F2b: Separation of variables

Example 1

F2b: Separation of variables (ver. 1)

Find the solution to the given IVP.

\[y'=( 6 \, t )y\hspace{1em} y(0)= -e^{\left(-5\right)}\]

Answer.

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

Example 2

F2b: Separation of variables (ver. 2)

Find the solution to the given IVP.

\[y'=( -6 \, t - 3 )y\hspace{1em} y(0)= -3 \, e^{\left(-1\right)}\]

Answer.

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

Example 3

F2b: Separation of variables (ver. 3)

Find the solution to the given IVP.

\[y'=( -3 \, \cos\left(t\right) )y\hspace{1em} y(0)= 4\]

Answer.

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

Example 4

F2b: Separation of variables (ver. 4)

Find the solution to the given IVP.

\[y'=( -4 \, t - 2 )y\hspace{1em} y(0)= 2 \, e\]

Answer.

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

Example 5

F2b: Separation of variables (ver. 5)

Find the solution to the given IVP.

\[y'=( 2 \, t - 2 )y\hspace{1em} y(0)= 4 \, e^{\left(-2\right)}\]

Answer.

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

Example 6

F2b: Separation of variables (ver. 6)

Find the solution to the given IVP.

\[y'=( 6 \, t - 1 )y\hspace{1em} y(0)= -3 \, e^{\left(-2\right)}\]

Answer.

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

Example 7

F2b: Separation of variables (ver. 7)

Find the solution to the given IVP.

\[y'=( 3 \, \cos\left(t\right) )y\hspace{1em} y(0)= -1\]

Answer.

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

Example 8

F2b: Separation of variables (ver. 8)

Find the solution to the given IVP.

\[y'=( 6 \, t + 1 )y\hspace{1em} y(0)= 3 \, e^{\left(-4\right)}\]

Answer.

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

Example 9

F2b: Separation of variables (ver. 9)

Find the solution to the given IVP.

\[y'=( 2 \, \cos\left(t\right) )y\hspace{1em} y(0)= 3\]

Answer.

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

Example 10

F2b: Separation of variables (ver. 10)

Find the solution to the given IVP.

\[y'=( -\cos\left(t\right) )y\hspace{1em} y(0)= -2\]

Answer.

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

Example 11

F2b: Separation of variables (ver. 11)

Find the solution to the given IVP.

\[y'=( \sin\left(t\right) )y\hspace{1em} y(0)= e^{\left(-1\right)}\]

Answer.

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

Example 12

F2b: Separation of variables (ver. 12)

Find the solution to the given IVP.

\[y'=( \cos\left(t\right) )y\hspace{1em} y(0)= 4\]

Answer.

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

Example 13

F2b: Separation of variables (ver. 13)

Find the solution to the given IVP.

\[y'=( \sin\left(t\right) )y\hspace{1em} y(0)= -3 \, e^{\left(-1\right)}\]

Answer.

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

Example 14

F2b: Separation of variables (ver. 14)

Find the solution to the given IVP.

\[y'=( \cos\left(t\right) )y\hspace{1em} y(0)= 1\]

Answer.

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

Example 15

F2b: Separation of variables (ver. 15)

Find the solution to the given IVP.

\[y'=( 2 \, \sin\left(t\right) )y\hspace{1em} y(0)= -2 \, e^{\left(-2\right)}\]

Answer.

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

Example 16

F2b: Separation of variables (ver. 16)

Find the solution to the given IVP.

\[y'=( -\sin\left(t\right) )y\hspace{1em} y(0)= -2 \, e\]

Answer.

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

Example 17

F2b: Separation of variables (ver. 17)

Find the solution to the given IVP.

\[y'=( -4 \, t )y\hspace{1em} y(0)= e^{4}\]

Answer.

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

Example 18

F2b: Separation of variables (ver. 18)

Find the solution to the given IVP.

\[y'=( -6 \, t + 1 )y\hspace{1em} y(0)= -4 \, e^{\left(-2\right)}\]

Answer.

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

Example 19

F2b: Separation of variables (ver. 19)

Find the solution to the given IVP.

\[y'=( \sin\left(t\right) )y\hspace{1em} y(0)= 2 \, e^{\left(-1\right)}\]

Answer.

