F6: Exact ODEs


Example 1

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 \(y( 0 )= 1\).

Answer.

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:

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


Example 2

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 \(y( 1 )= 0\).

Answer.

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:

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


Example 3

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 \(y( -1 )= 1\).

Answer.

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:

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


Example 4

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 \(y( 1 )= 0\).

Answer.

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:

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


Example 5

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 \(y( 1 )= 0\).

Answer.

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:

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


Example 6

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 \(y( 1 )= 0\).

Answer.

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:

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


Example 7

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 \(y( -1 )= 0\).

Answer.

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:

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


Example 8

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 \(y( 1 )= 0\).

Answer.

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:

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


Example 9

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 \(y( 1 )= -1\).

Answer.

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:

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


Example 10

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 \(y( -1 )= 0\).

Answer.

The following ODE is exact.

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

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

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


Example 11

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 \(y( 0 )= -1\).

Answer.

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:

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


Example 12

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 \(y( 0 )= 1\).

Answer.

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:

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


Example 13

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 \(y( 1 )= -1\).

Answer.

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:

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


Example 14

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 \(y( -1 )= 0\).

Answer.

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:

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


Example 15

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 \(y( 0 )= -1\).

Answer.

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:

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


Example 16

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 \(y( 0 )= -1\).

Answer.

The following ODE is exact.

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

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

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


Example 17

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 \(y( 1 )= -1\).

Answer.

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:

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


Example 18

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 \(y( 1 )= 0\).

Answer.

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:

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


Example 19

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 \(y( 0 )= 1\).

Answer.

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:

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


Example 20

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 \(y( 0 )= -1\).

Answer.

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:

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


Example 21

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 \(y( 1 )= -1\).

Answer.

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:

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


Example 22

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 \(y( 0 )= 1\).

Answer.

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:

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


Example 23

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 \(y( 1 )= 0\).

Answer.

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:

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


Example 24

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 \(y( 0 )= 1\).

Answer.

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:

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


Example 25

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 \(y( 0 )= 1\).

Answer.

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:

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


Example 26

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 \(y( -1 )= 0\).

Answer.

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:

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


Example 27

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 \(y( 1 )= 0\).

Answer.

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:

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


Example 28

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 \(y( 0 )= 1\).

Answer.

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:

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


Example 29

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 \(y( 1 )= 0\).

Answer.

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:

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


Example 30

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 \(y( 1 )= 0\).

Answer.

The following ODE is exact.

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

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

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


Example 31

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 \(y( 1 )= -1\).

Answer.

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:

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


Example 32

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 \(y( 1 )= -1\).

Answer.

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:

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


Example 33

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 \(y( -1 )= 1\).

Answer.

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:

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


Example 34

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 \(y( 1 )= -1\).

Answer.

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:

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


Example 35

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 \(y( 0 )= 1\).

Answer.

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:

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


Example 36

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 \(y( 0 )= -1\).

Answer.

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:

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


Example 37

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 \(y( -1 )= 0\).

Answer.

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:

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


Example 38

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 \(y( 1 )= -1\).

Answer.

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:

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


Example 39

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 \(y( 1 )= 0\).

Answer.

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:

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


Example 40

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 \(y( -1 )= 1\).

Answer.

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:

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


Example 41

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 \(y( 1 )= 0\).

Answer.

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:

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


Example 42

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 \(y( -1 )= 0\).

Answer.

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:

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


Example 43

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 \(y( 0 )= 1\).

Answer.

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:

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


Example 44

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 \(y( -1 )= 0\).

Answer.

The following ODE is exact.

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

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

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


Example 45

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 \(y( 0 )= -1\).

Answer.

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:

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


Example 46

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 \(y( -1 )= 0\).

Answer.

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:

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


Example 47

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 \(y( -1 )= 1\).

Answer.

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:

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


Example 48

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 \(y( -1 )= 0\).

Answer.

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:

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


Example 49

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 \(y( 1 )= -1\).

Answer.

The following ODE is exact.

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

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

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


Example 50

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 \(y( 0 )= 1\).

Answer.

The following ODE is exact.

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

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

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