C2 - Non-homogeneous first-order linear ODE

Example 1

C2 - Non-homogeneous first-order linear ODE (ver. 1)

Find the general solution to the given ODE.

\[y'-5y= -9 \, \cos\left(-3 \, t\right) e^{\left(5 \, t\right)}\]

Answer.

\[y= k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(-3 \, t\right)\]

Example 2

C2 - Non-homogeneous first-order linear ODE (ver. 2)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(-5 \, t\right)} - 2 \, e^{\left(-5 \, t\right)} \sin\left(-t\right)\]

Example 3

C2 - Non-homogeneous first-order linear ODE (ver. 3)

Find the general solution to the given ODE.

\[y'-3y= 15 \, e^{\left(3 \, t\right)} \sin\left(-5 \, t\right)\]

Answer.

\[y= k e^{\left(3 \, t\right)} + 3 \, \cos\left(-5 \, t\right) e^{\left(3 \, t\right)}\]

Example 4

C2 - Non-homogeneous first-order linear ODE (ver. 4)

Find the general solution to the given ODE.

\[y'+4y= 15 \, e^{t}\]

Answer.

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

Example 5

C2 - Non-homogeneous first-order linear ODE (ver. 5)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{t} + 2 \, t e^{t}\]

Example 6

C2 - Non-homogeneous first-order linear ODE (ver. 6)

Find the general solution to the given ODE.

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

Answer.

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

Example 7

C2 - Non-homogeneous first-order linear ODE (ver. 7)

Find the general solution to the given ODE.

\[y'-5y= 9 \, \cos\left(3 \, t\right) e^{\left(5 \, t\right)}\]

Answer.

\[y= k e^{\left(5 \, t\right)} + 3 \, e^{\left(5 \, t\right)} \sin\left(3 \, t\right)\]

Example 8

C2 - Non-homogeneous first-order linear ODE (ver. 8)

Find the general solution to the given ODE.

\[y'-5y= 9 \, e^{\left(5 \, t\right)} \sin\left(-3 \, t\right)\]

Answer.

\[y= k e^{\left(5 \, t\right)} + 3 \, \cos\left(-3 \, t\right) e^{\left(5 \, t\right)}\]

Example 9

C2 - Non-homogeneous first-order linear ODE (ver. 9)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(-4 \, t\right)} - 3 \, t e^{\left(-4 \, t\right)}\]

Example 10

C2 - Non-homogeneous first-order linear ODE (ver. 10)

Find the general solution to the given ODE.

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

Answer.

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

Example 11

C2 - Non-homogeneous first-order linear ODE (ver. 11)

Find the general solution to the given ODE.

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

Answer.

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

Example 12

C2 - Non-homogeneous first-order linear ODE (ver. 12)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(3 \, t\right)} + 2 \, \cos\left(2 \, t\right) e^{\left(3 \, t\right)}\]

Example 13

C2 - Non-homogeneous first-order linear ODE (ver. 13)

Find the general solution to the given ODE.

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

Answer.

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

Example 14

C2 - Non-homogeneous first-order linear ODE (ver. 14)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(-t\right)} - 2 \, e^{\left(-t\right)} \sin\left(-5 \, t\right)\]

Example 15

C2 - Non-homogeneous first-order linear ODE (ver. 15)

Find the general solution to the given ODE.

\[y'-4y= -6 \, e^{t}\]

Answer.

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

Example 16

C2 - Non-homogeneous first-order linear ODE (ver. 16)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(3 \, t\right)} - 3 \, \cos\left(t\right) e^{\left(3 \, t\right)}\]

Example 17

C2 - Non-homogeneous first-order linear ODE (ver. 17)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(-3 \, t\right)} - 2 \, e^{\left(-3 \, t\right)} \sin\left(3 \, t\right)\]

Example 18

C2 - Non-homogeneous first-order linear ODE (ver. 18)

Find the general solution to the given ODE.

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

Answer.

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

Example 19

C2 - Non-homogeneous first-order linear ODE (ver. 19)

Find the general solution to the given ODE.

