C6 - Non-homogeneous second-order linear ODEs


Example 1

C6 - Non-homogeneous second-order linear ODEs (ver. 1)

Find the solution to the given ODE.

\[{y''} - 2 \, {y'} - 15 \, {y} = 12 \, e^{\left(-t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(5 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} - e^{\left(-t\right)}\]


Example 2

C6 - Non-homogeneous second-order linear ODEs (ver. 2)

Find the solution to the given ODE.

\[{y''} + 6 \, {y'} + 5 \, {y} = 105 \, e^{\left(2 \, t\right)}\]

Answer.

\[{y} = k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-5 \, t\right)} + 5 \, e^{\left(2 \, t\right)}\]


Example 3

C6 - Non-homogeneous second-order linear ODEs (ver. 3)

Find the solution to the given ODE.

\[{y''} + 2 \, {y'} - 15 \, {y} = -36 \, e^{\left(4 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-5 \, t\right)} - 4 \, e^{\left(4 \, t\right)}\]


Example 4

C6 - Non-homogeneous second-order linear ODEs (ver. 4)

Find the solution to the given ODE.

\[{y''} + 3 \, {y'} - 4 \, {y} = -18 \, e^{\left(-5 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(-4 \, t\right)} + k_{2} e^{t} - 3 \, e^{\left(-5 \, t\right)}\]


Example 5

C6 - Non-homogeneous second-order linear ODEs (ver. 5)

Find the solution to the given ODE.

\[{y''} + {y'} - 2 \, {y} = -12 \, e^{\left(-3 \, t\right)}\]

Answer.

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


Example 6

C6 - Non-homogeneous second-order linear ODEs (ver. 6)

Find the solution to the given ODE.

\[{y''} - 3 \, {y'} - 10 \, {y} = 72 \, e^{\left(-4 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(5 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + 4 \, e^{\left(-4 \, t\right)}\]


Example 7

C6 - Non-homogeneous second-order linear ODEs (ver. 7)

Find the solution to the given ODE.

\[{y''} - 8 \, {y'} + 15 \, {y} = -4 \, e^{\left(4 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(5 \, t\right)} + k_{2} e^{\left(3 \, t\right)} + 4 \, e^{\left(4 \, t\right)}\]


Example 8

C6 - Non-homogeneous second-order linear ODEs (ver. 8)

Find the solution to the given ODE.

\[{y''} - 2 \, {y'} - 15 \, {y} = -12 \, e^{\left(3 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(5 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + e^{\left(3 \, t\right)}\]


Example 9

C6 - Non-homogeneous second-order linear ODEs (ver. 9)

Find the solution to the given ODE.

\[{y''} + 7 \, {y'} + 12 \, {y} = 6 \, e^{\left(-2 \, t\right)}\]

Answer.

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


Example 10

C6 - Non-homogeneous second-order linear ODEs (ver. 10)

Find the solution to the given ODE.

\[{y''} + 8 \, {y'} + 15 \, {y} = -12 \, e^{\left(-2 \, t\right)}\]

Answer.

\[{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{\left(-5 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}\]


Example 11

C6 - Non-homogeneous second-order linear ODEs (ver. 11)

Find the solution to the given ODE.

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

Answer.

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


Example 12

C6 - Non-homogeneous second-order linear ODEs (ver. 12)

Find the solution to the given ODE.

\[{y''} - 5 \, {y'} + 6 \, {y} = 8 \, e^{\left(4 \, t\right)}\]

Answer.

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


Example 13

C6 - Non-homogeneous second-order linear ODEs (ver. 13)

Find the solution to the given ODE.

\[{y''} - 3 \, {y'} + 2 \, {y} = -6 \, e^{\left(-t\right)}\]

Answer.

