C5 - Homogeneous second-order linear IVP

Example 1

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

Find the solution to the given IVP.

\[y''-2y'-15y = 0 \hspace{1em} y(0) = 8 , y'(0) = 16\]

Answer.

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

Example 2

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

Find the solution to the given IVP.

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

Answer.

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

Example 3

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

Find the solution to the given IVP.

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

Answer.

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

Example 4

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

Find the solution to the given IVP.

\[y''+3y'-4y = 0 \hspace{1em} y(0) = -1 , y'(0) = -21\]

Answer.

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

Example 5

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

Find the solution to the given IVP.

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

Answer.

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

Example 6

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

Find the solution to the given IVP.

\[y''-3y'-10y = 0 \hspace{1em} y(0) = 1 , y'(0) = 12\]

Answer.

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

Example 7

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

Find the solution to the given IVP.

\[y''-8y'+15y = 0 \hspace{1em} y(0) = 4 , y'(0) = 18\]

Answer.

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

Example 8

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

Find the solution to the given IVP.

\[y''-2y'-15y = 0 \hspace{1em} y(0) = 6 , y'(0) = 22\]

Answer.

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

Example 9

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

Find the solution to the given IVP.

\[y''+7y'+12y = 0 \hspace{1em} y(0) = -3 , y'(0) = 12\]

Answer.

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

Example 10

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

Find the solution to the given IVP.

\[y''+8y'+15y = 0 \hspace{1em} y(0) = 1 , y'(0) = -13\]

Answer.

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

Example 11

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

Find the solution to the given IVP.

\[y''+4y'+3y = 0 \hspace{1em} y(0) = -6 , y'(0) = 10\]

Answer.

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

Example 12

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

Find the solution to the given IVP.

\[y''-5y'+6y = 0 \hspace{1em} y(0) = 7 , y'(0) = 17\]

Answer.

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

Example 13

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

Find the solution to the given IVP.

\[y''-3y'+2y = 0 \hspace{1em} y(0) = -6 , y'(0) = -10\]

Answer.

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

Example 14

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

Find the solution to the given IVP.

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

Answer.

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

Example 15

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

Find the solution to the given IVP.

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

Answer.

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

Example 16

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

Find the solution to the given IVP.

\[y''-4y'+3y = 0 \hspace{1em} y(0) = -1 , y'(0) = -11\]

Answer.

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

Example 17

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

Find the solution to the given IVP.

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

Answer.

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

Example 18

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

Find the solution to the given IVP.

\[y''+8y'+15y = 0 \hspace{1em} y(0) = 5 , y'(0) = -25\]

Answer.

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

Example 19

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

Find the solution to the given IVP.

\[y''-7y'+10y = 0 \hspace{1em} y(0) = -2 , y'(0) = -1\]

Answer.

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

Example 20

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

Find the solution to the given IVP.

\[y''+2y'-15y = 0 \hspace{1em} y(0) = 1 , y'(0) = -29\]

Answer.

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

Example 21

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

Find the solution to the given IVP.

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

Answer.

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

Example 22

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

Find the solution to the given IVP.

\[y''+3y'+2y = 0 \hspace{1em} y(0) = 5 , y'(0) = -10\]

Answer.

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

Example 23

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

Find the solution to the given IVP.

\[y''+6y'+5y = 0 \hspace{1em} y(0) = -4 , y'(0) = 24\]

Answer.

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

Example 24

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

Find the solution to the given IVP.

\[y''-5y'+6y = 0 \hspace{1em} y(0) = 3 , y'(0) = 10\]

Answer.

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

Example 25

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

Find the solution to the given IVP.

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

Answer.

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

Example 26

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

Find the solution to the given IVP.

\[y''-6y'+5y = 0 \hspace{1em} y(0) = 6 , y'(0) = 10\]

Answer.

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

Example 27

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

Find the solution to the given IVP.

\[y''+4y'-5y = 0 \hspace{1em} y(0) = -4 , y'(0) = -10\]

Answer.

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

Example 28

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

Find the solution to the given IVP.

\[y''+2y'-15y = 0 \hspace{1em} y(0) = 0 , y'(0) = -24\]

Answer.

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

Example 29

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

Find the solution to the given IVP.

\[y''-3y'-10y = 0 \hspace{1em} y(0) = -2 , y'(0) = 11\]

Answer.

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

Example 30

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

Find the solution to the given IVP.

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

Answer.

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

Example 31

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

Find the solution to the given IVP.

\[y''+3y'-4y = 0 \hspace{1em} y(0) = 3 , y'(0) = 8\]

Answer.