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

Example 20

F2b: Separation of variables (ver. 20)

Find the solution to the given IVP.

\[y'=( 3 \, \sin\left(t\right) )y\hspace{1em} y(0)= 4 \, e^{\left(-3\right)}\]

Answer.

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

Example 21

F2b: Separation of variables (ver. 21)

Find the solution to the given IVP.

\[y'=( -2 \, t - 2 )y\hspace{1em} y(0)= -2 \, e\]

Answer.

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

Example 22

F2b: Separation of variables (ver. 22)

Find the solution to the given IVP.

\[y'=( -3 \, \sin\left(t\right) )y\hspace{1em} y(0)= -4 \, e^{3}\]

Answer.

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

Example 23

F2b: Separation of variables (ver. 23)

Find the solution to the given IVP.

\[y'=( 3 \, \sin\left(t\right) )y\hspace{1em} y(0)= -4 \, e^{\left(-3\right)}\]

Answer.

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

Example 24

F2b: Separation of variables (ver. 24)

Find the solution to the given IVP.

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

Answer.

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

Example 25

F2b: Separation of variables (ver. 25)

Find the solution to the given IVP.

\[y'=( 4 \, t + 3 )y\hspace{1em} y(0)= -2 \, e^{3}\]

Answer.

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

Example 26

F2b: Separation of variables (ver. 26)

Find the solution to the given IVP.

\[y'=( -2 \, \sin\left(t\right) )y\hspace{1em} y(0)= -2 \, e^{2}\]

Answer.

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

Example 27

F2b: Separation of variables (ver. 27)

Find the solution to the given IVP.

\[y'=( -\cos\left(t\right) )y\hspace{1em} y(0)= -2\]

Answer.

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

Example 28

F2b: Separation of variables (ver. 28)

Find the solution to the given IVP.

\[y'=( -\sin\left(t\right) )y\hspace{1em} y(0)= e\]

Answer.

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

Example 29

F2b: Separation of variables (ver. 29)

Find the solution to the given IVP.

\[y'=( 6 \, t - 1 )y\hspace{1em} y(0)= 4 \, e^{\left(-5\right)}\]

Answer.

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

Example 30

F2b: Separation of variables (ver. 30)

Find the solution to the given IVP.

\[y'=( 3 \, \sin\left(t\right) )y\hspace{1em} y(0)= -e^{\left(-3\right)}\]

Answer.

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

Example 31

F2b: Separation of variables (ver. 31)

Find the solution to the given IVP.

\[y'=( 3 \, \sin\left(t\right) )y\hspace{1em} y(0)= e^{\left(-3\right)}\]

Answer.

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

Example 32

F2b: Separation of variables (ver. 32)

Find the solution to the given IVP.

\[y'=( 4 \, t + 2 )y\hspace{1em} y(0)= 3 \, e^{3}\]

Answer.

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

Example 33

F2b: Separation of variables (ver. 33)

Find the solution to the given IVP.

\[y'=( 2 \, \cos\left(t\right) )y\hspace{1em} y(0)= -1\]

Answer.

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

Example 34

F2b: Separation of variables (ver. 34)

Find the solution to the given IVP.

\[y'=( -3 \, \sin\left(t\right) )y\hspace{1em} y(0)= -2 \, e^{3}\]

Answer.

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

Example 35

F2b: Separation of variables (ver. 35)

Find the solution to the given IVP.

\[y'=( -3 \, \sin\left(t\right) )y\hspace{1em} y(0)= -4 \, e^{3}\]

Answer.

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

Example 36

F2b: Separation of variables (ver. 36)

Find the solution to the given IVP.

\[y'=( -6 \, t )y\hspace{1em} y(0)= 3 \, e^{\left(-3\right)}\]

Answer.

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

Example 37

F2b: Separation of variables (ver. 37)

Find the solution to the given IVP.

\[y'=( 6 \, t + 3 )y\hspace{1em} y(0)= 3 \, e^{\left(-2\right)}\]

Answer.

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

Example 38

F2b: Separation of variables (ver. 38)

Find the solution to the given IVP.

\[y'=( -2 \, t - 3 )y\hspace{1em} y(0)= -3 \, e^{3}\]

Answer.

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

Example 39

F2b: Separation of variables (ver. 39)

Find the solution to the given IVP.

\[y'=( -6 \, t + 1 )y\hspace{1em} y(0)= 2 \, e^{\left(-1\right)}\]

Answer.

