C5 - Homogeneous second-order linear IVP


Example 1

C5 - Homogeneous second-order linear IVP (ver. 1)

Find the solution to the given IVP.

\[{y''} + 49 \, {y} = 0 \hspace{1em} y(0) = -5 , y'(0) = -7\]

Answer.

\[{y} = -5 \, \cos\left(7 \, t\right) - \sin\left(7 \, t\right)\]


Example 2

C5 - Homogeneous second-order linear IVP (ver. 2)

Find the solution to the given IVP.

\[{y''} + 36 \, {y} = 0 \hspace{1em} y(0) = -1 , y'(0) = -24\]

Answer.

\[{y} = -\cos\left(6 \, t\right) - 4 \, \sin\left(6 \, t\right)\]


Example 3

C5 - Homogeneous second-order linear IVP (ver. 3)

Find the solution to the given IVP.

\[{y''} - 8 \, {y'} + 15 \, {y} = 0 \hspace{1em} y(0) = 5 , y'(0) = 21\]

Answer.

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


Example 4

C5 - Homogeneous second-order linear IVP (ver. 4)

Find the solution to the given IVP.

\[{y''} + 16 \, {y} = 0 \hspace{1em} y(0) = 1 , y'(0) = -16\]

Answer.

\[{y} = \cos\left(4 \, t\right) - 4 \, \sin\left(4 \, t\right)\]


Example 5

C5 - Homogeneous second-order linear IVP (ver. 5)

Find the solution to the given IVP.

\[{y''} + {y'} - 6 \, {y} = 0 \hspace{1em} y(0) = -3 , y'(0) = 4\]

Answer.

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


Example 6

C5 - Homogeneous second-order linear IVP (ver. 6)

Find the solution to the given IVP.

\[{y''} + 100 \, {y} = 0 \hspace{1em} y(0) = 2 , y'(0) = -20\]

Answer.

\[{y} = 2 \, \cos\left(10 \, t\right) - 2 \, \sin\left(10 \, t\right)\]


Example 7

C5 - Homogeneous second-order linear IVP (ver. 7)

Find the solution to the given IVP.

\[{y''} + 64 \, {y} = 0 \hspace{1em} y(0) = 5 , y'(0) = 0\]

Answer.

\[{y} = 5 \, \cos\left(8 \, t\right)\]


Example 8

C5 - Homogeneous second-order linear IVP (ver. 8)

Find the solution to the given IVP.

\[{y''} + 100 \, {y} = 0 \hspace{1em} y(0) = -4 , y'(0) = 10\]

Answer.

\[{y} = -4 \, \cos\left(10 \, t\right) + \sin\left(10 \, t\right)\]


Example 9

C5 - Homogeneous second-order linear IVP (ver. 9)

Find the solution to the given IVP.

\[{y''} + 16 \, {y} = 0 \hspace{1em} y(0) = -2 , y'(0) = 4\]

Answer.

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


Example 10

C5 - Homogeneous second-order linear IVP (ver. 10)

Find the solution to the given IVP.

\[{y''} + 25 \, {y} = 0 \hspace{1em} y(0) = -5 , y'(0) = -5\]

Answer.

\[{y} = -5 \, \cos\left(5 \, t\right) - \sin\left(5 \, t\right)\]


Example 11

C5 - Homogeneous second-order linear IVP (ver. 11)

Find the solution to the given IVP.

\[{y''} + 4 \, {y'} + 3 \, {y} = 0 \hspace{1em} y(0) = -6 , y'(0) = 14\]

Answer.

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


Example 12

C5 - Homogeneous second-order linear IVP (ver. 12)

Find the solution to the given IVP.

\[{y''} + 8 \, {y'} + 15 \, {y} = 0 \hspace{1em} y(0) = 2 , y'(0) = -2\]

Answer.

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


Example 13

C5 - Homogeneous second-order linear IVP (ver. 13)

Find the solution to the given IVP.

\[{y''} + 9 \, {y'} + 20 \, {y} = 0 \hspace{1em} y(0) = -7 , y'(0) = 30\]

Answer.

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


Example 14

C5 - Homogeneous second-order linear IVP (ver. 14)

Find the solution to the given IVP.

\[{y''} + {y'} - 2 \, {y} = 0 \hspace{1em} y(0) = -1 , y'(0) = 5\]

Answer.

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


Example 15

C5 - Homogeneous second-order linear IVP (ver. 15)

Find the solution to the given IVP.

\[{y''} + 64 \, {y} = 0 \hspace{1em} y(0) = 2 , y'(0) = -24\]

Answer.

\[{y} = 2 \, \cos\left(8 \, t\right) - 3 \, \sin\left(8 \, t\right)\]


Example 16

C5 - Homogeneous second-order linear IVP (ver. 16)

Find the solution to the given IVP.

\[{y''} + 7 \, {y'} + 12 \, {y} = 0 \hspace{1em} y(0) = -1 , y'(0) = 8\]

Answer.