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

Answer.

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

Example 20

C2 - Non-homogeneous first-order linear ODE (ver. 20)

Find the general solution to the given ODE.

\[y'-3y= -15 \, e^{\left(3 \, t\right)} \sin\left(-5 \, t\right)\]

Answer.

\[y= k e^{\left(3 \, t\right)} - 3 \, \cos\left(-5 \, t\right) e^{\left(3 \, t\right)}\]

Example 21

C2 - Non-homogeneous first-order linear ODE (ver. 21)

Find the general solution to the given ODE.

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

Answer.

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

Example 22

C2 - Non-homogeneous first-order linear ODE (ver. 22)

Find the general solution to the given ODE.

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

Answer.

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

Example 23

C2 - Non-homogeneous first-order linear ODE (ver. 23)

Find the general solution to the given ODE.

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

Answer.

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

Example 24

C2 - Non-homogeneous first-order linear ODE (ver. 24)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(3 \, t\right)} + 3 \, t e^{\left(3 \, t\right)}\]

Example 25

C2 - Non-homogeneous first-order linear ODE (ver. 25)

Find the general solution to the given ODE.

\[y'-3y= 10 \, e^{\left(3 \, t\right)} \sin\left(5 \, t\right)\]

Answer.

\[y= k e^{\left(3 \, t\right)} - 2 \, \cos\left(5 \, t\right) e^{\left(3 \, t\right)}\]

Example 26

C2 - Non-homogeneous first-order linear ODE (ver. 26)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(5 \, t\right)} + 2 \, \cos\left(t\right) e^{\left(5 \, t\right)}\]

Example 27

C2 - Non-homogeneous first-order linear ODE (ver. 27)

Find the general solution to the given ODE.

\[y'-1y= 15 \, \cos\left(-5 \, t\right) e^{t}\]

Answer.

\[y= k e^{t} - 3 \, e^{t} \sin\left(-5 \, t\right)\]

Example 28

C2 - Non-homogeneous first-order linear ODE (ver. 28)

Find the general solution to the given ODE.

\[y'-3y= -15 \, e^{\left(3 \, t\right)} \sin\left(-5 \, t\right)\]

Answer.

\[y= k e^{\left(3 \, t\right)} - 3 \, \cos\left(-5 \, t\right) e^{\left(3 \, t\right)}\]

Example 29

C2 - Non-homogeneous first-order linear ODE (ver. 29)

Find the general solution to the given ODE.

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

Answer.

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

Example 30

C2 - Non-homogeneous first-order linear ODE (ver. 30)

Find the general solution to the given ODE.

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

Answer.

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

Example 31

C2 - Non-homogeneous first-order linear ODE (ver. 31)

Find the general solution to the given ODE.

\[y'-1y= 3 \, e^{t}\]

Answer.

\[y= k e^{t} + 3 \, t e^{t}\]

Example 32

C2 - Non-homogeneous first-order linear ODE (ver. 32)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(3 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(3 \, t\right)}\]

Example 33

C2 - Non-homogeneous first-order linear ODE (ver. 33)

Find the general solution to the given ODE.

\[y'+y= -15 \, \cos\left(5 \, t\right) e^{\left(-t\right)}\]

Answer.

\[y= k e^{\left(-t\right)} - 3 \, e^{\left(-t\right)} \sin\left(5 \, t\right)\]

Example 34

C2 - Non-homogeneous first-order linear ODE (ver. 34)

Find the general solution to the given ODE.

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

Answer.

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

Example 35

C2 - Non-homogeneous first-order linear ODE (ver. 35)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(2 \, t\right)} + 2 \, \cos\left(3 \, t\right) e^{\left(2 \, t\right)}\]

Example 36

C2 - Non-homogeneous first-order linear ODE (ver. 36)

Find the general solution to the given ODE.

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

Answer.