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

Example 40

F2b: Separation of variables (ver. 40)

Find the solution to the given IVP.

\[y'=( -\cos\left(t\right) )y\hspace{1em} y(0)= 3\]

Answer.

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

Example 41

F2b: Separation of variables (ver. 41)

Find the solution to the given IVP.

\[y'=( -2 \, \sin\left(t\right) )y\hspace{1em} y(0)= -3 \, e^{2}\]

Answer.

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

Example 42

F2b: Separation of variables (ver. 42)

Find the solution to the given IVP.

\[y'=( 2 \, \sin\left(t\right) )y\hspace{1em} y(0)= 4 \, e^{\left(-2\right)}\]

Answer.

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

Example 43

F2b: Separation of variables (ver. 43)

Find the solution to the given IVP.

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

Answer.

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

Example 44

F2b: Separation of variables (ver. 44)

Find the solution to the given IVP.

\[y'=( -2 \, t - 3 )y\hspace{1em} y(0)= 3 \, e^{\left(-5\right)}\]

Answer.

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

Example 45

F2b: Separation of variables (ver. 45)

Find the solution to the given IVP.

\[y'=( \sin\left(t\right) )y\hspace{1em} y(0)= -2 \, e^{\left(-1\right)}\]

Answer.

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

Example 46

F2b: Separation of variables (ver. 46)

Find the solution to the given IVP.

\[y'=( 3 \, \sin\left(t\right) )y\hspace{1em} y(0)= 4 \, e^{\left(-3\right)}\]

Answer.

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

Example 47

F2b: Separation of variables (ver. 47)

Find the solution to the given IVP.

\[y'=( \sin\left(t\right) )y\hspace{1em} y(0)= 2 \, e^{\left(-1\right)}\]

Answer.

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

Example 48

F2b: Separation of variables (ver. 48)

Find the solution to the given IVP.

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

Answer.

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

Example 49

F2b: Separation of variables (ver. 49)

Find the solution to the given IVP.

\[y'=( -3 \, \cos\left(t\right) )y\hspace{1em} y(0)= -3\]

Answer.

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

Example 50

F2b: Separation of variables (ver. 50)

Find the solution to the given IVP.

\[y'=( 2 \, \cos\left(t\right) )y\hspace{1em} y(0)= 2\]

Answer.

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

Example 51

F2b: Separation of variables (ver. 51)

Find the solution to the given IVP.

\[y'=( -3 \, \sin\left(t\right) )y\hspace{1em} y(0)= 2 \, e^{3}\]

Answer.

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

Example 52

F2b: Separation of variables (ver. 52)

Find the solution to the given IVP.

\[y'=( 4 \, t )y\hspace{1em} y(0)= 2 \, e^{5}\]

Answer.

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

Example 53

F2b: Separation of variables (ver. 53)

Find the solution to the given IVP.

\[y'=( 2 \, \sin\left(t\right) )y\hspace{1em} y(0)= 2 \, e^{\left(-2\right)}\]

Answer.

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

Example 54

F2b: Separation of variables (ver. 54)

Find the solution to the given IVP.

\[y'=( 3 \, \sin\left(t\right) )y\hspace{1em} y(0)= -4 \, e^{\left(-3\right)}\]

Answer.

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

Example 55

F2b: Separation of variables (ver. 55)

Find the solution to the given IVP.

\[y'=( -3 \, \cos\left(t\right) )y\hspace{1em} y(0)= 3\]

Answer.

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

Example 56

F2b: Separation of variables (ver. 56)

Find the solution to the given IVP.

\[y'=( -4 \, t + 2 )y\hspace{1em} y(0)= -3 \, e\]

Answer.

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

Example 57

F2b: Separation of variables (ver. 57)

Find the solution to the given IVP.

\[y'=( 2 \, \cos\left(t\right) )y\hspace{1em} y(0)= -4\]

Answer.

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

Example 58

F2b: Separation of variables (ver. 58)

Find the solution to the given IVP.

\[y'=( -\cos\left(t\right) )y\hspace{1em} y(0)= -2\]

Answer.

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

Example 59

F2b: Separation of variables (ver. 59)

Find the solution to the given IVP.

\[y'=( -\cos\left(t\right) )y\hspace{1em} y(0)= -4\]

Answer.