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


Example 14

C6 - Non-homogeneous second-order linear ODEs (ver. 14)

Find the solution to the given ODE.

\[{y''} + 6 \, {y'} + 5 \, {y} = 64 \, e^{\left(3 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-5 \, t\right)} + 2 \, e^{\left(3 \, t\right)}\]


Example 15

C6 - Non-homogeneous second-order linear ODEs (ver. 15)

Find the solution to the given ODE.

\[{y''} - 5 \, {y'} + 4 \, {y} = -160 \, e^{\left(-4 \, t\right)}\]

Answer.

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


Example 16

C6 - Non-homogeneous second-order linear ODEs (ver. 16)

Find the solution to the given ODE.

\[{y''} - 4 \, {y'} + 3 \, {y} = -24 \, e^{\left(-t\right)}\]

Answer.

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


Example 17

C6 - Non-homogeneous second-order linear ODEs (ver. 17)

Find the solution to the given ODE.

\[{y''} - 9 \, {y} = 5 \, e^{\left(2 \, t\right)}\]

Answer.

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


Example 18

C6 - Non-homogeneous second-order linear ODEs (ver. 18)

Find the solution to the given ODE.

\[{y''} + 8 \, {y'} + 15 \, {y} = -e^{\left(-4 \, t\right)}\]

Answer.

\[{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{\left(-5 \, t\right)} + e^{\left(-4 \, t\right)}\]


Example 19

C6 - Non-homogeneous second-order linear ODEs (ver. 19)

Find the solution to the given ODE.

\[{y''} - 7 \, {y'} + 10 \, {y} = -112 \, e^{\left(-2 \, t\right)}\]

Answer.

\[{y} = k_{2} e^{\left(5 \, t\right)} + k_{1} e^{\left(2 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}\]


Example 20

C6 - Non-homogeneous second-order linear ODEs (ver. 20)

Find the solution to the given ODE.

\[{y''} + 2 \, {y'} - 15 \, {y} = -21 \, e^{\left(2 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-5 \, t\right)} + 3 \, e^{\left(2 \, t\right)}\]


Example 21

C6 - Non-homogeneous second-order linear ODEs (ver. 21)

Find the solution to the given ODE.

\[{y''} - 4 \, {y} = -84 \, e^{\left(-5 \, t\right)}\]

Answer.

\[{y} = k_{2} e^{\left(2 \, t\right)} + k_{1} e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}\]


Example 22

C6 - Non-homogeneous second-order linear ODEs (ver. 22)

Find the solution to the given ODE.

\[{y''} + 3 \, {y'} + 2 \, {y} = 126 \, e^{\left(5 \, t\right)}\]

Answer.

\[{y} = k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}\]


Example 23

C6 - Non-homogeneous second-order linear ODEs (ver. 23)

Find the solution to the given ODE.

\[{y''} + 6 \, {y'} + 5 \, {y} = -16 \, e^{\left(-3 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(-t\right)} + k_{2} e^{\left(-5 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}\]


Example 24

C6 - Non-homogeneous second-order linear ODEs (ver. 24)

Find the solution to the given ODE.

\[{y''} - 5 \, {y'} + 6 \, {y} = -112 \, e^{\left(-5 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(2 \, t\right)} - 2 \, e^{\left(-5 \, t\right)}\]


Example 25

C6 - Non-homogeneous second-order linear ODEs (ver. 25)

Find the solution to the given ODE.

\[{y''} - 8 \, {y'} + 15 \, {y} = -6 \, e^{\left(2 \, t\right)}\]

Answer.

\[{y} = k_{2} e^{\left(5 \, t\right)} + k_{1} e^{\left(3 \, t\right)} - 2 \, e^{\left(2 \, t\right)}\]


Example 26

C6 - Non-homogeneous second-order linear ODEs (ver. 26)

Find the solution to the given ODE.

\[{y''} - 6 \, {y'} + 5 \, {y} = -20 \, e^{\left(3 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(5 \, t\right)} + k_{2} e^{t} + 5 \, e^{\left(3 \, t\right)}\]


