F5: First order linear IVP

Example 1

F5: First order linear IVP (ver. 1)

Find the solution to the given IVP.

\[-\frac{24}{t} +ty'= 5 y \hspace{1em} y( -1 ) = 3\]

Answer.

\[y= 1 t^{ 5 } -4 \frac{1}{t}\]

Example 2

F5: First order linear IVP (ver. 2)

Find the solution to the given IVP.

\[-\frac{4}{t^{4}} +ty'= -5 y \hspace{1em} y( -1 ) = 6\]

Answer.

\[y= -2 t^{ -5 } +4 \frac{1}{t^{4}}\]

Example 3

F5: First order linear IVP (ver. 3)

Find the solution to the given IVP.

\[8 \, t +ty'= 3 y \hspace{1em} y( 1 ) = 6\]

Answer.

\[y= 2 t^{ 3 } +4 t\]

Example 4

F5: First order linear IVP (ver. 4)

Find the solution to the given IVP.

\[-\frac{3}{t} +ty'= -4 y \hspace{1em} y( -1 ) = 2\]

Answer.

\[y= 3 t^{ -4 } + \frac{1}{t}\]

Example 5

F5: First order linear IVP (ver. 5)

Find the solution to the given IVP.

\[-\frac{8}{t^{2}} +ty'= 2 y \hspace{1em} y( -1 ) = 0\]

Answer.

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

Example 6

F5: First order linear IVP (ver. 6)

Find the solution to the given IVP.

\[\frac{27}{t^{4}} +ty'= 5 y \hspace{1em} y( -1 ) = 1\]

Answer.

\[y= 2 t^{ 5 } +3 \frac{1}{t^{4}}\]

Example 7

F5: First order linear IVP (ver. 7)

Find the solution to the given IVP.

\[6 \, t^{3} +ty'= 5 y \hspace{1em} y( 1 ) = 7\]

Answer.

\[y= 4 t^{ 5 } +3 t^{3}\]

Example 8

F5: First order linear IVP (ver. 8)

Find the solution to the given IVP.

\[\frac{40}{t^{3}} +ty'= 5 y \hspace{1em} y( 1 ) = 6\]

Answer.

\[y= 1 t^{ 5 } +5 \frac{1}{t^{3}}\]

Example 9

F5: First order linear IVP (ver. 9)

Find the solution to the given IVP.

\[-24 \, t^{4} +ty'= -4 y \hspace{1em} y( 1 ) = 1\]

Answer.

\[y= -2 t^{ -4 } +3 t^{4}\]

Example 10

F5: First order linear IVP (ver. 10)

Find the solution to the given IVP.

\[7 \, t^{2} +ty'= -5 y \hspace{1em} y( -1 ) = -2\]

Answer.

\[y= 1 t^{ -5 } -1 t^{2}\]

Example 11

F5: First order linear IVP (ver. 11)

Find the solution to the given IVP.

\[\frac{1}{t^{2}} +ty'= -3 y \hspace{1em} y( -1 ) = 0\]

Answer.

\[y= -1 t^{ -3 } -1 \frac{1}{t^{2}}\]

Example 12

F5: First order linear IVP (ver. 12)

Find the solution to the given IVP.

\[5 \, t^{4} +ty'= 3 y \hspace{1em} y( -1 ) = -1\]

Answer.

\[y= -4 t^{ 3 } -5 t^{4}\]

Example 13

F5: First order linear IVP (ver. 13)

Find the solution to the given IVP.

\[-\frac{10}{t^{3}} +ty'= 2 y \hspace{1em} y( -1 ) = -1\]

Answer.

\[y= -3 t^{ 2 } -2 \frac{1}{t^{3}}\]

Example 14

F5: First order linear IVP (ver. 14)

Find the solution to the given IVP.

\[18 \, t^{4} +ty'= -2 y \hspace{1em} y( 1 ) = -4\]

Answer.

\[y= -1 t^{ -2 } -3 t^{4}\]

Example 15

F5: First order linear IVP (ver. 15)

Find the solution to the given IVP.

\[3 \, t +ty'= 4 y \hspace{1em} y( -1 ) = 2\]

Answer.

