F2: Separation of variables

Example 1

F2: Separation of variables (ver. 1)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( -2 ) = -32\]

Answer.

\[y= 4 t^{ 3 }\]

Example 2

F2: Separation of variables (ver. 2)

Find the solution to the given IVP.

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

Answer.

\[y= 2 t^{ -3 }\]

Example 3

F2: Separation of variables (ver. 3)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( -4 ) = 64\]

Answer.

\[y= 4 t^{ 2 }\]

Example 4

F2: Separation of variables (ver. 4)

Find the solution to the given IVP.

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

Answer.

\[y= -4 t^{ -3 }\]

Example 5

F2: Separation of variables (ver. 5)

Find the solution to the given IVP.

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

Answer.

\[y= -3 t^{ 2 }\]

Example 6

F2: Separation of variables (ver. 6)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( -2 ) = 24\]

Answer.

\[y= -3 t^{ 3 }\]

Example 7

F2: Separation of variables (ver. 7)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( 3 ) = 108\]

Answer.

\[y= 4 t^{ 3 }\]

Example 8

F2: Separation of variables (ver. 8)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( -3 ) = -108\]

Answer.

\[y= 4 t^{ 3 }\]

Example 9

F2: Separation of variables (ver. 9)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( -3 ) = \frac{1}{27}\]

Answer.

\[y= -1 t^{ -3 }\]

Example 10

F2: Separation of variables (ver. 10)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( -2 ) = \frac{1}{2}\]

Answer.

\[y= -4 t^{ -3 }\]

Example 11

F2: Separation of variables (ver. 11)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( -1 ) = -2\]

Answer.

\[y= -2 t^{ -2 }\]

Example 12

F2: Separation of variables (ver. 12)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 2 ) = 8\]

Answer.

\[y= 2 t^{ 2 }\]

Example 13

F2: Separation of variables (ver. 13)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 1 ) = -1\]

Answer.

\[y= -1 t^{ 2 }\]

Example 14

F2: Separation of variables (ver. 14)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( -4 ) = \frac{1}{8}\]

Answer.

\[y= 2 t^{ -2 }\]

Example 15

F2: Separation of variables (ver. 15)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( 1 ) = -3\]

Answer.

\[y= -3 t^{ 3 }\]

Example 16

F2: Separation of variables (ver. 16)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 1 ) = 1\]

Answer.

\[y= 1 t^{ 2 }\]

Example 17

F2: Separation of variables (ver. 17)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( 2 ) = \frac{1}{2}\]

Answer.

\[y= 2 t^{ -2 }\]

Example 18

F2: Separation of variables (ver. 18)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( -3 ) = \frac{4}{27}\]

Answer.

\[y= -4 t^{ -3 }\]

Example 19

F2: Separation of variables (ver. 19)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 4 ) = -32\]

Answer.

\[y= -2 t^{ 2 }\]

Example 20

F2: Separation of variables (ver. 20)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 3 ) = -36\]

Answer.

\[y= -4 t^{ 2 }\]

Example 21

F2: Separation of variables (ver. 21)

Find the solution to the given IVP.

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

Answer.

\[y= -4 t^{ -2 }\]

Example 22

F2: Separation of variables (ver. 22)

Find the solution to the given IVP.

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

Answer.

\[y= 4 t^{ -2 }\]

Example 23

F2: Separation of variables (ver. 23)

Find the solution to the given IVP.

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

Answer.

\[y= -4 t^{ -2 }\]

Example 24

F2: Separation of variables (ver. 24)

Find the solution to the given IVP.

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

Answer.

\[y= -4 t^{ 2 }\]

Example 25

F2: Separation of variables (ver. 25)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 4 ) = 32\]

Answer.

\[y= 2 t^{ 2 }\]

Example 26

F2: Separation of variables (ver. 26)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( 1 ) = 3\]

Answer.

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

Example 27

F2: Separation of variables (ver. 27)

Find the solution to the given IVP.

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

Answer.

\[y= 1 t^{ 2 }\]

Example 28

F2: Separation of variables (ver. 28)

Find the solution to the given IVP.

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

Answer.

\[y= 1 t^{ 2 }\]

Example 29

F2: Separation of variables (ver. 29)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( -2 ) = -24\]

Answer.

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

Example 30

F2: Separation of variables (ver. 30)

Find the solution to the given IVP.

