C4 - Homogeneous second-order linear ODE


Example 1

C4 - Homogeneous second-order linear ODE (ver. 1)

Find the general solution to the given ODE.

\[{y''} + 6 \, {y'} + 10 \, {y} = 0\]

Answer.

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

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


Example 2

C4 - Homogeneous second-order linear ODE (ver. 2)

Find the general solution to the given ODE.

\[{y''} + 6 \, {y'} + 13 \, {y} = 0\]

Answer.

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

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


Example 3

C4 - Homogeneous second-order linear ODE (ver. 3)

Find the general solution to the given ODE.

\[{y''} - 18 \, {y'} + 81 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(9 \, t\right)} + k_{2} e^{\left(9 \, t\right)}\]


Example 4

C4 - Homogeneous second-order linear ODE (ver. 4)

Find the general solution to the given ODE.

\[{y''} + 4 \, {y'} + 13 \, {y} = 0\]

Answer.

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

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


Example 5

C4 - Homogeneous second-order linear ODE (ver. 5)

Find the general solution to the given ODE.

\[{y''} + 12 \, {y'} + 36 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}\]


Example 6

C4 - Homogeneous second-order linear ODE (ver. 6)

Find the general solution to the given ODE.

\[{y''} + 10 \, {y'} + 29 \, {y} = 0\]

Answer.

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

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


Example 7

C4 - Homogeneous second-order linear ODE (ver. 7)

Find the general solution to the given ODE.

\[{y''} - 8 \, {y'} + 41 \, {y} = 0\]

Answer.

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

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


Example 8

C4 - Homogeneous second-order linear ODE (ver. 8)

Find the general solution to the given ODE.

\[{y''} - 10 \, {y'} + 26 \, {y} = 0\]

Answer.

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

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


Example 9

C4 - Homogeneous second-order linear ODE (ver. 9)

Find the general solution to the given ODE.

\[{y''} - 4 \, {y'} + 8 \, {y} = 0\]

Answer.

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

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


Example 10

C4 - Homogeneous second-order linear ODE (ver. 10)

Find the general solution to the given ODE.

\[{y''} + 6 \, {y'} + 10 \, {y} = 0\]

Answer.

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

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


Example 11

C4 - Homogeneous second-order linear ODE (ver. 11)

Find the general solution to the given ODE.

\[{y''} + 10 \, {y'} + 25 \, {y} = 0\]

Answer.

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


Example 12

C4 - Homogeneous second-order linear ODE (ver. 12)

Find the general solution to the given ODE.

\[{y''} + 12 \, {y'} + 36 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}\]


Example 13

C4 - Homogeneous second-order linear ODE (ver. 13)

Find the general solution to the given ODE.

\[{y''} + 18 \, {y'} + 81 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}\]


Example 14

C4 - Homogeneous second-order linear ODE (ver. 14)

Find the general solution to the given ODE.

\[{y''} - 4 \, {y'} + 4 \, {y} = 0\]

Answer.

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


Example 15

C4 - Homogeneous second-order linear ODE (ver. 15)

Find the general solution to the given ODE.

\[{y''} + 8 \, {y'} + 32 \, {y} = 0\]

Answer.

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

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


Example 16

C4 - Homogeneous second-order linear ODE (ver. 16)

Find the general solution to the given ODE.

\[{y''} + 12 \, {y'} + 36 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(-6 \, t\right)} + k_{2} e^{\left(-6 \, t\right)}\]


Example 17

C4 - Homogeneous second-order linear ODE (ver. 17)

Find the general solution to the given ODE.

\[{y''} + 10 \, {y'} + 25 \, {y} = 0\]

Answer.

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


Example 18

C4 - Homogeneous second-order linear ODE (ver. 18)

Find the general solution to the given ODE.

\[{y''} - 4 \, {y'} + 8 \, {y} = 0\]

Answer.

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

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


Example 19

C4 - Homogeneous second-order linear ODE (ver. 19)

Find the general solution to the given ODE.

\[{y''} - 20 \, {y'} + 100 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}\]


Example 20

C4 - Homogeneous second-order linear ODE (ver. 20)

Find the general solution to the given ODE.

\[{y''} - 14 \, {y'} + 49 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}\]


Example 21

C4 - Homogeneous second-order linear ODE (ver. 21)

Find the general solution to the given ODE.

\[{y''} + 2 \, {y'} + {y} = 0\]

Answer.

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


Example 22

C4 - Homogeneous second-order linear ODE (ver. 22)

Find the general solution to the given ODE.

\[{y''} + 10 \, {y'} + 25 \, {y} = 0\]

Answer.

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


Example 23

C4 - Homogeneous second-order linear ODE (ver. 23)

Find the general solution to the given ODE.

\[{y''} + 4 \, {y'} + 4 \, {y} = 0\]

Answer.

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


Example 24

C4 - Homogeneous second-order linear ODE (ver. 24)

Find the general solution to the given ODE.

\[{y''} + 18 \, {y'} + 81 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}\]


Example 25

C4 - Homogeneous second-order linear ODE (ver. 25)

Find the general solution to the given ODE.

\[{y''} - 16 \, {y'} + 64 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(8 \, t\right)} + k_{2} e^{\left(8 \, t\right)}\]


Example 26

C4 - Homogeneous second-order linear ODE (ver. 26)

Find the general solution to the given ODE.

\[{y''} + 8 \, {y'} + 32 \, {y} = 0\]

Answer.