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

Example 32

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

Find the solution to the given IVP.

\[y''-7y'+12y = 0 \hspace{1em} y(0) = 2 , y'(0) = 11\]

Answer.

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

Example 33

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

Find the solution to the given IVP.

\[y''-4y'-5y = 0 \hspace{1em} y(0) = -2 , y'(0) = 20\]

Answer.

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

Example 34

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

Find the solution to the given IVP.

\[y''-2y'-8y = 0 \hspace{1em} y(0) = 3 , y'(0) = 18\]

Answer.

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

Example 35

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

Find the solution to the given IVP.

\[y''-5y'+6y = 0 \hspace{1em} y(0) = 7 , y'(0) = 19\]

Answer.

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

Example 36

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

Find the solution to the given IVP.

\[y''+8y'+15y = 0 \hspace{1em} y(0) = -7 , y'(0) = 25\]

Answer.

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

Example 37

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

Find the solution to the given IVP.

\[y''+y'-20y = 0 \hspace{1em} y(0) = -1 , y'(0) = -4\]

Answer.

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

Example 38

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

Find the solution to the given IVP.

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

Answer.

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

Example 39

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

Find the solution to the given IVP.

\[y''+9y'+20y = 0 \hspace{1em} y(0) = -8 , y'(0) = 37\]

Answer.

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

Example 40

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

Find the solution to the given IVP.

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

Answer.

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

Example 41

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

Find the solution to the given IVP.

\[y''+3y'-10y = 0 \hspace{1em} y(0) = 5 , y'(0) = -25\]

Answer.

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

Example 42

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

Find the solution to the given IVP.

\[y''-6y'+8y = 0 \hspace{1em} y(0) = -6 , y'(0) = -22\]

Answer.

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

Example 43

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

Find the solution to the given IVP.

\[y''-5y'+6y = 0 \hspace{1em} y(0) = 5 , y'(0) = 14\]

Answer.

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

Example 44

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

Find the solution to the given IVP.

\[y''+4y'+3y = 0 \hspace{1em} y(0) = -2 , y'(0) = 12\]

Answer.

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

Example 45

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

Find the solution to the given IVP.

\[y''-6y'+5y = 0 \hspace{1em} y(0) = -6 , y'(0) = -22\]

Answer.

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

Example 46

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

Find the solution to the given IVP.

\[y''+y'-6y = 0 \hspace{1em} y(0) = 1 , y'(0) = 22\]

Answer.

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

Example 47

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

Find the solution to the given IVP.

\[y''+6y'+5y = 0 \hspace{1em} y(0) = -8 , y'(0) = 28\]

Answer.

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

Example 48

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

Find the solution to the given IVP.

\[y''-5y'+4y = 0 \hspace{1em} y(0) = 4 , y'(0) = 7\]

Answer.

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

Example 49

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

Find the solution to the given IVP.

\[y''+5y'+6y = 0 \hspace{1em} y(0) = 1 , y'(0) = 2\]

Answer.

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

Example 50

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

Find the solution to the given IVP.

\[y''-2y'-15y = 0 \hspace{1em} y(0) = -3 , y'(0) = -7\]

Answer.

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

Example 51

C5 - Homogeneous second-order linear IVP (ver. 51)

Find the solution to the given IVP.

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

Answer.

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

Example 52

C5 - Homogeneous second-order linear IVP (ver. 52)

Find the solution to the given IVP.

\[y''-5y'+6y = 0 \hspace{1em} y(0) = 1 , y'(0) = 2\]

Answer.

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

Example 53

C5 - Homogeneous second-order linear IVP (ver. 53)

Find the solution to the given IVP.

\[y''+y'-20y = 0 \hspace{1em} y(0) = -4 , y'(0) = 29\]

Answer.

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

Example 54

C5 - Homogeneous second-order linear IVP (ver. 54)

Find the solution to the given IVP.

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

Answer.

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

Example 55

C5 - Homogeneous second-order linear IVP (ver. 55)

Find the solution to the given IVP.

\[y''-4y'-5y = 0 \hspace{1em} y(0) = 2 , y'(0) = 10\]

Answer.

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

Example 56

C5 - Homogeneous second-order linear IVP (ver. 56)

Find the solution to the given IVP.

\[y''-1y'-12y = 0 \hspace{1em} y(0) = 1 , y'(0) = 18\]

Answer.

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

Example 57

C5 - Homogeneous second-order linear IVP (ver. 57)

Find the solution to the given IVP.

\[y''+0y'-16y = 0 \hspace{1em} y(0) = 3 , y'(0) = 4\]

Answer.