Example 27

C6 - Non-homogeneous second-order linear ODEs (ver. 27)

Find the solution to the given ODE.

\[{y''} + 4 \, {y'} - 5 \, {y} = 32 \, e^{\left(-t\right)}\]

Answer.

\[{y} = k_{2} e^{\left(-5 \, t\right)} + k_{1} e^{t} - 4 \, e^{\left(-t\right)}\]


Example 28

C6 - Non-homogeneous second-order linear ODEs (ver. 28)

Find the solution to the given ODE.

\[{y''} + 2 \, {y'} - 15 \, {y} = -28 \, e^{\left(2 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(3 \, t\right)} + k_{2} e^{\left(-5 \, t\right)} + 4 \, e^{\left(2 \, t\right)}\]


Example 29

C6 - Non-homogeneous second-order linear ODEs (ver. 29)

Find the solution to the given ODE.

\[{y''} - 3 \, {y'} - 10 \, {y} = -20 \, e^{\left(3 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(5 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + 2 \, e^{\left(3 \, t\right)}\]


Example 30

C6 - Non-homogeneous second-order linear ODEs (ver. 30)

Find the solution to the given ODE.

\[{y''} - 3 \, {y'} + 2 \, {y} = -120 \, e^{\left(-4 \, t\right)}\]

Answer.

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


Example 31

C6 - Non-homogeneous second-order linear ODEs (ver. 31)

Find the solution to the given ODE.

\[{y''} + 3 \, {y'} - 4 \, {y} = -18 \, e^{\left(-5 \, t\right)}\]

Answer.

\[{y} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{t} - 3 \, e^{\left(-5 \, t\right)}\]


Example 32

C6 - Non-homogeneous second-order linear ODEs (ver. 32)

Find the solution to the given ODE.

\[{y''} - 7 \, {y'} + 12 \, {y} = 8 \, e^{\left(2 \, t\right)}\]

Answer.

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


Example 33

C6 - Non-homogeneous second-order linear ODEs (ver. 33)

Find the solution to the given ODE.

\[{y''} - 4 \, {y'} - 5 \, {y} = -32 \, e^{t}\]

Answer.

\[{y} = k_{2} e^{\left(5 \, t\right)} + k_{1} e^{\left(-t\right)} + 4 \, e^{t}\]


Example 34

C6 - Non-homogeneous second-order linear ODEs (ver. 34)

Find the solution to the given ODE.

\[{y''} - 2 \, {y'} - 8 \, {y} = 21 \, e^{\left(5 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-2 \, t\right)} + 3 \, e^{\left(5 \, t\right)}\]


Example 35

C6 - Non-homogeneous second-order linear ODEs (ver. 35)

Find the solution to the given ODE.

\[{y''} - 5 \, {y'} + 6 \, {y} = 8 \, e^{\left(4 \, t\right)}\]

Answer.

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


Example 36

C6 - Non-homogeneous second-order linear ODEs (ver. 36)

Find the solution to the given ODE.

\[{y''} + 8 \, {y'} + 15 \, {y} = 9 \, e^{\left(-2 \, t\right)}\]

Answer.

\[{y} = k_{2} e^{\left(-3 \, t\right)} + k_{1} e^{\left(-5 \, t\right)} + 3 \, e^{\left(-2 \, t\right)}\]


Example 37

C6 - Non-homogeneous second-order linear ODEs (ver. 37)

Find the solution to the given ODE.

\[{y''} + {y'} - 20 \, {y} = -14 \, e^{\left(2 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(4 \, t\right)} + k_{2} e^{\left(-5 \, t\right)} + e^{\left(2 \, t\right)}\]


Example 38

C6 - Non-homogeneous second-order linear ODEs (ver. 38)

Find the solution to the given ODE.

\[{y''} + {y'} - 2 \, {y} = -4 \, e^{\left(-t\right)}\]

Answer.