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

Example 37

C2 - Non-homogeneous first-order linear ODE (ver. 37)

Find the general solution to the given ODE.

\[y'-4y= 10 \, \cos\left(-5 \, t\right) e^{\left(4 \, t\right)}\]

Answer.

\[y= k e^{\left(4 \, t\right)} - 2 \, e^{\left(4 \, t\right)} \sin\left(-5 \, t\right)\]

Example 38

C2 - Non-homogeneous first-order linear ODE (ver. 38)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(-2 \, t\right)} - 3 \, \cos\left(t\right) e^{\left(-2 \, t\right)}\]

Example 39

C2 - Non-homogeneous first-order linear ODE (ver. 39)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(-5 \, t\right)} - 3 \, e^{\left(-4 \, t\right)}\]

Example 40

C2 - Non-homogeneous first-order linear ODE (ver. 40)

Find the general solution to the given ODE.

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

Answer.

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

Example 41

C2 - Non-homogeneous first-order linear ODE (ver. 41)

Find the general solution to the given ODE.

\[y'-2y= -10 \, e^{\left(2 \, t\right)} \sin\left(-5 \, t\right)\]

Answer.

\[y= k e^{\left(2 \, t\right)} - 2 \, \cos\left(-5 \, t\right) e^{\left(2 \, t\right)}\]

Example 42

C2 - Non-homogeneous first-order linear ODE (ver. 42)

Find the general solution to the given ODE.

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

Answer.

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

Example 43

C2 - Non-homogeneous first-order linear ODE (ver. 43)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(2 \, t\right)} + 2 \, \cos\left(3 \, t\right) e^{\left(2 \, t\right)}\]

Example 44

C2 - Non-homogeneous first-order linear ODE (ver. 44)

Find the general solution to the given ODE.

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

Answer.

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

Example 45

C2 - Non-homogeneous first-order linear ODE (ver. 45)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(5 \, t\right)} - 3 \, t e^{\left(5 \, t\right)}\]

Example 46

C2 - Non-homogeneous first-order linear ODE (ver. 46)

Find the general solution to the given ODE.

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

Answer.

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

Example 47

C2 - Non-homogeneous first-order linear ODE (ver. 47)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(-5 \, t\right)} - 3 \, t e^{\left(-5 \, t\right)}\]

Example 48

C2 - Non-homogeneous first-order linear ODE (ver. 48)

Find the general solution to the given ODE.

\[y'-1y= 8 \, \cos\left(4 \, t\right) e^{t}\]

Answer.

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

Example 49

C2 - Non-homogeneous first-order linear ODE (ver. 49)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(-3 \, t\right)} - 3 \, \cos\left(-2 \, t\right) e^{\left(-3 \, t\right)}\]

Example 50

C2 - Non-homogeneous first-order linear ODE (ver. 50)

Find the general solution to the given ODE.

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

Answer.

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

Example 51

C2 - Non-homogeneous first-order linear ODE (ver. 51)

Find the general solution to the given ODE.

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

Answer.

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

Example 52

C2 - Non-homogeneous first-order linear ODE (ver. 52)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(2 \, t\right)} + 2 \, e^{\left(2 \, t\right)} \sin\left(3 \, t\right)\]

Example 53

C2 - Non-homogeneous first-order linear ODE (ver. 53)

Find the general solution to the given ODE.

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

Answer.

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

Example 54

C2 - Non-homogeneous first-order linear ODE (ver. 54)

Find the general solution to the given ODE.

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

Answer.

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

Example 55

C2 - Non-homogeneous first-order linear ODE (ver. 55)

Find the general solution to the given ODE.

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

Answer.

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

Example 56

C2 - Non-homogeneous first-order linear ODE (ver. 56)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(-3 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(-3 \, t\right)}\]

Example 57

C2 - Non-homogeneous first-order linear ODE (ver. 57)

Find the general solution to the given ODE.

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

Answer.

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

Example 58

C2 - Non-homogeneous first-order linear ODE (ver. 58)

Find the general solution to the given ODE.

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

Answer.

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

Example 59

C2 - Non-homogeneous first-order linear ODE (ver. 59)

Find the general solution to the given ODE.

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

Answer.

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

Example 60

C2 - Non-homogeneous first-order linear ODE (ver. 60)

Find the general solution to the given ODE.

