C4b - Homogeneous second-order linear ODE

Example 1

C4b - Homogeneous second-order linear ODE (ver. 1)

Find the general solution to the given ODE.

\[y''-10y'+34y = 0\]

Answer.

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

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

Example 2

C4b - Homogeneous second-order linear ODE (ver. 2)

Find the general solution to the given ODE.

\[y''+10y'+26y = 0\]

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 3

C4b - Homogeneous second-order linear ODE (ver. 3)

Find the general solution to the given ODE.

\[y''-6y'+34y = 0\]

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 4

C4b - Homogeneous second-order linear ODE (ver. 4)

Find the general solution to the given ODE.

\[y''+8y'+17y = 0\]

Answer.

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

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

Example 5

C4b - Homogeneous second-order linear ODE (ver. 5)

Find the general solution to the given ODE.

\[y''-2y'+5y = 0\]

Answer.

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

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

Example 6

C4b - Homogeneous second-order linear ODE (ver. 6)

Find the general solution to the given ODE.

\[y''-10y'+29y = 0\]

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

C4b - Homogeneous second-order linear ODE (ver. 7)

Find the general solution to the given ODE.

\[y''-10y'+34y = 0\]

Answer.

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

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

Example 8

C4b - Homogeneous second-order linear ODE (ver. 8)

Find the general solution to the given ODE.

\[y''-10y'+34y = 0\]

Answer.

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

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

Example 9

C4b - Homogeneous second-order linear ODE (ver. 9)

Find the general solution to the given ODE.

\[y''+8y'+25y = 0\]

Answer.

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

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

Example 10

C4b - Homogeneous second-order linear ODE (ver. 10)

Find the general solution to the given ODE.

\[y''+10y'+34y = 0\]

Answer.

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

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

Example 11

C4b - Homogeneous second-order linear ODE (ver. 11)

Find the general solution to the given ODE.

\[y''+6y'+10y = 0\]

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 12

C4b - Homogeneous second-order linear ODE (ver. 12)

Find the general solution to the given ODE.

\[y''-6y'+18y = 0\]

Answer.

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

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

Example 13

C4b - Homogeneous second-order linear ODE (ver. 13)

Find the general solution to the given ODE.

\[y''-4y'+5y = 0\]

Answer.

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

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

Example 14

C4b - Homogeneous second-order linear ODE (ver. 14)

Find the general solution to the given ODE.

\[y''+2y'+26y = 0\]

Answer.

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

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

Example 15

C4b - Homogeneous second-order linear ODE (ver. 15)

Find the general solution to the given ODE.

\[y''-8y'+17y = 0\]

Answer.

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

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

Example 16

C4b - Homogeneous second-order linear ODE (ver. 16)

Find the general solution to the given ODE.

\[y''-6y'+10y = 0\]

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 17

C4b - Homogeneous second-order linear ODE (ver. 17)

Find the general solution to the given ODE.

\[y''+6y'+18y = 0\]

Answer.

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

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

Example 18

C4b - Homogeneous second-order linear ODE (ver. 18)

Find the general solution to the given ODE.

\[y''+10y'+34y = 0\]

Answer.

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

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

Example 19

C4b - Homogeneous second-order linear ODE (ver. 19)

Find the general solution to the given ODE.

\[y''-4y'+29y = 0\]

Answer.

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

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

Example 20

C4b - Homogeneous second-order linear ODE (ver. 20)

Find the general solution to the given ODE.

\[y''-6y'+18y = 0\]

Answer.

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

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

Example 21

C4b - Homogeneous second-order linear ODE (ver. 21)

Find the general solution to the given ODE.

\[y''+4y'+8y = 0\]

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 22

C4b - Homogeneous second-order linear ODE (ver. 22)

Find the general solution to the given ODE.

\[y''+4y'+5y = 0\]

Answer.

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

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

Example 23

C4b - Homogeneous second-order linear ODE (ver. 23)

Find the general solution to the given ODE.

\[y''+2y'+2y = 0\]

Answer.