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

Answer.

\[y= -3 t^{ 2 }\]

Example 31

F2: Separation of variables (ver. 31)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( -3 ) = -36\]

Answer.

\[y= -4 t^{ 2 }\]

Example 32

F2: Separation of variables (ver. 32)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( 4 ) = 128\]

Answer.

\[y= 2 t^{ 3 }\]

Example 33

F2: Separation of variables (ver. 33)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( 4 ) = \frac{1}{16}\]

Answer.

\[y= 1 t^{ -2 }\]

Example 34

F2: Separation of variables (ver. 34)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( -2 ) = -32\]

Answer.

\[y= 4 t^{ 3 }\]

Example 35

F2: Separation of variables (ver. 35)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 3 ) = 27\]

Answer.

\[y= 3 t^{ 2 }\]

Example 36

F2: Separation of variables (ver. 36)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( -3 ) = \frac{2}{27}\]

Answer.

\[y= -2 t^{ -3 }\]

Example 37

F2: Separation of variables (ver. 37)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( -4 ) = -128\]

Answer.

\[y= 2 t^{ 3 }\]

Example 38

F2: Separation of variables (ver. 38)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( 1 ) = 1\]

Answer.

\[y= 1 t^{ -2 }\]

Example 39

F2: Separation of variables (ver. 39)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( -3 ) = \frac{1}{27}\]

Answer.

\[y= -1 t^{ -3 }\]

Example 40

F2: Separation of variables (ver. 40)

Find the solution to the given IVP.

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

Answer.

\[y= 3 t^{ -2 }\]

Example 41

F2: Separation of variables (ver. 41)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( -4 ) = 32\]

Answer.

\[y= 2 t^{ 2 }\]

Example 42

F2: Separation of variables (ver. 42)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 4 ) = -32\]

Answer.

\[y= -2 t^{ 2 }\]

Example 43

F2: Separation of variables (ver. 43)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 2 ) = 12\]

Answer.

\[y= 3 t^{ 2 }\]

Example 44

F2: Separation of variables (ver. 44)

Find the solution to the given IVP.

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

Answer.

\[y= -3 t^{ -2 }\]

Example 45

F2: Separation of variables (ver. 45)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( 1 ) = -1\]

Answer.

\[y= -1 t^{ 3 }\]

Example 46

F2: Separation of variables (ver. 46)

Find the solution to the given IVP.

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

Answer.

\[y= -4 t^{ 2 }\]

Example 47

F2: Separation of variables (ver. 47)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( -1 ) = 1\]

Answer.

\[y= -1 t^{ -3 }\]

Example 48

F2: Separation of variables (ver. 48)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 3 ) = 27\]

Answer.

\[y= 3 t^{ 2 }\]

Example 49

F2: Separation of variables (ver. 49)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( -1 ) = 1\]

Answer.

\[y= 1 t^{ -2 }\]

Example 50

F2: Separation of variables (ver. 50)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( 4 ) = -\frac{1}{32}\]

Answer.

\[y= -2 t^{ -3 }\]

Example 51

F2: Separation of variables (ver. 51)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( 3 ) = \frac{4}{9}\]

Answer.

\[y= 4 t^{ -2 }\]

Example 52

F2: Separation of variables (ver. 52)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 3 ) = 27\]

Answer.

\[y= 3 t^{ 2 }\]

Example 53

F2: Separation of variables (ver. 53)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( -4 ) = 192\]

Answer.

\[y= -3 t^{ 3 }\]

Example 54

F2: Separation of variables (ver. 54)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( 3 ) = -108\]

Answer.

\[y= -4 t^{ 3 }\]

Example 55

F2: Separation of variables (ver. 55)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( -1 ) = 3\]

Answer.

\[y= -3 t^{ 3 }\]

Example 56

F2: Separation of variables (ver. 56)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( 3 ) = \frac{2}{27}\]

Answer.

\[y= 2 t^{ -3 }\]

Example 57

F2: Separation of variables (ver. 57)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( 4 ) = \frac{3}{64}\]

Answer.

\[y= 3 t^{ -3 }\]

Example 58

F2: Separation of variables (ver. 58)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( -2 ) = \frac{1}{4}\]

Answer.

\[y= -2 t^{ -3 }\]

Example 59

F2: Separation of variables (ver. 59)

Find the solution to the given IVP.

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

Answer.

\[y= -4 t^{ -2 }\]

Example 60

F2: Separation of variables (ver. 60)

Find the solution to the given IVP.

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

Answer.

\[y= 3 t^{ -2 }\]

Example 61

F2: Separation of variables (ver. 61)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 2 ) = 12\]

Answer.

\[y= 3 t^{ 2 }\]

Example 62

F2: Separation of variables (ver. 62)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( 2 ) = -\frac{1}{2}\]

Answer.

\[y= -2 t^{ -2 }\]

Example 63

F2: Separation of variables (ver. 63)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( 2 ) = -\frac{1}{4}\]

Answer.

\[y= -2 t^{ -3 }\]

Example 64

F2: Separation of variables (ver. 64)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( 1 ) = 2\]

Answer.

\[y= 2 t^{ 3 }\]

Example 65

F2: Separation of variables (ver. 65)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( -3 ) = -\frac{4}{27}\]

Answer.

\[y= 4 t^{ -3 }\]

Example 66

F2: Separation of variables (ver. 66)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( -4 ) = -\frac{1}{16}\]

Answer.

\[y= -1 t^{ -2 }\]

Example 67

F2: Separation of variables (ver. 67)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( -4 ) = 256\]

Answer.

\[y= -4 t^{ 3 }\]

Example 68

F2: Separation of variables (ver. 68)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 4 ) = 16\]

Answer.

\[y= 1 t^{ 2 }\]

Example 69

F2: Separation of variables (ver. 69)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( -2 ) = -12\]