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

Example 60

F2b: Separation of variables (ver. 60)

Find the solution to the given IVP.

\[y'=( -2 \, t - 3 )y\hspace{1em} y(0)= 2 \, e^{\left(-2\right)}\]

Answer.

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

Example 61

F2b: Separation of variables (ver. 61)

Find the solution to the given IVP.

\[y'=( -3 \, \sin\left(t\right) )y\hspace{1em} y(0)= -3 \, e^{3}\]

Answer.

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

Example 62

F2b: Separation of variables (ver. 62)

Find the solution to the given IVP.

\[y'=( -2 \, t - 2 )y\hspace{1em} y(0)= -e\]

Answer.

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

Example 63

F2b: Separation of variables (ver. 63)

Find the solution to the given IVP.

\[y'=( \sin\left(t\right) )y\hspace{1em} y(0)= 4 \, e^{\left(-1\right)}\]

Answer.

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

Example 64

F2b: Separation of variables (ver. 64)

Find the solution to the given IVP.

\[y'=( 6 \, t - 3 )y\hspace{1em} y(0)= 3 \, e^{3}\]

Answer.

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

Example 65

F2b: Separation of variables (ver. 65)

Find the solution to the given IVP.

\[y'=( -6 \, t )y\hspace{1em} y(0)= e^{\left(-2\right)}\]

Answer.

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

Example 66

F2b: Separation of variables (ver. 66)

Find the solution to the given IVP.

\[y'=( 2 \, \cos\left(t\right) )y\hspace{1em} y(0)= -3\]

Answer.

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

Example 67

F2b: Separation of variables (ver. 67)

Find the solution to the given IVP.

\[y'=( 6 \, t + 3 )y\hspace{1em} y(0)= 2 \, e^{\left(-4\right)}\]

Answer.

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

Example 68

F2b: Separation of variables (ver. 68)

Find the solution to the given IVP.

\[y'=( -\sin\left(t\right) )y\hspace{1em} y(0)= -2 \, e\]

Answer.

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

Example 69

F2b: Separation of variables (ver. 69)

Find the solution to the given IVP.

\[y'=( 2 \, \sin\left(t\right) )y\hspace{1em} y(0)= e^{\left(-2\right)}\]

Answer.

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

Example 70

F2b: Separation of variables (ver. 70)

Find the solution to the given IVP.

\[y'=( -6 \, t - 3 )y\hspace{1em} y(0)= -2 \, e^{4}\]

Answer.

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

Example 71

F2b: Separation of variables (ver. 71)

Find the solution to the given IVP.

\[y'=( 2 \, \cos\left(t\right) )y\hspace{1em} y(0)= 1\]

Answer.

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

Example 72

F2b: Separation of variables (ver. 72)

Find the solution to the given IVP.

\[y'=( 2 \, \sin\left(t\right) )y\hspace{1em} y(0)= 4 \, e^{\left(-2\right)}\]

Answer.

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

Example 73

F2b: Separation of variables (ver. 73)

Find the solution to the given IVP.

\[y'=( 6 \, t )y\hspace{1em} y(0)= 1\]

Answer.

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

Example 74

F2b: Separation of variables (ver. 74)

Find the solution to the given IVP.

\[y'=( -3 \, \cos\left(t\right) )y\hspace{1em} y(0)= 2\]

Answer.

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

Example 75

F2b: Separation of variables (ver. 75)

Find the solution to the given IVP.

\[y'=( -4 \, t + 2 )y\hspace{1em} y(0)= e^{\left(-3\right)}\]

Answer.

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

Example 76

F2b: Separation of variables (ver. 76)

Find the solution to the given IVP.

\[y'=( -2 \, \sin\left(t\right) )y\hspace{1em} y(0)= 3 \, e^{2}\]

Answer.

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

Example 77

F2b: Separation of variables (ver. 77)

Find the solution to the given IVP.

\[y'=( -2 \, \cos\left(t\right) )y\hspace{1em} y(0)= 1\]

Answer.

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

Example 78

F2b: Separation of variables (ver. 78)

Find the solution to the given IVP.

\[y'=( \sin\left(t\right) )y\hspace{1em} y(0)= 4 \, e^{\left(-1\right)}\]

Answer.

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

Example 79

F2b: Separation of variables (ver. 79)

Find the solution to the given IVP.

\[y'=( 2 \, \sin\left(t\right) )y\hspace{1em} y(0)= -4 \, e^{\left(-2\right)}\]

Answer.

