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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

\[y''+0y'-9y = 5 \, e^{\left(2 \, t\right)}\]

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.

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

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.

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

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.

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

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.

\[y''+0y'-4y = -84 \, e^{\left(-5 \, t\right)}\]

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

Answer.

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

Example 51

C6 - Non-homogeneous second-order linear ODEs (ver. 51)

Find the solution to the given ODE.

\[y''-2y'-8y = 20 \, e^{\left(-t\right)}\]

Answer.

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

Example 52

C6 - Non-homogeneous second-order linear ODEs (ver. 52)

Find the solution to the given ODE.

\[y''-5y'+6y = -60 \, e^{\left(-3 \, t\right)}\]

Answer.

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

Example 53

C6 - Non-homogeneous second-order linear ODEs (ver. 53)

Find the solution to the given ODE.

\[y''+y'-20y = -32 \, e^{\left(-4 \, t\right)}\]

Answer.

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

Example 54

C6 - Non-homogeneous second-order linear ODEs (ver. 54)

Find the solution to the given ODE.

\[y''-7y'+12y = 216 \, e^{\left(-5 \, t\right)}\]

Answer.

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

Example 55

C6 - Non-homogeneous second-order linear ODEs (ver. 55)

Find the solution to the given ODE.

\[y''-4y'-5y = -135 \, e^{\left(-4 \, t\right)}\]

Answer.

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

Example 56

C6 - Non-homogeneous second-order linear ODEs (ver. 56)

Find the solution to the given ODE.

\[y''-1y'-12y = 10 \, e^{\left(2 \, t\right)}\]

Answer.

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

Example 57

C6 - Non-homogeneous second-order linear ODEs (ver. 57)

Find the solution to the given ODE.

\[y''+0y'-16y = -28 \, e^{\left(3 \, t\right)}\]

Answer.

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

Example 58

C6 - Non-homogeneous second-order linear ODEs (ver. 58)

Find the solution to the given ODE.

\[y''+6y'+8y = -12 \, e^{\left(-t\right)}\]

Answer.

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

Example 59

C6 - Non-homogeneous second-order linear ODEs (ver. 59)

Find the solution to the given ODE.

\[y''+6y'+8y = -3 \, e^{\left(-5 \, t\right)}\]

Answer.

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

Example 60

C6 - Non-homogeneous second-order linear ODEs (ver. 60)

Find the solution to the given ODE.

\[y''+3y'+2y = -60 \, e^{\left(3 \, t\right)}\]

Answer.

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

Example 61

C6 - Non-homogeneous second-order linear ODEs (ver. 61)

Find the solution to the given ODE.

\[y''-3y'+2y = 12 \, e^{\left(4 \, t\right)}\]

Answer.

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

Example 62

C6 - Non-homogeneous second-order linear ODEs (ver. 62)

Find the solution to the given ODE.

\[y''-1y'-2y = 8 \, e^{\left(-2 \, t\right)}\]

Answer.

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

Example 63

C6 - Non-homogeneous second-order linear ODEs (ver. 63)

Find the solution to the given ODE.

\[y''+3y'-10y = 12 \, e^{\left(-2 \, t\right)}\]

Answer.

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

Example 64

C6 - Non-homogeneous second-order linear ODEs (ver. 64)

Find the solution to the given ODE.

\[y''-6y'+5y = 4 \, e^{\left(3 \, t\right)}\]

Answer.

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

Example 65

C6 - Non-homogeneous second-order linear ODEs (ver. 65)

Find the solution to the given ODE.

\[y''+7y'+12y = -288 \, e^{\left(5 \, t\right)}\]

Answer.

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

Example 66

C6 - Non-homogeneous second-order linear ODEs (ver. 66)

Find the solution to the given ODE.

\[y''+8y'+15y = 24 \, e^{\left(-t\right)}\]

Answer.

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

Example 67

C6 - Non-homogeneous second-order linear ODEs (ver. 67)

Find the solution to the given ODE.

\[y''+0y'-25y = 18 \, e^{\left(4 \, t\right)}\]

Answer.

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

Example 68

C6 - Non-homogeneous second-order linear ODEs (ver. 68)

Find the solution to the given ODE.

\[y''-6y'+5y = -12 \, e^{\left(2 \, t\right)}\]

Answer.

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

Example 69

C6 - Non-homogeneous second-order linear ODEs (ver. 69)

Find the solution to the given ODE.

\[y''+0y'-9y = 16 \, e^{\left(5 \, t\right)}\]

Answer.

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

Example 70

C6 - Non-homogeneous second-order linear ODEs (ver. 70)

Find the solution to the given ODE.

\[y''+4y'-5y = -9 \, e^{\left(-2 \, t\right)}\]

Answer.

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

Example 71

C6 - Non-homogeneous second-order linear ODEs (ver. 71)

Find the solution to the given ODE.

\[y''+7y'+10y = -270 \, e^{\left(4 \, t\right)}\]

Answer.

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

Example 72

C6 - Non-homogeneous second-order linear ODEs (ver. 72)

Find the solution to the given ODE.

\[y''+0y'-16y = 14 \, e^{\left(-3 \, t\right)}\]

Answer.

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

Example 73

C6 - Non-homogeneous second-order linear ODEs (ver. 73)

Find the solution to the given ODE.

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

Answer.

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

Example 74

C6 - Non-homogeneous second-order linear ODEs (ver. 74)

Find the solution to the given ODE.

\[y''+y'-6y = -70 \, e^{\left(-5 \, t\right)}\]

Answer.