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


Example 39

C6 - Non-homogeneous second-order linear ODEs (ver. 39)

Find the solution to the given ODE.

\[{y''} + 9 \, {y'} + 20 \, {y} = -60 \, e^{\left(-t\right)}\]

Answer.

\[{y} = k_{2} e^{\left(-4 \, t\right)} + k_{1} e^{\left(-5 \, t\right)} - 5 \, e^{\left(-t\right)}\]


Example 40

C6 - Non-homogeneous second-order linear ODEs (ver. 40)

Find the solution to the given ODE.

\[{y''} + 3 \, {y'} + 2 \, {y} = -20 \, e^{\left(3 \, t\right)}\]

Answer.

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


Example 41

C6 - Non-homogeneous second-order linear ODEs (ver. 41)

Find the solution to the given ODE.

\[{y''} + 3 \, {y'} - 10 \, {y} = 32 \, e^{\left(3 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-5 \, t\right)} + 4 \, e^{\left(3 \, t\right)}\]


Example 42

C6 - Non-homogeneous second-order linear ODEs (ver. 42)

Find the solution to the given ODE.

\[{y''} - 6 \, {y'} + 8 \, {y} = -24 \, e^{\left(-2 \, t\right)}\]

Answer.

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


Example 43

C6 - Non-homogeneous second-order linear ODEs (ver. 43)

Find the solution to the given ODE.

\[{y''} - 5 \, {y'} + 6 \, {y} = -6 \, e^{\left(5 \, t\right)}\]

Answer.

\[{y} = k_{2} e^{\left(3 \, t\right)} + k_{1} e^{\left(2 \, t\right)} - e^{\left(5 \, t\right)}\]


Example 44

C6 - Non-homogeneous second-order linear ODEs (ver. 44)

Find the solution to the given ODE.

\[{y''} + 4 \, {y'} + 3 \, {y} = 3 \, e^{\left(-4 \, t\right)}\]

Answer.

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


Example 45

C6 - Non-homogeneous second-order linear ODEs (ver. 45)

Find the solution to the given ODE.

\[{y''} - 6 \, {y'} + 5 \, {y} = 24 \, e^{\left(-t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(5 \, t\right)} + k_{2} e^{t} + 2 \, e^{\left(-t\right)}\]


Example 46

C6 - Non-homogeneous second-order linear ODEs (ver. 46)

Find the solution to the given ODE.

\[{y''} + {y'} - 6 \, {y} = 14 \, e^{\left(-5 \, t\right)}\]

Answer.

\[{y} = k_{1} e^{\left(2 \, t\right)} + k_{2} e^{\left(-3 \, t\right)} + e^{\left(-5 \, t\right)}\]


Example 47

C6 - Non-homogeneous second-order linear ODEs (ver. 47)

Find the solution to the given ODE.

\[{y''} + 6 \, {y'} + 5 \, {y} = -36 \, e^{t}\]

Answer.

\[{y} = k_{2} e^{\left(-t\right)} + k_{1} e^{\left(-5 \, t\right)} - 3 \, e^{t}\]


Example 48

C6 - Non-homogeneous second-order linear ODEs (ver. 48)

Find the solution to the given ODE.

\[{y''} - 5 \, {y'} + 4 \, {y} = -20 \, e^{\left(-t\right)}\]

Answer.

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


Example 49

C6 - Non-homogeneous second-order linear ODEs (ver. 49)

Find the solution to the given ODE.

\[{y''} + 5 \, {y'} + 6 \, {y} = -48 \, e^{t}\]

Answer.

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


Example 50

C6 - Non-homogeneous second-order linear ODEs (ver. 50)

Find the solution to the given ODE.

\[{y''} - 2 \, {y'} - 15 \, {y} = 75 \, e^{\left(2 \, t\right)}\]

Answer.

\[{y} = k_{2} e^{\left(5 \, t\right)} + k_{1} e^{\left(-3 \, t\right)} - 5 \, e^{\left(2 \, t\right)}\]