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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

Answer.

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