t - 2 \, y )+( 5 \, t^{2} + 2 \, t y + 9 \, y^{2} )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= 1$$.

The following ODE is exact.

$( 8 \, t y^{2} + y^{2} - 2 \, y )+( 8 \, t^{2} y + 2 \, t y - 2 \, t )y'=0$

Its implicit solution satisfying $$y( 0 )= 1$$ is:

$4 \, t^{2} y^{2} + t y^{2} - 2 \, t y = 0$

## F6: Exact ODEs (ver. 25)

Determine which of the following ODEs is exact.

$( 12 \, t^{3} - y^{2} )+( -8 \, t^{2} y - t^{2} - 2 \, t )y'=0$

$( 12 \, t^{3} - 8 \, t y^{2} )+( -8 \, t^{2} y - 16 \, y^{3} )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= 1$$.

The following ODE is exact.

$( 12 \, t^{3} - 8 \, t y^{2} )+( -8 \, t^{2} y - 16 \, y^{3} )y'=0$

Its implicit solution satisfying $$y( 0 )= 1$$ is:

$3 \, t^{4} - 4 \, t^{2} y^{2} - 4 \, y^{4} = -4$

## F6: Exact ODEs (ver. 26)

Determine which of the following ODEs is exact.

$( -10 \, t y^{2} - 4 \, t y )+( -10 \, t^{2} y + 16 \, y^{3} - 2 \, t^{2} )y'=0$

$( -4 \, t y + 5 \, y )+( -10 \, t^{2} y - 6 \, t y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( -1 )= 0$$.

The following ODE is exact.

$( -10 \, t y^{2} - 4 \, t y )+( -10 \, t^{2} y + 16 \, y^{3} - 2 \, t^{2} )y'=0$

Its implicit solution satisfying $$y( -1 )= 0$$ is:

$-5 \, t^{2} y^{2} + 4 \, y^{4} - 2 \, t^{2} y = 0$

## F6: Exact ODEs (ver. 27)

Determine which of the following ODEs is exact.

$( 8 \, t y^{2} - 4 \, y^{2} )+( 8 \, t^{2} y - 8 \, t y - 6 \, y^{2} )y'=0$

$( 4 \, t^{3} + 8 \, t y^{2} - 4 \, y^{2} )+( -2 \, t^{2} - 6 \, y^{2} - t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= 0$$.

The following ODE is exact.

$( 8 \, t y^{2} - 4 \, y^{2} )+( 8 \, t^{2} y - 8 \, t y - 6 \, y^{2} )y'=0$

Its implicit solution satisfying $$y( 1 )= 0$$ is:

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

## F6: Exact ODEs (ver. 28)

Determine which of the following ODEs is exact.

$( 12 \, t^{3} - 2 \, t y^{2} + 4 \, y )+( -2 \, t^{2} y + 4 \, t )y'=0$

$( 12 \, t^{3} + 4 \, y )+( -2 \, t^{2} y - 5 \, t^{2} - 4 \, t y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= 1$$.

The following ODE is exact.

$( 12 \, t^{3} - 2 \, t y^{2} + 4 \, y )+( -2 \, t^{2} y + 4 \, t )y'=0$

Its implicit solution satisfying $$y( 0 )= 1$$ is:

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

## F6: Exact ODEs (ver. 29)

Determine which of the following ODEs is exact.

$( 6 \, t y + 10 \, t + 2 \, y )+( 2 \, t^{2} y + 2 \, t y + 3 \, y^{2} )y'=0$

$( 6 \, t y + 2 \, y )+( 3 \, t^{2} + 3 \, y^{2} + 2 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= 0$$.

The following ODE is exact.

$( 6 \, t y + 2 \, y )+( 3 \, t^{2} + 3 \, y^{2} + 2 \, t )y'=0$

Its implicit solution satisfying $$y( 1 )= 0$$ is:

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

## F6: Exact ODEs (ver. 30)

Determine which of the following ODEs is exact.

$( -2 \, t y - 4 \, y )+( -10 \, t^{2} y - 6 \, t y )y'=0$

$( 4 \, t - 4 \, y )+( -4 \, t + 8 \, y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= 0$$.

The following ODE is exact.

$( 4 \, t - 4 \, y )+( -4 \, t + 8 \, y )y'=0$

Its implicit solution satisfying $$y( 1 )= 0$$ is:

$2 \, t^{2} - 4 \, t y + 4 \, y^{2} = 2$

## F6: Exact ODEs (ver. 31)

Determine which of the following ODEs is exact.