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

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

Example 24

C4b - Homogeneous second-order linear ODE (ver. 24)

Find the general solution to the given ODE.

\[y''-6y'+13y = 0\]

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 25

C4b - Homogeneous second-order linear ODE (ver. 25)

Find the general solution to the given ODE.

\[y''-6y'+34y = 0\]

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 26

C4b - Homogeneous second-order linear ODE (ver. 26)

Find the general solution to the given ODE.

\[y''-10y'+26y = 0\]

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 27

C4b - Homogeneous second-order linear ODE (ver. 27)

Find the general solution to the given ODE.

\[y''-2y'+26y = 0\]

Answer.

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

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

Example 28

C4b - Homogeneous second-order linear ODE (ver. 28)

Find the general solution to the given ODE.

\[y''-6y'+34y = 0\]

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 29

C4b - Homogeneous second-order linear ODE (ver. 29)

Find the general solution to the given ODE.

\[y''-10y'+29y = 0\]

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 30

C4b - Homogeneous second-order linear ODE (ver. 30)

Find the general solution to the given ODE.

\[y''-4y'+5y = 0\]

Answer.

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

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

Example 31

C4b - Homogeneous second-order linear ODE (ver. 31)

Find the general solution to the given ODE.

\[y''-2y'+17y = 0\]

Answer.

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

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

Example 32

C4b - Homogeneous second-order linear ODE (ver. 32)

Find the general solution to the given ODE.

\[y''-6y'+25y = 0\]

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 33

C4b - Homogeneous second-order linear ODE (ver. 33)

Find the general solution to the given ODE.

\[y''+2y'+26y = 0\]

Answer.

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

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

Example 34

C4b - Homogeneous second-order linear ODE (ver. 34)

Find the general solution to the given ODE.

\[y''-8y'+20y = 0\]

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 35

C4b - Homogeneous second-order linear ODE (ver. 35)

Find the general solution to the given ODE.

\[y''-4y'+13y = 0\]

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 36

C4b - Homogeneous second-order linear ODE (ver. 36)

Find the general solution to the given ODE.

\[y''+10y'+34y = 0\]

Answer.

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

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

Example 37

C4b - Homogeneous second-order linear ODE (ver. 37)

Find the general solution to the given ODE.

\[y''-8y'+41y = 0\]

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 38

C4b - Homogeneous second-order linear ODE (ver. 38)

Find the general solution to the given ODE.

\[y''+4y'+5y = 0\]

Answer.

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

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

Example 39

C4b - Homogeneous second-order linear ODE (ver. 39)

Find the general solution to the given ODE.

\[y''+10y'+41y = 0\]

Answer.

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

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

Example 40

C4b - Homogeneous second-order linear ODE (ver. 40)

Find the general solution to the given ODE.

\[y''+4y'+5y = 0\]

Answer.

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

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

Example 41

C4b - Homogeneous second-order linear ODE (ver. 41)

Find the general solution to the given ODE.

\[y''-4y'+29y = 0\]

Answer.

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

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

Example 42

C4b - Homogeneous second-order linear ODE (ver. 42)

Find the general solution to the given ODE.

\[y''-4y'+20y = 0\]

Answer.

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

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

Example 43

C4b - Homogeneous second-order linear ODE (ver. 43)

Find the general solution to the given ODE.

\[y''-4y'+13y = 0\]

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 44

C4b - Homogeneous second-order linear ODE (ver. 44)

Find the general solution to the given ODE.

\[y''+2y'+2y = 0\]

Answer.

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

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

Example 45

C4b - Homogeneous second-order linear ODE (ver. 45)

Find the general solution to the given ODE.

\[y''-10y'+26y = 0\]

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 46

C4b - Homogeneous second-order linear ODE (ver. 46)

Find the general solution to the given ODE.

\[y''-4y'+13y = 0\]

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 47

C4b - Homogeneous second-order linear ODE (ver. 47)

Find the general solution to the given ODE.

\[y''+10y'+26y = 0\]

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 48

C4b - Homogeneous second-order linear ODE (ver. 48)

Find the general solution to the given ODE.

\[y''-2y'+17y = 0\]

Answer.

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

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

Example 49

C4b - Homogeneous second-order linear ODE (ver. 49)

Find the general solution to the given ODE.

\[y''+6y'+13y = 0\]

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 50

C4b - Homogeneous second-order linear ODE (ver. 50)

Find the general solution to the given ODE.

\[y''+6y'+34y = 0\]

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 51

C4b - Homogeneous second-order linear ODE (ver. 51)

Find the general solution to the given ODE.

\[y''+4y'+20y = 0\]

Answer.