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

Answer.

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

Example 61

C2 - Non-homogeneous first-order linear ODE (ver. 61)

Find the general solution to the given ODE.

\[y'-1y= 4 \, \cos\left(2 \, t\right) e^{t}\]

Answer.

\[y= k e^{t} + 2 \, e^{t} \sin\left(2 \, t\right)\]

Example 62

C2 - Non-homogeneous first-order linear ODE (ver. 62)

Find the general solution to the given ODE.

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

Answer.

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

Example 63

C2 - Non-homogeneous first-order linear ODE (ver. 63)

Find the general solution to the given ODE.

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

Answer.

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

Example 64

C2 - Non-homogeneous first-order linear ODE (ver. 64)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(5 \, t\right)} - 2 \, e^{\left(5 \, t\right)} \sin\left(t\right)\]

Example 65

C2 - Non-homogeneous first-order linear ODE (ver. 65)

Find the general solution to the given ODE.

\[y'+4y= 9 \, e^{\left(-4 \, t\right)} \sin\left(-3 \, t\right)\]

Answer.

\[y= k e^{\left(-4 \, t\right)} + 3 \, \cos\left(-3 \, t\right) e^{\left(-4 \, t\right)}\]

Example 66

C2 - Non-homogeneous first-order linear ODE (ver. 66)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(-3 \, t\right)} - 3 \, t e^{\left(-3 \, t\right)}\]

Example 67

C2 - Non-homogeneous first-order linear ODE (ver. 67)

Find the general solution to the given ODE.

\[y'-5y= -30 \, e^{\left(-5 \, t\right)}\]

Answer.

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

Example 68

C2 - Non-homogeneous first-order linear ODE (ver. 68)

Find the general solution to the given ODE.

\[y'-1y= 15 \, e^{t} \sin\left(5 \, t\right)\]

Answer.

\[y= k e^{t} - 3 \, \cos\left(5 \, t\right) e^{t}\]

Example 69

C2 - Non-homogeneous first-order linear ODE (ver. 69)

Find the general solution to the given ODE.

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

Answer.

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

Example 70

C2 - Non-homogeneous first-order linear ODE (ver. 70)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(-5 \, t\right)} - 3 \, e^{\left(-5 \, t\right)} \sin\left(t\right)\]

Example 71

C2 - Non-homogeneous first-order linear ODE (ver. 71)

Find the general solution to the given ODE.

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

Answer.

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

Example 72

C2 - Non-homogeneous first-order linear ODE (ver. 72)

Find the general solution to the given ODE.

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

Answer.

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

Example 73

C2 - Non-homogeneous first-order linear ODE (ver. 73)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(4 \, t\right)} - 2 \, \cos\left(3 \, t\right) e^{\left(4 \, t\right)}\]

Example 74

C2 - Non-homogeneous first-order linear ODE (ver. 74)

Find the general solution to the given ODE.

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

Answer.

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

Example 75

C2 - Non-homogeneous first-order linear ODE (ver. 75)

Find the general solution to the given ODE.

\[y'+2y= -15 \, \cos\left(-5 \, t\right) e^{\left(-2 \, t\right)}\]

Answer.

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

Example 76

C2 - Non-homogeneous first-order linear ODE (ver. 76)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(5 \, t\right)} - 2 \, \cos\left(4 \, t\right) e^{\left(5 \, t\right)}\]

Example 77

C2 - Non-homogeneous first-order linear ODE (ver. 77)

Find the general solution to the given ODE.

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

Answer.

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

Example 78

C2 - Non-homogeneous first-order linear ODE (ver. 78)

Find the general solution to the given ODE.

\[y'-2y= 18 \, e^{\left(-4 \, t\right)}\]

Answer.

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

Example 79

C2 - Non-homogeneous first-order linear ODE (ver. 79)

Find the general solution to the given ODE.

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

Answer.