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

Example 16

F5: First order linear IVP (ver. 16)

Find the solution to the given IVP.

\[-10 \, t +ty'= 3 y \hspace{1em} y( -1 ) = 9\]

Answer.

\[y= -4 t^{ 3 } -5 t\]

Example 17

F5: First order linear IVP (ver. 17)

Find the solution to the given IVP.

\[-\frac{4}{t^{4}} +ty'= -3 y \hspace{1em} y( -1 ) = 0\]

Answer.

\[y= -4 t^{ -3 } -4 \frac{1}{t^{4}}\]

Example 18

F5: First order linear IVP (ver. 18)

Find the solution to the given IVP.

\[-\frac{12}{t} +ty'= -5 y \hspace{1em} y( 1 ) = 5\]

Answer.

\[y= 2 t^{ -5 } +3 \frac{1}{t}\]

Example 19

F5: First order linear IVP (ver. 19)

Find the solution to the given IVP.

\[\frac{10}{t^{2}} +ty'= 3 y \hspace{1em} y( 1 ) = -1\]

Answer.

\[y= -3 t^{ 3 } +2 \frac{1}{t^{2}}\]

Example 20

F5: First order linear IVP (ver. 20)

Find the solution to the given IVP.

\[t^{4} +ty'= 3 y \hspace{1em} y( 1 ) = 3\]

Answer.

\[y= 4 t^{ 3 } -1 t^{4}\]

Example 21

F5: First order linear IVP (ver. 21)

Find the solution to the given IVP.

\[-12 \, t^{4} +ty'= -2 y \hspace{1em} y( -1 ) = 5\]

Answer.

\[y= 3 t^{ -2 } +2 t^{4}\]

Example 22

F5: First order linear IVP (ver. 22)

Find the solution to the given IVP.

\[-30 \, t^{3} +ty'= -3 y \hspace{1em} y( -1 ) = -6\]

Answer.

\[y= 1 t^{ -3 } +5 t^{3}\]

Example 23

F5: First order linear IVP (ver. 23)

Find the solution to the given IVP.

\[-3 \, t +ty'= -2 y \hspace{1em} y( -1 ) = 1\]

Answer.

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

Example 24

F5: First order linear IVP (ver. 24)

Find the solution to the given IVP.

\[-\frac{10}{t^{2}} +ty'= 3 y \hspace{1em} y( -1 ) = -5\]

Answer.

\[y= 3 t^{ 3 } -2 \frac{1}{t^{2}}\]

Example 25

F5: First order linear IVP (ver. 25)

Find the solution to the given IVP.

\[-\frac{20}{t} +ty'= 3 y \hspace{1em} y( 1 ) = -8\]

Answer.

\[y= -3 t^{ 3 } -5 \frac{1}{t}\]

Example 26

F5: First order linear IVP (ver. 26)

Find the solution to the given IVP.

\[\frac{45}{t^{4}} +ty'= 5 y \hspace{1em} y( -1 ) = 2\]

Answer.

\[y= 3 t^{ 5 } +5 \frac{1}{t^{4}}\]

Example 27

F5: First order linear IVP (ver. 27)

Find the solution to the given IVP.

\[-\frac{16}{t^{2}} +ty'= 2 y \hspace{1em} y( 1 ) = -6\]

Answer.

\[y= -2 t^{ 2 } -4 \frac{1}{t^{2}}\]

Example 28

F5: First order linear IVP (ver. 28)

Find the solution to the given IVP.

\[\frac{24}{t^{3}} +ty'= 3 y \hspace{1em} y( 1 ) = 2\]

Answer.

\[y= -2 t^{ 3 } +4 \frac{1}{t^{3}}\]

Example 29

F5: First order linear IVP (ver. 29)

Find the solution to the given IVP.

\[-\frac{24}{t^{3}} +ty'= 5 y \hspace{1em} y( -1 ) = 2\]

Answer.

\[y= 1 t^{ 5 } -3 \frac{1}{t^{3}}\]

Example 30

F5: First order linear IVP (ver. 30)

Find the solution to the given IVP.

\[2 \, t +ty'= 3 y \hspace{1em} y( -1 ) = -4\]

Answer.