Answer.

\[y= -3 t^{ 2 }\]

Example 70

F2: Separation of variables (ver. 70)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( 1 ) = 1\]

Answer.

\[y= 1 t^{ -3 }\]

Example 71

F2: Separation of variables (ver. 71)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( -4 ) = \frac{1}{32}\]

Answer.

\[y= -2 t^{ -3 }\]

Example 72

F2: Separation of variables (ver. 72)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( -3 ) = 54\]

Answer.

\[y= -2 t^{ 3 }\]

Example 73

F2: Separation of variables (ver. 73)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( 2 ) = 16\]

Answer.

\[y= 2 t^{ 3 }\]

Example 74

F2: Separation of variables (ver. 74)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( -3 ) = -36\]

Answer.

\[y= -4 t^{ 2 }\]

Example 75

F2: Separation of variables (ver. 75)

Find the solution to the given IVP.

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

Answer.

\[y= 4 t^{ -2 }\]

Example 76

F2: Separation of variables (ver. 76)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( 4 ) = 128\]

Answer.

\[y= 2 t^{ 3 }\]

Example 77

F2: Separation of variables (ver. 77)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( 4 ) = -192\]

Answer.

\[y= -3 t^{ 3 }\]

Example 78

F2: Separation of variables (ver. 78)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( -3 ) = -9\]

Answer.

\[y= -1 t^{ 2 }\]

Example 79

F2: Separation of variables (ver. 79)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( -2 ) = -\frac{3}{4}\]

Answer.

\[y= -3 t^{ -2 }\]

Example 80

F2: Separation of variables (ver. 80)

Find the solution to the given IVP.

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

Answer.

\[y= 2 t^{ 3 }\]

Example 81

F2: Separation of variables (ver. 81)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( -3 ) = \frac{1}{27}\]

Answer.

\[y= -1 t^{ -3 }\]

Example 82

F2: Separation of variables (ver. 82)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( -1 ) = -3\]

Answer.

\[y= 3 t^{ -3 }\]

Example 83

F2: Separation of variables (ver. 83)

Find the solution to the given IVP.

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

Answer.

\[y= -1 t^{ -2 }\]

Example 84

F2: Separation of variables (ver. 84)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( -3 ) = -\frac{1}{27}\]

Answer.

\[y= 1 t^{ -3 }\]

Example 85

F2: Separation of variables (ver. 85)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( -3 ) = \frac{1}{3}\]

Answer.

\[y= 3 t^{ -2 }\]

Example 86

F2: Separation of variables (ver. 86)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( 2 ) = 1\]

Answer.

\[y= 4 t^{ -2 }\]

Example 87

F2: Separation of variables (ver. 87)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 4 ) = -64\]

Answer.

\[y= -4 t^{ 2 }\]

Example 88

F2: Separation of variables (ver. 88)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( 2 ) = \frac{3}{8}\]

Answer.

\[y= 3 t^{ -3 }\]

Example 89

F2: Separation of variables (ver. 89)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( 3 ) = \frac{1}{27}\]

Answer.

\[y= 1 t^{ -3 }\]

Example 90

F2: Separation of variables (ver. 90)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( 1 ) = -2\]

Answer.

\[y= -2 t^{ 2 }\]

Example 91

F2: Separation of variables (ver. 91)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( -3 ) = -\frac{4}{27}\]

Answer.

\[y= 4 t^{ -3 }\]

Example 92

F2: Separation of variables (ver. 92)

Find the solution to the given IVP.

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

Answer.

\[y= 4 t^{ 2 }\]

Example 93

F2: Separation of variables (ver. 93)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( -3 ) = 9\]

Answer.

\[y= 1 t^{ 2 }\]

Example 94

F2: Separation of variables (ver. 94)

Find the solution to the given IVP.

\[ty'= -3 y \hspace{1em} y( 3 ) = \frac{1}{27}\]

Answer.

\[y= 1 t^{ -3 }\]

Example 95

F2: Separation of variables (ver. 95)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( -4 ) = -48\]

Answer.

\[y= -3 t^{ 2 }\]

Example 96

F2: Separation of variables (ver. 96)

Find the solution to the given IVP.

\[ty'= 3 y \hspace{1em} y( 2 ) = 8\]

Answer.

\[y= 1 t^{ 3 }\]

Example 97

F2: Separation of variables (ver. 97)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( 4 ) = \frac{3}{16}\]

Answer.

\[y= 3 t^{ -2 }\]

Example 98

F2: Separation of variables (ver. 98)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( -2 ) = -\frac{3}{4}\]

Answer.

\[y= -3 t^{ -2 }\]

Example 99

F2: Separation of variables (ver. 99)

Find the solution to the given IVP.

\[ty'= -2 y \hspace{1em} y( 2 ) = -1\]

Answer.

\[y= -4 t^{ -2 }\]

Example 100

F2: Separation of variables (ver. 100)

Find the solution to the given IVP.

\[ty'= 2 y \hspace{1em} y( -1 ) = -1\]

Answer.

\[y= -1 t^{ 2 }\]