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

Example 58

C5 - Homogeneous second-order linear IVP (ver. 58)

Find the solution to the given IVP.

\[y''+6y'+8y = 0 \hspace{1em} y(0) = -7 , y'(0) = 18\]

Answer.

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

Example 59

C5 - Homogeneous second-order linear IVP (ver. 59)

Find the solution to the given IVP.

\[y''+6y'+8y = 0 \hspace{1em} y(0) = 1 , y'(0) = 2\]

Answer.

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

Example 60

C5 - Homogeneous second-order linear IVP (ver. 60)

Find the solution to the given IVP.

\[y''+3y'+2y = 0 \hspace{1em} y(0) = 6 , y'(0) = -7\]

Answer.

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

Example 61

C5 - Homogeneous second-order linear IVP (ver. 61)

Find the solution to the given IVP.

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

Answer.

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

Example 62

C5 - Homogeneous second-order linear IVP (ver. 62)

Find the solution to the given IVP.

\[y''-1y'-2y = 0 \hspace{1em} y(0) = -5 , y'(0) = -4\]

Answer.

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

Example 63

C5 - Homogeneous second-order linear IVP (ver. 63)

Find the solution to the given IVP.

\[y''+3y'-10y = 0 \hspace{1em} y(0) = -5 , y'(0) = 4\]

Answer.

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

Example 64

C5 - Homogeneous second-order linear IVP (ver. 64)

Find the solution to the given IVP.

\[y''-6y'+5y = 0 \hspace{1em} y(0) = 2 , y'(0) = -2\]

Answer.

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

Example 65

C5 - Homogeneous second-order linear IVP (ver. 65)

Find the solution to the given IVP.

\[y''+7y'+12y = 0 \hspace{1em} y(0) = 8 , y'(0) = -29\]

Answer.

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

Example 66

C5 - Homogeneous second-order linear IVP (ver. 66)

Find the solution to the given IVP.

\[y''+8y'+15y = 0 \hspace{1em} y(0) = -5 , y'(0) = 17\]

Answer.

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

Example 67

C5 - Homogeneous second-order linear IVP (ver. 67)

Find the solution to the given IVP.

\[y''+0y'-25y = 0 \hspace{1em} y(0) = -2 , y'(0) = 40\]

Answer.

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

Example 68

C5 - Homogeneous second-order linear IVP (ver. 68)

Find the solution to the given IVP.

\[y''-6y'+5y = 0 \hspace{1em} y(0) = 0 , y'(0) = 12\]

Answer.

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

Example 69

C5 - Homogeneous second-order linear IVP (ver. 69)

Find the solution to the given IVP.

\[y''+0y'-9y = 0 \hspace{1em} y(0) = -4 , y'(0) = 12\]

Answer.

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

Example 70

C5 - Homogeneous second-order linear IVP (ver. 70)

Find the solution to the given IVP.

\[y''+4y'-5y = 0 \hspace{1em} y(0) = -2 , y'(0) = 28\]

Answer.

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

Example 71

C5 - Homogeneous second-order linear IVP (ver. 71)

Find the solution to the given IVP.

\[y''+7y'+10y = 0 \hspace{1em} y(0) = 3 , y'(0) = -9\]

Answer.

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

Example 72

C5 - Homogeneous second-order linear IVP (ver. 72)

Find the solution to the given IVP.

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

Answer.

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

Example 73

C5 - Homogeneous second-order linear IVP (ver. 73)

Find the solution to the given IVP.

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

Answer.

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

Example 74

C5 - Homogeneous second-order linear IVP (ver. 74)

Find the solution to the given IVP.

\[y''+y'-6y = 0 \hspace{1em} y(0) = 4 , y'(0) = 8\]

Answer.

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

Example 75

C5 - Homogeneous second-order linear IVP (ver. 75)

Find the solution to the given IVP.

\[y''+7y'+10y = 0 \hspace{1em} y(0) = 6 , y'(0) = -15\]

Answer.

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

Example 76

C5 - Homogeneous second-order linear IVP (ver. 76)

Find the solution to the given IVP.

\[y''-9y'+20y = 0 \hspace{1em} y(0) = 5 , y'(0) = 20\]

Answer.

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

Example 77

C5 - Homogeneous second-order linear IVP (ver. 77)

Find the solution to the given IVP.

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

Answer.

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

Example 78

C5 - Homogeneous second-order linear IVP (ver. 78)

Find the solution to the given IVP.

\[y''+2y'-8y = 0 \hspace{1em} y(0) = 2 , y'(0) = -26\]

Answer.

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

Example 79

C5 - Homogeneous second-order linear IVP (ver. 79)

Find the solution to the given IVP.

\[y''+6y'+8y = 0 \hspace{1em} y(0) = -8 , y'(0) = 26\]

Answer.