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

Example 31

F5: First order linear IVP (ver. 31)

Find the solution to the given IVP.

\[-\frac{12}{t^{2}} +ty'= 2 y \hspace{1em} y( 1 ) = -2\]

Answer.

\[y= 1 t^{ 2 } -3 \frac{1}{t^{2}}\]

Example 32

F5: First order linear IVP (ver. 32)

Find the solution to the given IVP.

\[\frac{35}{t^{3}} +ty'= 4 y \hspace{1em} y( 1 ) = 1\]

Answer.

\[y= -4 t^{ 4 } +5 \frac{1}{t^{3}}\]

Example 33

F5: First order linear IVP (ver. 33)

Find the solution to the given IVP.

\[-24 \, t^{4} +ty'= -2 y \hspace{1em} y( 1 ) = 0\]

Answer.

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

Example 34

F5: First order linear IVP (ver. 34)

Find the solution to the given IVP.

\[-\frac{30}{t^{2}} +ty'= 4 y \hspace{1em} y( -1 ) = -3\]

Answer.

\[y= 2 t^{ 4 } -5 \frac{1}{t^{2}}\]

Example 35

F5: First order linear IVP (ver. 35)

Find the solution to the given IVP.

\[\frac{35}{t^{4}} +ty'= 3 y \hspace{1em} y( 1 ) = 9\]

Answer.

\[y= 4 t^{ 3 } +5 \frac{1}{t^{4}}\]

Example 36

F5: First order linear IVP (ver. 36)

Find the solution to the given IVP.

\[\frac{1}{t^{4}} +ty'= -5 y \hspace{1em} y( 1 ) = -2\]

Answer.

\[y= -1 t^{ -5 } -1 \frac{1}{t^{4}}\]

Example 37

F5: First order linear IVP (ver. 37)

Find the solution to the given IVP.

\[-\frac{24}{t^{3}} +ty'= 5 y \hspace{1em} y( 1 ) = -5\]

Answer.

\[y= -2 t^{ 5 } -3 \frac{1}{t^{3}}\]

Example 38

F5: First order linear IVP (ver. 38)

Find the solution to the given IVP.

\[\frac{5}{t} +ty'= -2 y \hspace{1em} y( -1 ) = 2\]

Answer.

\[y= -3 t^{ -2 } -5 \frac{1}{t}\]

Example 39

F5: First order linear IVP (ver. 39)

Find the solution to the given IVP.

\[-\frac{6}{t^{2}} +ty'= -5 y \hspace{1em} y( 1 ) = 0\]

Answer.

\[y= -2 t^{ -5 } +2 \frac{1}{t^{2}}\]

Example 40

F5: First order linear IVP (ver. 40)

Find the solution to the given IVP.

\[20 \, t^{2} +ty'= -3 y \hspace{1em} y( -1 ) = -5\]

Answer.

\[y= 1 t^{ -3 } -4 t^{2}\]

Example 41

F5: First order linear IVP (ver. 41)

Find the solution to the given IVP.

\[-5 \, t^{4} +ty'= 3 y \hspace{1em} y( 1 ) = 4\]

Answer.

\[y= -1 t^{ 3 } +5 t^{4}\]

Example 42

F5: First order linear IVP (ver. 42)

Find the solution to the given IVP.

\[-\frac{8}{t} +ty'= 3 y \hspace{1em} y( 1 ) = -5\]

Answer.

\[y= -3 t^{ 3 } -2 \frac{1}{t}\]

Example 43

F5: First order linear IVP (ver. 43)

Find the solution to the given IVP.

\[\frac{30}{t^{4}} +ty'= 2 y \hspace{1em} y( -1 ) = 8\]

Answer.

\[y= 3 t^{ 2 } +5 \frac{1}{t^{4}}\]

Example 44

F5: First order linear IVP (ver. 44)

Find the solution to the given IVP.

\[-\frac{2}{t} +ty'= -2 y \hspace{1em} y( -1 ) = -1\]

Answer.