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

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

Example 52

C4b - Homogeneous second-order linear ODE (ver. 52)

Find the general solution to the given ODE.

\[y''-4y'+13y = 0\]

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 53

C4b - Homogeneous second-order linear ODE (ver. 53)

Find the general solution to the given ODE.

\[y''-8y'+41y = 0\]

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 54

C4b - Homogeneous second-order linear ODE (ver. 54)

Find the general solution to the given ODE.

\[y''-8y'+25y = 0\]

Answer.

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

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

Example 55

C4b - Homogeneous second-order linear ODE (ver. 55)

Find the general solution to the given ODE.

\[y''-10y'+26y = 0\]

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 56

C4b - Homogeneous second-order linear ODE (ver. 56)

Find the general solution to the given ODE.

\[y''+6y'+25y = 0\]

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 57

C4b - Homogeneous second-order linear ODE (ver. 57)

Find the general solution to the given ODE.

\[y''+8y'+32y = 0\]

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 58

C4b - Homogeneous second-order linear ODE (ver. 58)

Find the general solution to the given ODE.

\[y''+8y'+20y = 0\]

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 59

C4b - Homogeneous second-order linear ODE (ver. 59)

Find the general solution to the given ODE.

\[y''+4y'+20y = 0\]

Answer.

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

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

Example 60

C4b - Homogeneous second-order linear ODE (ver. 60)

Find the general solution to the given ODE.

\[y''+4y'+5y = 0\]

Answer.

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

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

Example 61

C4b - Homogeneous second-order linear ODE (ver. 61)

Find the general solution to the given ODE.

\[y''-2y'+5y = 0\]

Answer.

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

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

Example 62

C4b - Homogeneous second-order linear ODE (ver. 62)

Find the general solution to the given ODE.

\[y''+2y'+5y = 0\]

Answer.

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

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

Example 63

C4b - Homogeneous second-order linear ODE (ver. 63)

Find the general solution to the given ODE.

\[y''+10y'+29y = 0\]

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 64

C4b - Homogeneous second-order linear ODE (ver. 64)

Find the general solution to the given ODE.

\[y''-10y'+26y = 0\]

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 65

C4b - Homogeneous second-order linear ODE (ver. 65)

Find the general solution to the given ODE.

\[y''+8y'+25y = 0\]

Answer.

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

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

Example 66

C4b - Homogeneous second-order linear ODE (ver. 66)

Find the general solution to the given ODE.

\[y''+6y'+34y = 0\]

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 67

C4b - Homogeneous second-order linear ODE (ver. 67)

Find the general solution to the given ODE.

\[y''-10y'+50y = 0\]

Answer.

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

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

Example 68

C4b - Homogeneous second-order linear ODE (ver. 68)

Find the general solution to the given ODE.

\[y''-2y'+26y = 0\]

Answer.

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

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

Example 69

C4b - Homogeneous second-order linear ODE (ver. 69)

Find the general solution to the given ODE.

\[y''-6y'+18y = 0\]

Answer.

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

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

Example 70

C4b - Homogeneous second-order linear ODE (ver. 70)

Find the general solution to the given ODE.

\[y''+10y'+26y = 0\]

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 71

C4b - Homogeneous second-order linear ODE (ver. 71)

Find the general solution to the given ODE.

\[y''+10y'+50y = 0\]

Answer.

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

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

Example 72

C4b - Homogeneous second-order linear ODE (ver. 72)

Find the general solution to the given ODE.

\[y''-8y'+32y = 0\]

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 73

C4b - Homogeneous second-order linear ODE (ver. 73)

Find the general solution to the given ODE.

\[y''-8y'+25y = 0\]

Answer.

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

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

Example 74

C4b - Homogeneous second-order linear ODE (ver. 74)

Find the general solution to the given ODE.

\[y''-4y'+13y = 0\]

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 75

C4b - Homogeneous second-order linear ODE (ver. 75)

Find the general solution to the given ODE.

\[y''+4y'+29y = 0\]

Answer.