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


Example 17

C5 - Homogeneous second-order linear IVP (ver. 17)

Find the solution to the given IVP.

\[{y''} + 8 \, {y'} + 15 \, {y} = 0 \hspace{1em} y(0) = -1 , y'(0) = 9\]

Answer.

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


Example 18

C5 - Homogeneous second-order linear IVP (ver. 18)

Find the solution to the given IVP.

\[{y''} + 25 \, {y} = 0 \hspace{1em} y(0) = -2 , y'(0) = 5\]

Answer.

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


Example 19

C5 - Homogeneous second-order linear IVP (ver. 19)

Find the solution to the given IVP.

\[{y''} - 2 \, {y'} - 15 \, {y} = 0 \hspace{1em} y(0) = -2 , y'(0) = -18\]

Answer.

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


Example 20

C5 - Homogeneous second-order linear IVP (ver. 20)

Find the solution to the given IVP.

\[{y''} - 3 \, {y'} - 4 \, {y} = 0 \hspace{1em} y(0) = 5 , y'(0) = 15\]

Answer.

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


Example 21

C5 - Homogeneous second-order linear IVP (ver. 21)

Find the solution to the given IVP.

\[{y''} - 3 \, {y'} - 4 \, {y} = 0 \hspace{1em} y(0) = 0 , y'(0) = 10\]

Answer.

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


Example 22

C5 - Homogeneous second-order linear IVP (ver. 22)

Find the solution to the given IVP.

\[{y''} + 2 \, {y'} - 3 \, {y} = 0 \hspace{1em} y(0) = 5 , y'(0) = -15\]

Answer.

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


Example 23

C5 - Homogeneous second-order linear IVP (ver. 23)

Find the solution to the given IVP.

\[{y''} - {y'} - 2 \, {y} = 0 \hspace{1em} y(0) = -3 , y'(0) = 6\]

Answer.

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


Example 24

C5 - Homogeneous second-order linear IVP (ver. 24)

Find the solution to the given IVP.

\[{y''} + 2 \, {y'} - 15 \, {y} = 0 \hspace{1em} y(0) = -4 , y'(0) = -4\]

Answer.

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


Example 25

C5 - Homogeneous second-order linear IVP (ver. 25)

Find the solution to the given IVP.

\[{y''} - 16 \, {y} = 0 \hspace{1em} y(0) = 2 , y'(0) = 32\]

Answer.

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


Example 26

C5 - Homogeneous second-order linear IVP (ver. 26)

Find the solution to the given IVP.

\[{y''} + 64 \, {y} = 0 \hspace{1em} y(0) = 3 , y'(0) = -40\]

Answer.

\[{y} = 3 \, \cos\left(8 \, t\right) - 5 \, \sin\left(8 \, t\right)\]


Example 27

C5 - Homogeneous second-order linear IVP (ver. 27)

Find the solution to the given IVP.

\[{y''} - 3 \, {y'} - 10 \, {y} = 0 \hspace{1em} y(0) = -4 , y'(0) = 15\]

Answer.

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


Example 28

C5 - Homogeneous second-order linear IVP (ver. 28)

Find the solution to the given IVP.

\[{y''} - 3 \, {y'} - 10 \, {y} = 0 \hspace{1em} y(0) = 4 , y'(0) = 13\]

Answer.

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


Example 29

C5 - Homogeneous second-order linear IVP (ver. 29)

Find the solution to the given IVP.

\[{y''} + 36 \, {y} = 0 \hspace{1em} y(0) = -5 , y'(0) = -12\]

Answer.

\[{y} = -5 \, \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right)\]


Example 30

C5 - Homogeneous second-order linear IVP (ver. 30)

Find the solution to the given IVP.

\[{y''} - 16 \, {y} = 0 \hspace{1em} y(0) = -2 , y'(0) = 32\]

Answer.

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


Example 31

C5 - Homogeneous second-order linear IVP (ver. 31)

Find the solution to the given IVP.

\[{y''} - 5 \, {y'} + 4 \, {y} = 0 \hspace{1em} y(0) = -1 , y'(0) = -4\]

Answer.

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


Example 32

C5 - Homogeneous second-order linear IVP (ver. 32)

Find the solution to the given IVP.

\[{y''} + 49 \, {y} = 0 \hspace{1em} y(0) = -3 , y'(0) = 21\]

Answer.

\[{y} = -3 \, \cos\left(7 \, t\right) + 3 \, \sin\left(7 \, t\right)\]


Example 33

C5 - Homogeneous second-order linear IVP (ver. 33)

Find the solution to the given IVP.

\[{y''} + {y'} - 12 \, {y} = 0 \hspace{1em} y(0) = 4 , y'(0) = 5\]

Answer.