$( 8 \, t y^{2} + 3 \, t^{2} )+( 5 \, t^{2} - 6 \, t y - 3 \, t )y'=0$

$( 8 \, t y^{2} + 3 \, t^{2} - 3 \, y )+( 8 \, t^{2} y - 3 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= -1$$.

The following ODE is exact.

$( 8 \, t y^{2} + 3 \, t^{2} - 3 \, y )+( 8 \, t^{2} y - 3 \, t )y'=0$

Its implicit solution satisfying $$y( 1 )= -1$$ is:

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

## F6: Exact ODEs (ver. 32)

Determine which of the following ODEs is exact.

$( -8 \, t y + y^{2} + 4 \, y )+( 4 \, t^{2} y - 20 \, y^{3} )y'=0$

$( 12 \, t^{3} + y^{2} + 4 \, y )+( 2 \, t y + 4 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= -1$$.

The following ODE is exact.

$( 12 \, t^{3} + y^{2} + 4 \, y )+( 2 \, t y + 4 \, t )y'=0$

Its implicit solution satisfying $$y( 1 )= -1$$ is:

$3 \, t^{4} + t y^{2} + 4 \, t y = 0$

## F6: Exact ODEs (ver. 33)

Determine which of the following ODEs is exact.

$( 8 \, t y^{2} - 4 \, t y )+( 8 \, t^{2} y - 16 \, y^{3} - 2 \, t^{2} )y'=0$

$( 8 \, t y^{2} + 4 \, y )+( -2 \, t^{2} - 10 \, t y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( -1 )= 1$$.

The following ODE is exact.

$( 8 \, t y^{2} - 4 \, t y )+( 8 \, t^{2} y - 16 \, y^{3} - 2 \, t^{2} )y'=0$

Its implicit solution satisfying $$y( -1 )= 1$$ is:

$4 \, t^{2} y^{2} - 4 \, y^{4} - 2 \, t^{2} y = -2$

## F6: Exact ODEs (ver. 34)

Determine which of the following ODEs is exact.

$( -2 \, t y^{2} - 2 \, t y + 3 \, y )+( 8 \, y^{3} - 6 \, t y )y'=0$

$( -2 \, t y - 3 \, y^{2} + 3 \, y )+( -t^{2} - 6 \, t y + 3 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= -1$$.

The following ODE is exact.

$( -2 \, t y - 3 \, y^{2} + 3 \, y )+( -t^{2} - 6 \, t y + 3 \, t )y'=0$

Its implicit solution satisfying $$y( 1 )= -1$$ is:

$-t^{2} y - 3 \, t y^{2} + 3 \, t y = -5$

## F6: Exact ODEs (ver. 35)

Determine which of the following ODEs is exact.

$( 6 \, t^{2} - 4 \, t y + 4 \, y )+( -10 \, t^{2} y + 20 \, y^{3} + 6 \, t y )y'=0$

$( 6 \, t^{2} + 3 \, y^{2} + 4 \, y )+( 6 \, t y + 4 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= 1$$.

The following ODE is exact.

$( 6 \, t^{2} + 3 \, y^{2} + 4 \, y )+( 6 \, t y + 4 \, t )y'=0$

Its implicit solution satisfying $$y( 0 )= 1$$ is:

$2 \, t^{3} + 3 \, t y^{2} + 4 \, t y = 0$

## F6: Exact ODEs (ver. 36)

Determine which of the following ODEs is exact.

$( 6 \, t y^{2} - 15 \, t^{2} )+( 4 \, t^{2} + 2 \, t y + 3 \, t )y'=0$

$( 6 \, t y^{2} - 15 \, t^{2} + 8 \, t y )+( 6 \, t^{2} y + 4 \, t^{2} )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= -1$$.

The following ODE is exact.

$( 6 \, t y^{2} - 15 \, t^{2} + 8 \, t y )+( 6 \, t^{2} y + 4 \, t^{2} )y'=0$

Its implicit solution satisfying $$y( 0 )= -1$$ is:

$3 \, t^{2} y^{2} - 5 \, t^{3} + 4 \, t^{2} y = 0$

## F6: Exact ODEs (ver. 37)

Determine which of the following ODEs is exact.

$( 16 \, t^{3} + 8 \, t y^{2} )+( 8 \, t^{2} y - 6 \, y^{2} )y'=0$

$( 16 \, t^{3} - 8 \, t y )+( 8 \, t^{2} y + 4 \, t y + t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( -1 )= 0$$.

The following ODE is exact.