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

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

Example 76

C4b - Homogeneous second-order linear ODE (ver. 76)

Find the general solution to the given ODE.

\[y''-10y'+41y = 0\]

Answer.

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

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

Example 77

C4b - Homogeneous second-order linear ODE (ver. 77)

Find the general solution to the given ODE.

\[y''-10y'+50y = 0\]

Answer.

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

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

Example 78

C4b - Homogeneous second-order linear ODE (ver. 78)

Find the general solution to the given ODE.

\[y''-4y'+20y = 0\]

Answer.

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

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

Example 79

C4b - Homogeneous second-order linear ODE (ver. 79)

Find the general solution to the given ODE.

\[y''+4y'+8y = 0\]

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 80

C4b - Homogeneous second-order linear ODE (ver. 80)

Find the general solution to the given ODE.

\[y''-8y'+17y = 0\]

Answer.

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

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

Example 81

C4b - Homogeneous second-order linear ODE (ver. 81)

Find the general solution to the given ODE.

\[y''+10y'+41y = 0\]

Answer.

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

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

Example 82

C4b - Homogeneous second-order linear ODE (ver. 82)

Find the general solution to the given ODE.

\[y''+10y'+29y = 0\]

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 83

C4b - Homogeneous second-order linear ODE (ver. 83)

Find the general solution to the given ODE.

\[y''+2y'+10y = 0\]

Answer.

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

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

Example 84

C4b - Homogeneous second-order linear ODE (ver. 84)

Find the general solution to the given ODE.

\[y''+8y'+32y = 0\]

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 85

C4b - Homogeneous second-order linear ODE (ver. 85)

Find the general solution to the given ODE.

\[y''+4y'+20y = 0\]

Answer.

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

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

Example 86

C4b - Homogeneous second-order linear ODE (ver. 86)

Find the general solution to the given ODE.

\[y''+4y'+8y = 0\]

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 87

C4b - Homogeneous second-order linear ODE (ver. 87)

Find the general solution to the given ODE.

\[y''-6y'+25y = 0\]

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 88

C4b - Homogeneous second-order linear ODE (ver. 88)

Find the general solution to the given ODE.

\[y''+8y'+20y = 0\]

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 89

C4b - Homogeneous second-order linear ODE (ver. 89)

Find the general solution to the given ODE.

\[y''+10y'+41y = 0\]

Answer.

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

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

Example 90

C4b - Homogeneous second-order linear ODE (ver. 90)

Find the general solution to the given ODE.

\[y''-6y'+10y = 0\]

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 91

C4b - Homogeneous second-order linear ODE (ver. 91)

Find the general solution to the given ODE.

\[y''+8y'+32y = 0\]

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 92

C4b - Homogeneous second-order linear ODE (ver. 92)

Find the general solution to the given ODE.

\[y''-2y'+2y = 0\]

Answer.

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

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

Example 93

C4b - Homogeneous second-order linear ODE (ver. 93)

Find the general solution to the given ODE.

\[y''-2y'+17y = 0\]

Answer.

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

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

Example 94

C4b - Homogeneous second-order linear ODE (ver. 94)

Find the general solution to the given ODE.

\[y''+8y'+25y = 0\]

Answer.

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

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

Example 95

C4b - Homogeneous second-order linear ODE (ver. 95)

Find the general solution to the given ODE.

\[y''-4y'+20y = 0\]

Answer.

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

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

Example 96

C4b - Homogeneous second-order linear ODE (ver. 96)

Find the general solution to the given ODE.

\[y''-8y'+25y = 0\]

Answer.

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

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

Example 97

C4b - Homogeneous second-order linear ODE (ver. 97)

Find the general solution to the given ODE.

\[y''+2y'+26y = 0\]

Answer.

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

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

Example 98

C4b - Homogeneous second-order linear ODE (ver. 98)

Find the general solution to the given ODE.

\[y''+4y'+8y = 0\]

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 99

C4b - Homogeneous second-order linear ODE (ver. 99)

Find the general solution to the given ODE.

\[y''+2y'+10y = 0\]

Answer.

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

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

Example 100

C4b - Homogeneous second-order linear ODE (ver. 100)

Find the general solution to the given ODE.

\[y''-4y'+5y = 0\]

Answer.

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

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