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

Example 80

F2b: Separation of variables (ver. 80)

Find the solution to the given IVP.

\[y'=( 6 \, t - 3 )y\hspace{1em} y(0)= -4\]

Answer.

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

Example 81

F2b: Separation of variables (ver. 81)

Find the solution to the given IVP.

\[y'=( \sin\left(t\right) )y\hspace{1em} y(0)= 2 \, e^{\left(-1\right)}\]

Answer.

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

Example 82

F2b: Separation of variables (ver. 82)

Find the solution to the given IVP.

\[y'=( -2 \, \sin\left(t\right) )y\hspace{1em} y(0)= -2 \, e^{2}\]

Answer.

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

Example 83

F2b: Separation of variables (ver. 83)

Find the solution to the given IVP.

\[y'=( -\cos\left(t\right) )y\hspace{1em} y(0)= -1\]

Answer.

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

Example 84

F2b: Separation of variables (ver. 84)

Find the solution to the given IVP.

\[y'=( -6 \, t + 2 )y\hspace{1em} y(0)= -3 \, e^{\left(-3\right)}\]

Answer.

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

Example 85

F2b: Separation of variables (ver. 85)

Find the solution to the given IVP.

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

Answer.

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

Example 86

F2b: Separation of variables (ver. 86)

Find the solution to the given IVP.

\[y'=( -2 \, t - 2 )y\hspace{1em} y(0)= 3 \, e^{2}\]

Answer.

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

Example 87

F2b: Separation of variables (ver. 87)

Find the solution to the given IVP.

\[y'=( 4 \, t + 2 )y\hspace{1em} y(0)= 3 \, e^{3}\]

Answer.

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

Example 88

F2b: Separation of variables (ver. 88)

Find the solution to the given IVP.

\[y'=( 3 \, \cos\left(t\right) )y\hspace{1em} y(0)= 2\]

Answer.

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

Example 89

F2b: Separation of variables (ver. 89)

Find the solution to the given IVP.

\[y'=( -\sin\left(t\right) )y\hspace{1em} y(0)= -4 \, e\]

Answer.

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

Example 90

F2b: Separation of variables (ver. 90)

Find the solution to the given IVP.

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

Answer.

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

Example 91

F2b: Separation of variables (ver. 91)

Find the solution to the given IVP.

\[y'=( 3 \, \cos\left(t\right) )y\hspace{1em} y(0)= 1\]

Answer.

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

Example 92

F2b: Separation of variables (ver. 92)

Find the solution to the given IVP.

\[y'=( -3 \, \sin\left(t\right) )y\hspace{1em} y(0)= 3 \, e^{3}\]

Answer.

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

Example 93

F2b: Separation of variables (ver. 93)

Find the solution to the given IVP.

\[y'=( -\sin\left(t\right) )y\hspace{1em} y(0)= -3 \, e\]

Answer.

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

Example 94

F2b: Separation of variables (ver. 94)

Find the solution to the given IVP.

\[y'=( -6 \, t )y\hspace{1em} y(0)= e^{4}\]

Answer.

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

Example 95

F2b: Separation of variables (ver. 95)

Find the solution to the given IVP.

\[y'=( 2 \, \sin\left(t\right) )y\hspace{1em} y(0)= 3 \, e^{\left(-2\right)}\]

Answer.

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

Example 96

F2b: Separation of variables (ver. 96)

Find the solution to the given IVP.

\[y'=( -\sin\left(t\right) )y\hspace{1em} y(0)= -e\]

Answer.

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

Example 97

F2b: Separation of variables (ver. 97)

Find the solution to the given IVP.

\[y'=( -2 \, \sin\left(t\right) )y\hspace{1em} y(0)= 2 \, e^{2}\]

Answer.

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

Example 98

F2b: Separation of variables (ver. 98)

Find the solution to the given IVP.

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

Answer.

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

Example 99

F2b: Separation of variables (ver. 99)

Find the solution to the given IVP.

\[y'=( -2 \, t )y\hspace{1em} y(0)= -e^{3}\]

Answer.

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

Example 100

F2b: Separation of variables (ver. 100)

Find the solution to the given IVP.

\[y'=( 4 \, t - 3 )y\hspace{1em} y(0)= -4 \, e^{\left(-5\right)}\]

Answer.

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