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

Example 45

F5: First order linear IVP (ver. 45)

Find the solution to the given IVP.

\[-2 \, t^{4} +ty'= 5 y \hspace{1em} y( -1 ) = 1\]

Answer.

\[y= -3 t^{ 5 } -2 t^{4}\]

Example 46

F5: First order linear IVP (ver. 46)

Find the solution to the given IVP.

\[-\frac{7}{t^{4}} +ty'= 3 y \hspace{1em} y( -1 ) = -3\]

Answer.

\[y= 2 t^{ 3 } -1 \frac{1}{t^{4}}\]

Example 47

F5: First order linear IVP (ver. 47)

Find the solution to the given IVP.

\[\frac{2}{t^{4}} +ty'= -5 y \hspace{1em} y( -1 ) = 0\]

Answer.

\[y= -2 t^{ -5 } -2 \frac{1}{t^{4}}\]

Example 48

F5: First order linear IVP (ver. 48)

Find the solution to the given IVP.

\[-\frac{16}{t^{2}} +ty'= 2 y \hspace{1em} y( 1 ) = 0\]

Answer.

\[y= 4 t^{ 2 } -4 \frac{1}{t^{2}}\]

Example 49

F5: First order linear IVP (ver. 49)

Find the solution to the given IVP.

\[-25 \, t^{2} +ty'= -3 y \hspace{1em} y( -1 ) = 7\]

Answer.

\[y= -2 t^{ -3 } +5 t^{2}\]

Example 50

F5: First order linear IVP (ver. 50)

Find the solution to the given IVP.

\[14 \, t^{3} +ty'= -4 y \hspace{1em} y( 1 ) = -5\]

Answer.

\[y= -3 t^{ -4 } -2 t^{3}\]

Example 51

F5: First order linear IVP (ver. 51)

Find the solution to the given IVP.

\[25 \, t^{2} +ty'= -3 y \hspace{1em} y( 1 ) = -2\]

Answer.

\[y= 3 t^{ -3 } -5 t^{2}\]

Example 52

F5: First order linear IVP (ver. 52)

Find the solution to the given IVP.

\[\frac{12}{t} +ty'= 3 y \hspace{1em} y( 1 ) = 7\]

Answer.

\[y= 4 t^{ 3 } +3 \frac{1}{t}\]

Example 53

F5: First order linear IVP (ver. 53)

Find the solution to the given IVP.

\[t^{3} +ty'= 4 y \hspace{1em} y( 1 ) = 2\]

Answer.

\[y= 1 t^{ 4 } + t^{3}\]

Example 54

F5: First order linear IVP (ver. 54)

Find the solution to the given IVP.

\[t^{3} +ty'= 4 y \hspace{1em} y( 1 ) = 4\]

Answer.

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

Example 55

F5: First order linear IVP (ver. 55)

Find the solution to the given IVP.

\[9 \, t^{2} +ty'= 5 y \hspace{1em} y( -1 ) = 1\]

Answer.

\[y= 2 t^{ 5 } +3 t^{2}\]

Example 56

F5: First order linear IVP (ver. 56)

Find the solution to the given IVP.

\[\frac{8}{t^{2}} +ty'= -4 y \hspace{1em} y( 1 ) = -7\]

Answer.

\[y= -3 t^{ -4 } -4 \frac{1}{t^{2}}\]

Example 57

F5: First order linear IVP (ver. 57)

Find the solution to the given IVP.

\[-28 \, t^{3} +ty'= -4 y \hspace{1em} y( 1 ) = 7\]

Answer.