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

Example 75

C6 - Non-homogeneous second-order linear ODEs (ver. 75)

Find the solution to the given ODE.

\[y''+7y'+10y = 210 \, e^{\left(5 \, t\right)}\]

Answer.

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

Example 76

C6 - Non-homogeneous second-order linear ODEs (ver. 76)

Find the solution to the given ODE.

\[y''-9y'+20y = 10 \, e^{\left(3 \, t\right)}\]

Answer.

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

Example 77

C6 - Non-homogeneous second-order linear ODEs (ver. 77)

Find the solution to the given ODE.

\[y''-1y'-20y = 70 \, e^{\left(-2 \, t\right)}\]

Answer.

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

Example 78

C6 - Non-homogeneous second-order linear ODEs (ver. 78)

Find the solution to the given ODE.

\[y''+2y'-8y = -40 \, e^{\left(-2 \, t\right)}\]

Answer.

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

Example 79

C6 - Non-homogeneous second-order linear ODEs (ver. 79)

Find the solution to the given ODE.

\[y''+6y'+8y = -35 \, e^{\left(3 \, t\right)}\]

Answer.

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

Example 80

C6 - Non-homogeneous second-order linear ODEs (ver. 80)

Find the solution to the given ODE.

\[y''-3y'-4y = -20 \, e^{\left(3 \, t\right)}\]

Answer.

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

Example 81

C6 - Non-homogeneous second-order linear ODEs (ver. 81)

Find the solution to the given ODE.

\[y''+9y'+20y = -24 \, e^{\left(-t\right)}\]

Answer.

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

Example 82

C6 - Non-homogeneous second-order linear ODEs (ver. 82)

Find the solution to the given ODE.

\[y''+7y'+10y = 54 \, e^{\left(4 \, t\right)}\]

Answer.

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

Example 83

C6 - Non-homogeneous second-order linear ODEs (ver. 83)

Find the solution to the given ODE.

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

Answer.

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

Example 84

C6 - Non-homogeneous second-order linear ODEs (ver. 84)

Find the solution to the given ODE.

\[y''+3y'-4y = -6 \, e^{\left(-2 \, t\right)}\]

Answer.

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

Example 85

C6 - Non-homogeneous second-order linear ODEs (ver. 85)

Find the solution to the given ODE.

\[y''+6y'+8y = -140 \, e^{\left(3 \, t\right)}\]

Answer.

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

Example 86

C6 - Non-homogeneous second-order linear ODEs (ver. 86)

Find the solution to the given ODE.

\[y''+0y'-4y = 48 \, e^{\left(4 \, t\right)}\]

Answer.

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

Example 87

C6 - Non-homogeneous second-order linear ODEs (ver. 87)

Find the solution to the given ODE.

\[y''-7y'+12y = 216 \, e^{\left(-5 \, t\right)}\]

Answer.

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

Example 88

C6 - Non-homogeneous second-order linear ODEs (ver. 88)

Find the solution to the given ODE.

\[y''+2y'-8y = 64 \, e^{\left(4 \, t\right)}\]

Answer.

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

Example 89

C6 - Non-homogeneous second-order linear ODEs (ver. 89)

Find the solution to the given ODE.

\[y''+y'-20y = -54 \, e^{t}\]

Answer.

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

Example 90

C6 - Non-homogeneous second-order linear ODEs (ver. 90)

Find the solution to the given ODE.

\[y''-4y'+3y = -45 \, e^{\left(-2 \, t\right)}\]

Answer.

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

Example 91

C6 - Non-homogeneous second-order linear ODEs (ver. 91)

Find the solution to the given ODE.

\[y''-1y'-20y = -32 \, e^{\left(-3 \, t\right)}\]

Answer.

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

Example 92

C6 - Non-homogeneous second-order linear ODEs (ver. 92)

Find the solution to the given ODE.

\[y''+0y'-1y = 24 \, e^{\left(5 \, t\right)}\]

Answer.

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

Example 93

C6 - Non-homogeneous second-order linear ODEs (ver. 93)

Find the solution to the given ODE.

\[y''+3y'-4y = 28 \, e^{\left(3 \, t\right)}\]

Answer.

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

Example 94

C6 - Non-homogeneous second-order linear ODEs (ver. 94)

Find the solution to the given ODE.

\[y''+y'-12y = 24 \, e^{\left(2 \, t\right)}\]

Answer.

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

Example 95

C6 - Non-homogeneous second-order linear ODEs (ver. 95)

Find the solution to the given ODE.

\[y''+2y'-8y = 25 \, e^{\left(-3 \, t\right)}\]

Answer.

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

Example 96

C6 - Non-homogeneous second-order linear ODEs (ver. 96)

Find the solution to the given ODE.

\[y''-7y'+12y = -24 \, e^{t}\]

Answer.

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

Example 97

C6 - Non-homogeneous second-order linear ODEs (ver. 97)

Find the solution to the given ODE.

\[y''-4y'-5y = -8 \, e^{\left(3 \, t\right)}\]

Answer.

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

Example 98

C6 - Non-homogeneous second-order linear ODEs (ver. 98)

Find the solution to the given ODE.

\[y''+6y'+8y = -35 \, e^{\left(3 \, t\right)}\]

Answer.

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

Example 99

C6 - Non-homogeneous second-order linear ODEs (ver. 99)

Find the solution to the given ODE.

\[y''-2y'-3y = -160 \, e^{\left(-5 \, t\right)}\]

Answer.

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

Example 100

C6 - Non-homogeneous second-order linear ODEs (ver. 100)

Find the solution to the given ODE.

\[y''-1y'-2y = -4 \, e^{t}\]

Answer.

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