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

Example 80

C2 - Non-homogeneous first-order linear ODE (ver. 80)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(4 \, t\right)} - 2 \, \cos\left(-t\right) e^{\left(4 \, t\right)}\]

Example 81

C2 - Non-homogeneous first-order linear ODE (ver. 81)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(-5 \, t\right)} - 3 \, e^{\left(-4 \, t\right)}\]

Example 82

C2 - Non-homogeneous first-order linear ODE (ver. 82)

Find the general solution to the given ODE.

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

Answer.

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

Example 83

C2 - Non-homogeneous first-order linear ODE (ver. 83)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{\left(-t\right)} - 3 \, e^{\left(-3 \, t\right)}\]

Example 84

C2 - Non-homogeneous first-order linear ODE (ver. 84)

Find the general solution to the given ODE.

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

Answer.

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

Example 85

C2 - Non-homogeneous first-order linear ODE (ver. 85)

Find the general solution to the given ODE.

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

Answer.

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

Example 86

C2 - Non-homogeneous first-order linear ODE (ver. 86)

Find the general solution to the given ODE.

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

Answer.

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

Example 87

C2 - Non-homogeneous first-order linear ODE (ver. 87)

Find the general solution to the given ODE.

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

Answer.

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

Example 88

C2 - Non-homogeneous first-order linear ODE (ver. 88)

Find the general solution to the given ODE.

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

Answer.

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

Example 89

C2 - Non-homogeneous first-order linear ODE (ver. 89)

Find the general solution to the given ODE.

\[y'+5y= -12 \, \cos\left(4 \, t\right) e^{\left(-5 \, t\right)}\]

Answer.

\[y= k e^{\left(-5 \, t\right)} - 3 \, e^{\left(-5 \, t\right)} \sin\left(4 \, t\right)\]

Example 90

C2 - Non-homogeneous first-order linear ODE (ver. 90)

Find the general solution to the given ODE.

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

Answer.

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

Example 91

C2 - Non-homogeneous first-order linear ODE (ver. 91)

Find the general solution to the given ODE.

\[y'+4y= 15 \, \cos\left(5 \, t\right) e^{\left(-4 \, t\right)}\]

Answer.

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

Example 92

C2 - Non-homogeneous first-order linear ODE (ver. 92)

Find the general solution to the given ODE.

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

Answer.

\[y= k e^{t} + 3 \, e^{t} \sin\left(-t\right)\]

Example 93

C2 - Non-homogeneous first-order linear ODE (ver. 93)

Find the general solution to the given ODE.

\[y'-1y= -12 \, e^{t} \sin\left(-4 \, t\right)\]

Answer.

\[y= k e^{t} - 3 \, \cos\left(-4 \, t\right) e^{t}\]

Example 94

C2 - Non-homogeneous first-order linear ODE (ver. 94)

Find the general solution to the given ODE.

\[y'+4y= -9 \, \cos\left(3 \, t\right) e^{\left(-4 \, t\right)}\]

Answer.

\[y= k e^{\left(-4 \, t\right)} - 3 \, e^{\left(-4 \, t\right)} \sin\left(3 \, t\right)\]

Example 95

C2 - Non-homogeneous first-order linear ODE (ver. 95)

Find the general solution to the given ODE.

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

Answer.

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

Example 96

C2 - Non-homogeneous first-order linear ODE (ver. 96)

Find the general solution to the given ODE.

\[y'-4y= 9 \, e^{\left(4 \, t\right)} \sin\left(3 \, t\right)\]

Answer.

\[y= k e^{\left(4 \, t\right)} - 3 \, \cos\left(3 \, t\right) e^{\left(4 \, t\right)}\]

Example 97

C2 - Non-homogeneous first-order linear ODE (ver. 97)

Find the general solution to the given ODE.

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

Answer.

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

Example 98

C2 - Non-homogeneous first-order linear ODE (ver. 98)

Find the general solution to the given ODE.

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

Answer.

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

Example 99

C2 - Non-homogeneous first-order linear ODE (ver. 99)

Find the general solution to the given ODE.

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

Answer.

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

Example 100

C2 - Non-homogeneous first-order linear ODE (ver. 100)

Find the general solution to the given ODE.

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

Answer.

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