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

Example 58

F5: First order linear IVP (ver. 58)

Find the solution to the given IVP.

\[5 \, t +ty'= -4 y \hspace{1em} y( -1 ) = 0\]

Answer.

\[y= -1 t^{ -4 } -1 t\]

Example 59

F5: First order linear IVP (ver. 59)

Find the solution to the given IVP.

\[8 \, t^{2} +ty'= -2 y \hspace{1em} y( 1 ) = 0\]

Answer.

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

Example 60

F5: First order linear IVP (ver. 60)

Find the solution to the given IVP.

\[-\frac{10}{t} +ty'= -3 y \hspace{1em} y( -1 ) = -7\]

Answer.

\[y= 2 t^{ -3 } +5 \frac{1}{t}\]

Example 61

F5: First order linear IVP (ver. 61)

Find the solution to the given IVP.

\[6 \, t^{4} +ty'= 2 y \hspace{1em} y( -1 ) = 1\]

Answer.

\[y= 4 t^{ 2 } -3 t^{4}\]

Example 62

F5: First order linear IVP (ver. 62)

Find the solution to the given IVP.

\[-\frac{4}{t^{4}} +ty'= -2 y \hspace{1em} y( -1 ) = -5\]

Answer.

\[y= -3 t^{ -2 } -2 \frac{1}{t^{4}}\]

Example 63

F5: First order linear IVP (ver. 63)

Find the solution to the given IVP.

\[\frac{6}{t^{2}} +ty'= -5 y \hspace{1em} y( -1 ) = 1\]

Answer.

\[y= -3 t^{ -5 } -2 \frac{1}{t^{2}}\]

Example 64

F5: First order linear IVP (ver. 64)

Find the solution to the given IVP.

\[-\frac{28}{t^{2}} +ty'= 5 y \hspace{1em} y( -1 ) = 0\]

Answer.

\[y= -4 t^{ 5 } -4 \frac{1}{t^{2}}\]

Example 65

F5: First order linear IVP (ver. 65)

Find the solution to the given IVP.

\[-\frac{20}{t} +ty'= -5 y \hspace{1em} y( 1 ) = 7\]

Answer.

\[y= 2 t^{ -5 } +5 \frac{1}{t}\]

Example 66

F5: First order linear IVP (ver. 66)

Find the solution to the given IVP.

\[18 \, t^{3} +ty'= -3 y \hspace{1em} y( 1 ) = -4\]

Answer.

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

Example 67

F5: First order linear IVP (ver. 67)

Find the solution to the given IVP.

\[\frac{9}{t^{4}} +ty'= 5 y \hspace{1em} y( 1 ) = 2\]

Answer.

\[y= 1 t^{ 5 } + \frac{1}{t^{4}}\]

Example 68

F5: First order linear IVP (ver. 68)

Find the solution to the given IVP.

\[-8 \, t^{4} +ty'= 2 y \hspace{1em} y( 1 ) = 1\]

Answer.

\[y= -3 t^{ 2 } +4 t^{4}\]

Example 69

F5: First order linear IVP (ver. 69)

Find the solution to the given IVP.

\[-\frac{7}{t^{4}} +ty'= 3 y \hspace{1em} y( -1 ) = -2\]

Answer.

\[y= 1 t^{ 3 } -1 \frac{1}{t^{4}}\]

Example 70

F5: First order linear IVP (ver. 70)

Find the solution to the given IVP.

\[\frac{8}{t^{3}} +ty'= -5 y \hspace{1em} y( -1 ) = 8\]

Answer.

\[y= -4 t^{ -5 } -4 \frac{1}{t^{3}}\]

Example 71

F5: First order linear IVP (ver. 71)

Find the solution to the given IVP.

\[-8 \, t^{3} +ty'= -5 y \hspace{1em} y( 1 ) = 0\]

Answer.

\[y= -1 t^{ -5 } + t^{3}\]

Example 72

F5: First order linear IVP (ver. 72)