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


Example 34

C5 - Homogeneous second-order linear IVP (ver. 34)

Find the solution to the given IVP.

\[{y''} + 81 \, {y} = 0 \hspace{1em} y(0) = -1 , y'(0) = -18\]

Answer.

\[{y} = -\cos\left(9 \, t\right) - 2 \, \sin\left(9 \, t\right)\]


Example 35

C5 - Homogeneous second-order linear IVP (ver. 35)

Find the solution to the given IVP.

\[{y''} - 8 \, {y'} + 15 \, {y} = 0 \hspace{1em} y(0) = 2 , y'(0) = 14\]

Answer.

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


Example 36

C5 - Homogeneous second-order linear IVP (ver. 36)

Find the solution to the given IVP.

\[{y''} + 25 \, {y} = 0 \hspace{1em} y(0) = -3 , y'(0) = 0\]

Answer.

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


Example 37

C5 - Homogeneous second-order linear IVP (ver. 37)

Find the solution to the given IVP.

\[{y''} + 64 \, {y} = 0 \hspace{1em} y(0) = -2 , y'(0) = 40\]

Answer.

\[{y} = -2 \, \cos\left(8 \, t\right) + 5 \, \sin\left(8 \, t\right)\]


Example 38

C5 - Homogeneous second-order linear IVP (ver. 38)

Find the solution to the given IVP.

\[{y''} + 5 \, {y'} + 4 \, {y} = 0 \hspace{1em} y(0) = 0 , y'(0) = 15\]

Answer.

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


Example 39

C5 - Homogeneous second-order linear IVP (ver. 39)

Find the solution to the given IVP.

\[{y''} + 9 \, {y} = 0 \hspace{1em} y(0) = -1 , y'(0) = 6\]

Answer.

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


Example 40

C5 - Homogeneous second-order linear IVP (ver. 40)

Find the solution to the given IVP.

\[{y''} + {y'} - 2 \, {y} = 0 \hspace{1em} y(0) = -4 , y'(0) = -7\]

Answer.

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


Example 41

C5 - Homogeneous second-order linear IVP (ver. 41)

Find the solution to the given IVP.

\[{y''} - 3 \, {y'} - 10 \, {y} = 0 \hspace{1em} y(0) = 7 , y'(0) = 21\]

Answer.

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


Example 42

C5 - Homogeneous second-order linear IVP (ver. 42)

Find the solution to the given IVP.

\[{y''} - {y'} - 20 \, {y} = 0 \hspace{1em} y(0) = -6 , y'(0) = 15\]

Answer.

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


Example 43

C5 - Homogeneous second-order linear IVP (ver. 43)

Find the solution to the given IVP.

\[{y''} - 16 \, {y} = 0 \hspace{1em} y(0) = 9 , y'(0) = 4\]

Answer.

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


Example 44

C5 - Homogeneous second-order linear IVP (ver. 44)

Find the solution to the given IVP.

\[{y''} - {y} = 0 \hspace{1em} y(0) = 2 , y'(0) = 4\]

Answer.

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


Example 45

C5 - Homogeneous second-order linear IVP (ver. 45)

Find the solution to the given IVP.

\[{y''} + 49 \, {y} = 0 \hspace{1em} y(0) = 2 , y'(0) = -35\]

Answer.

\[{y} = 2 \, \cos\left(7 \, t\right) - 5 \, \sin\left(7 \, t\right)\]


Example 46

C5 - Homogeneous second-order linear IVP (ver. 46)

Find the solution to the given IVP.

\[{y''} + 2 \, {y'} - 8 \, {y} = 0 \hspace{1em} y(0) = 0 , y'(0) = 24\]

Answer.

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


Example 47

C5 - Homogeneous second-order linear IVP (ver. 47)

Find the solution to the given IVP.

\[{y''} + 25 \, {y} = 0 \hspace{1em} y(0) = -2 , y'(0) = -25\]

Answer.

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


Example 48

C5 - Homogeneous second-order linear IVP (ver. 48)

Find the solution to the given IVP.

\[{y''} - 8 \, {y'} + 15 \, {y} = 0 \hspace{1em} y(0) = -4 , y'(0) = -22\]

Answer.

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


Example 49

C5 - Homogeneous second-order linear IVP (ver. 49)

Find the solution to the given IVP.

\[{y''} + 5 \, {y'} + 6 \, {y} = 0 \hspace{1em} y(0) = 8 , y'(0) = -19\]

Answer.

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


Example 50

C5 - Homogeneous second-order linear IVP (ver. 50)

Find the solution to the given IVP.

\[{y''} + 4 \, {y} = 0 \hspace{1em} y(0) = 1 , y'(0) = 8\]

Answer.

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