$( 16 \, t^{3} + 8 \, t y^{2} )+( 8 \, t^{2} y - 6 \, y^{2} )y'=0$

Its implicit solution satisfying $$y( -1 )= 0$$ is:

$4 \, t^{4} + 4 \, t^{2} y^{2} - 2 \, y^{3} = 4$

## F6: Exact ODEs (ver. 38)

Determine which of the following ODEs is exact.

$( -4 \, t - y )+( 10 \, t^{2} y + 2 \, t^{2} + 4 \, t y )y'=0$

$( 4 \, t y - 4 \, t )+( -16 \, y^{3} + 2 \, t^{2} )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= -1$$.

The following ODE is exact.

$( 4 \, t y - 4 \, t )+( -16 \, y^{3} + 2 \, t^{2} )y'=0$

Its implicit solution satisfying $$y( 1 )= -1$$ is:

$-4 \, y^{4} + 2 \, t^{2} y - 2 \, t^{2} = -8$

## F6: Exact ODEs (ver. 39)

Determine which of the following ODEs is exact.

$( -20 \, t^{3} - 4 \, t y^{2} - y^{2} )+( -4 \, t^{2} y - 2 \, t y )y'=0$

$( -20 \, t^{3} - y^{2} )+( -4 \, t^{2} y + 5 \, t^{2} - 5 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= 0$$.

The following ODE is exact.

$( -20 \, t^{3} - 4 \, t y^{2} - y^{2} )+( -4 \, t^{2} y - 2 \, t y )y'=0$

Its implicit solution satisfying $$y( 1 )= 0$$ is:

$-5 \, t^{4} - 2 \, t^{2} y^{2} - t y^{2} = -5$

## F6: Exact ODEs (ver. 40)

Determine which of the following ODEs is exact.

$( 2 \, t y^{2} - y )+( 2 \, t^{2} y + 3 \, y^{2} - t )y'=0$

$( 2 \, t y^{2} + 6 \, t y - y )+( 8 \, t y + 3 \, y^{2} )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( -1 )= 1$$.

The following ODE is exact.

$( 2 \, t y^{2} - y )+( 2 \, t^{2} y + 3 \, y^{2} - t )y'=0$

Its implicit solution satisfying $$y( -1 )= 1$$ is:

$t^{2} y^{2} + y^{3} - t y = 3$

## F6: Exact ODEs (ver. 41)

Determine which of the following ODEs is exact.

$( 2 \, t y + 3 \, y^{2} )+( -8 \, t^{2} y + 4 \, t )y'=0$

$( -8 \, t y^{2} + 2 \, t y )+( -8 \, t^{2} y - 8 \, y^{3} + t^{2} )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= 0$$.

The following ODE is exact.

$( -8 \, t y^{2} + 2 \, t y )+( -8 \, t^{2} y - 8 \, y^{3} + t^{2} )y'=0$

Its implicit solution satisfying $$y( 1 )= 0$$ is:

$-4 \, t^{2} y^{2} - 2 \, y^{4} + t^{2} y = 0$

## F6: Exact ODEs (ver. 42)

Determine which of the following ODEs is exact.

$( 8 \, t y^{2} + 3 \, y^{2} )+( 8 \, t^{2} y + 6 \, t y - 12 \, y^{2} )y'=0$

$( 8 \, t^{3} + 3 \, y^{2} )+( 8 \, t^{2} y - 3 \, t^{2} - t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( -1 )= 0$$.

The following ODE is exact.

$( 8 \, t y^{2} + 3 \, y^{2} )+( 8 \, t^{2} y + 6 \, t y - 12 \, y^{2} )y'=0$

Its implicit solution satisfying $$y( -1 )= 0$$ is:

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

## F6: Exact ODEs (ver. 43)

Determine which of the following ODEs is exact.

$( 4 \, t y^{2} - y^{2} )+( 3 \, t^{2} + 5 \, t )y'=0$

$( 4 \, t y^{2} + 5 \, y )+( 4 \, t^{2} y + 16 \, y^{3} + 5 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= 1$$.

The following ODE is exact.

$( 4 \, t y^{2} + 5 \, y )+( 4 \, t^{2} y + 16 \, y^{3} + 5 \, t )y'=0$

Its implicit solution satisfying $$y( 0 )= 1$$ is:

$2 \, t^{2} y^{2} + 4 \, y^{4} + 5 \, t y = 4$

## F6: Exact ODEs (ver. 44)

Determine which of the following ODEs is exact.