Find the solution to the given IVP.

\[-\frac{18}{t} +ty'= 5 y \hspace{1em} y( 1 ) = -4\]

Answer.

\[y= -1 t^{ 5 } -3 \frac{1}{t}\]

Example 73

F5: First order linear IVP (ver. 73)

Find the solution to the given IVP.

\[-\frac{32}{t^{4}} +ty'= 4 y \hspace{1em} y( -1 ) = -7\]

Answer.

\[y= -3 t^{ 4 } -4 \frac{1}{t^{4}}\]

Example 74

F5: First order linear IVP (ver. 74)

Find the solution to the given IVP.

\[3 \, t +ty'= 2 y \hspace{1em} y( 1 ) = 4\]

Answer.

\[y= 1 t^{ 2 } +3 t\]

Example 75

F5: First order linear IVP (ver. 75)

Find the solution to the given IVP.

\[-\frac{3}{t^{4}} +ty'= -3 y \hspace{1em} y( 1 ) = -1\]

Answer.

\[y= 2 t^{ -3 } -3 \frac{1}{t^{4}}\]

Example 76

F5: First order linear IVP (ver. 76)

Find the solution to the given IVP.

\[\frac{45}{t^{4}} +ty'= 5 y \hspace{1em} y( 1 ) = 2\]

Answer.

\[y= -3 t^{ 5 } +5 \frac{1}{t^{4}}\]

Example 77

F5: First order linear IVP (ver. 77)

Find the solution to the given IVP.

\[-4 \, t +ty'= 5 y \hspace{1em} y( 1 ) = 3\]

Answer.

\[y= 4 t^{ 5 } -1 t\]

Example 78

F5: First order linear IVP (ver. 78)

Find the solution to the given IVP.

\[-4 \, t^{4} +ty'= 2 y \hspace{1em} y( 1 ) = 1\]

Answer.

\[y= -1 t^{ 2 } +2 t^{4}\]

Example 79

F5: First order linear IVP (ver. 79)

Find the solution to the given IVP.

\[4 \, t +ty'= -3 y \hspace{1em} y( -1 ) = 0\]

Answer.

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

Example 80

F5: First order linear IVP (ver. 80)

Find the solution to the given IVP.

\[\frac{45}{t^{4}} +ty'= 5 y \hspace{1em} y( -1 ) = 7\]

Answer.

\[y= -2 t^{ 5 } +5 \frac{1}{t^{4}}\]

Example 81

F5: First order linear IVP (ver. 81)

Find the solution to the given IVP.

\[7 \, t^{2} +ty'= -5 y \hspace{1em} y( 1 ) = -2\]

Answer.

\[y= -1 t^{ -5 } -1 t^{2}\]

Example 82

F5: First order linear IVP (ver. 82)

Find the solution to the given IVP.

\[\frac{5}{t^{4}} +ty'= -5 y \hspace{1em} y( -1 ) = -7\]

Answer.

\[y= 2 t^{ -5 } -5 \frac{1}{t^{4}}\]

Example 83

F5: First order linear IVP (ver. 83)

Find the solution to the given IVP.

\[4 \, t^{2} +ty'= -2 y \hspace{1em} y( -1 ) = -3\]

Answer.

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

Example 84

F5: First order linear IVP (ver. 84)

Find the solution to the given IVP.

\[\frac{6}{t^{2}} +ty'= -4 y \hspace{1em} y( 1 ) = -4\]

Answer.

\[y= -1 t^{ -4 } -3 \frac{1}{t^{2}}\]

Example 85

F5: First order linear IVP (ver. 85)

Find the solution to the given IVP.

\[-\frac{10}{t} +ty'= -3 y \hspace{1em} y( 1 ) = 6\]

Answer.

\[y= 1 t^{ -3 } +5 \frac{1}{t}\]

Example 86

F5: First order linear IVP (ver. 86)

Find the solution to the given IVP.

\[-24 \, t^{3} +ty'= -3 y \hspace{1em} y( -1 ) = -7\]

Answer.