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

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


Example 27

C4 - Homogeneous second-order linear ODE (ver. 27)

Find the general solution to the given ODE.

\[{y''} - 18 \, {y'} + 81 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(9 \, t\right)} + k_{2} e^{\left(9 \, t\right)}\]


Example 28

C4 - Homogeneous second-order linear ODE (ver. 28)

Find the general solution to the given ODE.

\[{y''} - 20 \, {y'} + 100 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}\]


Example 29

C4 - Homogeneous second-order linear ODE (ver. 29)

Find the general solution to the given ODE.

\[{y''} + 6 \, {y'} + 10 \, {y} = 0\]

Answer.

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

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


Example 30

C4 - Homogeneous second-order linear ODE (ver. 30)

Find the general solution to the given ODE.

\[{y''} + 16 \, {y'} + 64 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}\]


Example 31

C4 - Homogeneous second-order linear ODE (ver. 31)

Find the general solution to the given ODE.

\[{y''} - 14 \, {y'} + 49 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(7 \, t\right)} + k_{2} e^{\left(7 \, t\right)}\]


Example 32

C4 - Homogeneous second-order linear ODE (ver. 32)

Find the general solution to the given ODE.

\[{y''} - 6 \, {y'} + 34 \, {y} = 0\]

Answer.

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

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


Example 33

C4 - Homogeneous second-order linear ODE (ver. 33)

Find the general solution to the given ODE.

\[{y''} - 10 \, {y'} + 25 \, {y} = 0\]

Answer.

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


Example 34

C4 - Homogeneous second-order linear ODE (ver. 34)

Find the general solution to the given ODE.

\[{y''} + 10 \, {y'} + 29 \, {y} = 0\]

Answer.

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

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


Example 35

C4 - Homogeneous second-order linear ODE (ver. 35)

Find the general solution to the given ODE.

\[{y''} - 18 \, {y'} + 81 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(9 \, t\right)} + k_{2} e^{\left(9 \, t\right)}\]


Example 36

C4 - Homogeneous second-order linear ODE (ver. 36)

Find the general solution to the given ODE.

\[{y''} - 4 \, {y'} + 8 \, {y} = 0\]

Answer.

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

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


Example 37

C4 - Homogeneous second-order linear ODE (ver. 37)

Find the general solution to the given ODE.

\[{y''} - 8 \, {y'} + 20 \, {y} = 0\]

Answer.

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

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


Example 38

C4 - Homogeneous second-order linear ODE (ver. 38)

Find the general solution to the given ODE.

\[{y''} + 4 \, {y'} + 4 \, {y} = 0\]

Answer.

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


Example 39

C4 - Homogeneous second-order linear ODE (ver. 39)

Find the general solution to the given ODE.

\[{y''} - 4 \, {y'} + 8 \, {y} = 0\]

Answer.

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

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


Example 40

C4 - Homogeneous second-order linear ODE (ver. 40)

Find the general solution to the given ODE.

\[{y''} + 8 \, {y'} + 16 \, {y} = 0\]

Answer.

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


Example 41

C4 - Homogeneous second-order linear ODE (ver. 41)

Find the general solution to the given ODE.

\[{y''} - 20 \, {y'} + 100 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}\]


Example 42

C4 - Homogeneous second-order linear ODE (ver. 42)

Find the general solution to the given ODE.

\[{y''} - 20 \, {y'} + 100 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(10 \, t\right)} + k_{2} e^{\left(10 \, t\right)}\]


Example 43

C4 - Homogeneous second-order linear ODE (ver. 43)

Find the general solution to the given ODE.

\[{y''} + 16 \, {y'} + 64 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}\]


Example 44

C4 - Homogeneous second-order linear ODE (ver. 44)

Find the general solution to the given ODE.

\[{y''} + 2 \, {y'} + {y} = 0\]

Answer.

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


Example 45

C4 - Homogeneous second-order linear ODE (ver. 45)

Find the general solution to the given ODE.

\[{y''} + 6 \, {y'} + 25 \, {y} = 0\]

Answer.

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

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


Example 46

C4 - Homogeneous second-order linear ODE (ver. 46)

Find the general solution to the given ODE.

\[{y''} + 16 \, {y'} + 64 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(-8 \, t\right)} + k_{2} e^{\left(-8 \, t\right)}\]


Example 47

C4 - Homogeneous second-order linear ODE (ver. 47)

Find the general solution to the given ODE.

\[{y''} + 4 \, {y'} + 8 \, {y} = 0\]

Answer.

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

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


Example 48

C4 - Homogeneous second-order linear ODE (ver. 48)

Find the general solution to the given ODE.

\[{y''} - 12 \, {y'} + 36 \, {y} = 0\]

Answer.

\[{y} = k_{1} t e^{\left(6 \, t\right)} + k_{2} e^{\left(6 \, t\right)}\]


Example 49

C4 - Homogeneous second-order linear ODE (ver. 49)

Find the general solution to the given ODE.

\[{y''} + 6 \, {y'} + 9 \, {y} = 0\]

Answer.

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


Example 50

C4 - Homogeneous second-order linear ODE (ver. 50)

Find the general solution to the given ODE.

\[{y''} - 2 \, {y'} + 17 \, {y} = 0\]

Answer.

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

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