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

Example 80

C5 - Homogeneous second-order linear IVP (ver. 80)

Find the solution to the given IVP.

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

Answer.

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

Example 81

C5 - Homogeneous second-order linear IVP (ver. 81)

Find the solution to the given IVP.

\[y''+9y'+20y = 0 \hspace{1em} y(0) = -9 , y'(0) = 41\]

Answer.

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

Example 82

C5 - Homogeneous second-order linear IVP (ver. 82)

Find the solution to the given IVP.

\[y''+7y'+10y = 0 \hspace{1em} y(0) = 4 , y'(0) = -11\]

Answer.

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

Example 83

C5 - Homogeneous second-order linear IVP (ver. 83)

Find the solution to the given IVP.

\[y''+4y'+3y = 0 \hspace{1em} y(0) = -7 , y'(0) = 15\]

Answer.

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

Example 84

C5 - Homogeneous second-order linear IVP (ver. 84)

Find the solution to the given IVP.

\[y''+3y'-4y = 0 \hspace{1em} y(0) = -2 , y'(0) = 8\]

Answer.

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

Example 85

C5 - Homogeneous second-order linear IVP (ver. 85)

Find the solution to the given IVP.

\[y''+6y'+8y = 0 \hspace{1em} y(0) = 5 , y'(0) = -18\]

Answer.

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

Example 86

C5 - Homogeneous second-order linear IVP (ver. 86)

Find the solution to the given IVP.

\[y''+0y'-4y = 0 \hspace{1em} y(0) = 5 , y'(0) = -2\]

Answer.

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

Example 87

C5 - Homogeneous second-order linear IVP (ver. 87)

Find the solution to the given IVP.

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

Answer.

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

Example 88

C5 - Homogeneous second-order linear IVP (ver. 88)

Find the solution to the given IVP.

\[y''+2y'-8y = 0 \hspace{1em} y(0) = 7 , y'(0) = -4\]

Answer.

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

Example 89

C5 - Homogeneous second-order linear IVP (ver. 89)

Find the solution to the given IVP.

\[y''+y'-20y = 0 \hspace{1em} y(0) = -2 , y'(0) = 28\]

Answer.

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

Example 90

C5 - Homogeneous second-order linear IVP (ver. 90)

Find the solution to the given IVP.

\[y''-4y'+3y = 0 \hspace{1em} y(0) = -5 , y'(0) = -9\]

Answer.

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

Example 91

C5 - Homogeneous second-order linear IVP (ver. 91)

Find the solution to the given IVP.

\[y''-1y'-20y = 0 \hspace{1em} y(0) = 5 , y'(0) = -20\]

Answer.

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

Example 92

C5 - Homogeneous second-order linear IVP (ver. 92)

Find the solution to the given IVP.

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

Answer.

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

Example 93

C5 - Homogeneous second-order linear IVP (ver. 93)

Find the solution to the given IVP.

\[y''+3y'-4y = 0 \hspace{1em} y(0) = 1 , y'(0) = -24\]

Answer.

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

Example 94

C5 - Homogeneous second-order linear IVP (ver. 94)

Find the solution to the given IVP.

\[y''+y'-12y = 0 \hspace{1em} y(0) = -3 , y'(0) = 12\]

Answer.

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

Example 95

C5 - Homogeneous second-order linear IVP (ver. 95)

Find the solution to the given IVP.

\[y''+2y'-8y = 0 \hspace{1em} y(0) = 0 , y'(0) = 6\]

Answer.

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

Example 96

C5 - Homogeneous second-order linear IVP (ver. 96)

Find the solution to the given IVP.

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

Answer.

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

Example 97

C5 - Homogeneous second-order linear IVP (ver. 97)

Find the solution to the given IVP.

\[y''-4y'-5y = 0 \hspace{1em} y(0) = 4 , y'(0) = 14\]

Answer.

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

Example 98

C5 - Homogeneous second-order linear IVP (ver. 98)

Find the solution to the given IVP.

\[y''+6y'+8y = 0 \hspace{1em} y(0) = -3 , y'(0) = 14\]

Answer.

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

Example 99

C5 - Homogeneous second-order linear IVP (ver. 99)

Find the solution to the given IVP.

\[y''-2y'-3y = 0 \hspace{1em} y(0) = 4 , y'(0) = -8\]

Answer.

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

Example 100

C5 - Homogeneous second-order linear IVP (ver. 100)

Find the solution to the given IVP.

\[y''-1y'-2y = 0 \hspace{1em} y(0) = -7 , y'(0) = -8\]

Answer.

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