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

Example 87

F5: First order linear IVP (ver. 87)

Find the solution to the given IVP.

\[-\frac{6}{t^{3}} +ty'= 3 y \hspace{1em} y( 1 ) = 2\]

Answer.

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

Example 88

F5: First order linear IVP (ver. 88)

Find the solution to the given IVP.

\[-40 \, t^{3} +ty'= -5 y \hspace{1em} y( -1 ) = -9\]

Answer.

\[y= 4 t^{ -5 } +5 t^{3}\]

Example 89

F5: First order linear IVP (ver. 89)

Find the solution to the given IVP.

\[32 \, t^{3} +ty'= -5 y \hspace{1em} y( 1 ) = -7\]

Answer.

\[y= -3 t^{ -5 } -4 t^{3}\]

Example 90

F5: First order linear IVP (ver. 90)

Find the solution to the given IVP.

\[\frac{10}{t^{2}} +ty'= 3 y \hspace{1em} y( -1 ) = 6\]

Answer.

\[y= -4 t^{ 3 } +2 \frac{1}{t^{2}}\]

Example 91

F5: First order linear IVP (ver. 91)

Find the solution to the given IVP.

\[-28 \, t^{3} +ty'= -4 y \hspace{1em} y( 1 ) = 6\]

Answer.

\[y= 2 t^{ -4 } +4 t^{3}\]

Example 92

F5: First order linear IVP (ver. 92)

Find the solution to the given IVP.

\[-\frac{24}{t^{4}} +ty'= 2 y \hspace{1em} y( -1 ) = -3\]

Answer.

\[y= 1 t^{ 2 } -4 \frac{1}{t^{4}}\]

Example 93

F5: First order linear IVP (ver. 93)

Find the solution to the given IVP.

\[-\frac{25}{t^{3}} +ty'= 2 y \hspace{1em} y( 1 ) = -7\]

Answer.

\[y= -2 t^{ 2 } -5 \frac{1}{t^{3}}\]

Example 94

F5: First order linear IVP (ver. 94)

Find the solution to the given IVP.

\[-\frac{6}{t^{2}} +ty'= -4 y \hspace{1em} y( 1 ) = -1\]

Answer.

\[y= -4 t^{ -4 } +3 \frac{1}{t^{2}}\]

Example 95

F5: First order linear IVP (ver. 95)

Find the solution to the given IVP.

\[6 \, t +ty'= 3 y \hspace{1em} y( 1 ) = 4\]

Answer.

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

Example 96

F5: First order linear IVP (ver. 96)

Find the solution to the given IVP.

\[12 \, t +ty'= 4 y \hspace{1em} y( -1 ) = -8\]

Answer.

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

Example 97

F5: First order linear IVP (ver. 97)

Find the solution to the given IVP.

\[-\frac{4}{t^{3}} +ty'= -2 y \hspace{1em} y( 1 ) = -1\]

Answer.

\[y= 3 t^{ -2 } -4 \frac{1}{t^{3}}\]

Example 98

F5: First order linear IVP (ver. 98)

Find the solution to the given IVP.

\[\frac{1}{t^{3}} +ty'= -2 y \hspace{1em} y( -1 ) = 1\]

Answer.

\[y= 2 t^{ -2 } + \frac{1}{t^{3}}\]

Example 99

F5: First order linear IVP (ver. 99)

Find the solution to the given IVP.

\[-\frac{3}{t} +ty'= -2 y \hspace{1em} y( -1 ) = 1\]

Answer.

\[y= 4 t^{ -2 } +3 \frac{1}{t}\]

Example 100

F5: First order linear IVP (ver. 100)

Find the solution to the given IVP.

\[-\frac{14}{t^{4}} +ty'= 3 y \hspace{1em} y( -1 ) = -1\]

Answer.

\[y= -1 t^{ 3 } -2 \frac{1}{t^{4}}\]