$( -2 \, t - 4 \, y )+( 3 \, y^{2} - 4 \, t )y'=0$

$( y^{2} - 2 \, t - 4 \, y )+( -6 \, t^{2} y - 3 \, t^{2} + 3 \, y^{2} )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( -1 )= 0$$.

The following ODE is exact.

$( -2 \, t - 4 \, y )+( 3 \, y^{2} - 4 \, t )y'=0$

Its implicit solution satisfying $$y( -1 )= 0$$ is:

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

## F6: Exact ODEs (ver. 45)

Determine which of the following ODEs is exact.

$( 4 \, t y^{2} - 3 \, y^{2} )+( 4 \, t^{2} y - 6 \, t y - 8 \, y )y'=0$

$( 4 \, t y^{2} + 2 \, t y )+( -6 \, t y + 2 \, t )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= -1$$.

The following ODE is exact.

$( 4 \, t y^{2} - 3 \, y^{2} )+( 4 \, t^{2} y - 6 \, t y - 8 \, y )y'=0$

Its implicit solution satisfying $$y( 0 )= -1$$ is:

$2 \, t^{2} y^{2} - 3 \, t y^{2} - 4 \, y^{2} = -4$

## F6: Exact ODEs (ver. 46)

Determine which of the following ODEs is exact.

$( 4 \, t y^{2} + 6 \, t^{2} + y )+( 4 \, t^{2} y + t )y'=0$

$( 4 \, t y^{2} + 3 \, y^{2} + y )+( 4 \, t^{2} + 4 \, y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( -1 )= 0$$.

The following ODE is exact.

$( 4 \, t y^{2} + 6 \, t^{2} + y )+( 4 \, t^{2} y + t )y'=0$

Its implicit solution satisfying $$y( -1 )= 0$$ is:

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

## F6: Exact ODEs (ver. 47)

Determine which of the following ODEs is exact.

$( 8 \, t y - 10 \, t )+( -4 \, t^{2} y - 6 \, t y + t )y'=0$

$( -4 \, t y^{2} - 10 \, t )+( -4 \, t^{2} y - 4 \, y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( -1 )= 1$$.

The following ODE is exact.

$( -4 \, t y^{2} - 10 \, t )+( -4 \, t^{2} y - 4 \, y )y'=0$

Its implicit solution satisfying $$y( -1 )= 1$$ is:

$-2 \, t^{2} y^{2} - 5 \, t^{2} - 2 \, y^{2} = -9$

## F6: Exact ODEs (ver. 48)

Determine which of the following ODEs is exact.

$( 10 \, t y - y )+( 5 \, t^{2} + 15 \, y^{2} - t )y'=0$

$( -2 \, y^{2} - y )+( 6 \, t^{2} y + 5 \, t^{2} )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( -1 )= 0$$.

The following ODE is exact.

$( 10 \, t y - y )+( 5 \, t^{2} + 15 \, y^{2} - t )y'=0$

Its implicit solution satisfying $$y( -1 )= 0$$ is:

$5 \, t^{2} y + 5 \, y^{3} - t y = 0$

## F6: Exact ODEs (ver. 49)

Determine which of the following ODEs is exact.

$( -6 \, t - 4 \, y )+( -12 \, y^{3} - 4 \, t )y'=0$

$( -4 \, t y - 6 \, t - 4 \, y )+( -6 \, t^{2} y - 12 \, y^{3} + 10 \, t y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 1 )= -1$$.

The following ODE is exact.

$( -6 \, t - 4 \, y )+( -12 \, y^{3} - 4 \, t )y'=0$

Its implicit solution satisfying $$y( 1 )= -1$$ is:

$-3 \, y^{4} - 3 \, t^{2} - 4 \, t y = -2$

## F6: Exact ODEs (ver. 50)

Determine which of the following ODEs is exact.

$( -8 \, t y^{2} - 5 \, y^{2} )+( 5 \, t^{2} + 2 \, t )y'=0$

$( -8 \, t y^{2} + 10 \, t y )+( -8 \, t^{2} y + 5 \, t^{2} - 8 \, y )y'=0$

Then find an implicit solution for this exact ODE satisfying the initial value $$y( 0 )= 1$$.

$( -8 \, t y^{2} + 10 \, t y )+( -8 \, t^{2} y + 5 \, t^{2} - 8 \, y )y'=0$
Its implicit solution satisfying $$y( 0 )= 1$$ is:
$-4 \, t^{2} y^{2} + 5 \, t^{2} y - 4 \, y^{2} = -4$