S1: System of IVPs

Example 1

S1: System of IVPs (ver. 1)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -3 \, e^{\left(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}\]

\[y= -e^{\left(6 \, t\right)} - 3 \, e^{\left(-7 \, t\right)}\]

Example 2

S1: System of IVPs (ver. 2)

Find the solution to the given system of IVPs.

\[x'= -4 \, x + 12 \, y\hspace{2em}x(0)=-7\]

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

Answer.

\[x= -4 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}\]

\[y= -3 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}\]

Example 3

S1: System of IVPs (ver. 3)

Find the solution to the given system of IVPs.

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

\[y'= -x + 9 \, y\hspace{2em}y(0)=0\]

Answer.

\[x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}\]

\[y= -e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}\]

Example 4

S1: System of IVPs (ver. 4)

Find the solution to the given system of IVPs.

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

\[y'= 2 \, x - 9 \, y\hspace{2em}y(0)=3\]

Answer.

\[x= 2 \, e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}\]

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

Example 5

S1: System of IVPs (ver. 5)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 6

S1: System of IVPs (ver. 6)

Find the solution to the given system of IVPs.

\[x'= 2 \, x - 9 \, y\hspace{2em}x(0)=8\]

\[y'= -4 \, x - 3 \, y\hspace{2em}y(0)=-5\]

Answer.

\[x= 9 \, e^{\left(6 \, t\right)} - e^{\left(-7 \, t\right)}\]

\[y= -4 \, e^{\left(6 \, t\right)} - e^{\left(-7 \, t\right)}\]

Example 7

S1: System of IVPs (ver. 7)

Find the solution to the given system of IVPs.

\[x'= 3 \, x + 9 \, y\hspace{2em}x(0)=-8\]

\[y'= 4 \, x + 8 \, y\hspace{2em}y(0)=5\]

Answer.

\[x= e^{\left(12 \, t\right)} - 9 \, e^{\left(-t\right)}\]

\[y= e^{\left(12 \, t\right)} + 4 \, e^{\left(-t\right)}\]

Example 8

S1: System of IVPs (ver. 8)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= 3 \, e^{\left(7 \, t\right)} - 4 \, e^{\left(-6 \, t\right)}\]

\[y= e^{\left(7 \, t\right)} + 3 \, e^{\left(-6 \, t\right)}\]

Example 9

S1: System of IVPs (ver. 9)

Find the solution to the given system of IVPs.

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

\[y'= 4 \, x - 7 \, y\hspace{2em}y(0)=-5\]

Answer.

\[x= -e^{\left(-3 \, t\right)} - e^{\left(-6 \, t\right)}\]

\[y= -e^{\left(-3 \, t\right)} - 4 \, e^{\left(-6 \, t\right)}\]

Example 10

S1: System of IVPs (ver. 10)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= 4 \, e^{\left(-12 \, t\right)} - 3 \, e^{t}\]

\[y= 3 \, e^{\left(-12 \, t\right)} + e^{t}\]

Example 11

S1: System of IVPs (ver. 11)

Find the solution to the given system of IVPs.

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

\[y'= 4 \, x - 9 \, y\hspace{2em}y(0)=5\]

Answer.

\[x= e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}\]

\[y= e^{\left(-5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}\]

Example 12

S1: System of IVPs (ver. 12)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -e^{\left(6 \, t\right)} + 2 \, e^{\left(3 \, t\right)}\]

\[y= -2 \, e^{\left(6 \, t\right)} + e^{\left(3 \, t\right)}\]

Example 13

S1: System of IVPs (ver. 13)

Find the solution to the given system of IVPs.

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

\[y'= -4 \, x + 7 \, y\hspace{2em}y(0)=5\]

Answer.

\[x= -e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}\]

\[y= 4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}\]

Example 14

S1: System of IVPs (ver. 14)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 15

S1: System of IVPs (ver. 15)

Find the solution to the given system of IVPs.

\[x'= x - 9 \, y\hspace{2em}x(0)=10\]

\[y'= -4 \, x + 6 \, y\hspace{2em}y(0)=3\]

Answer.

\[x= e^{\left(10 \, t\right)} + 9 \, e^{\left(-3 \, t\right)}\]

\[y= -e^{\left(10 \, t\right)} + 4 \, e^{\left(-3 \, t\right)}\]

Example 16

S1: System of IVPs (ver. 16)

Find the solution to the given system of IVPs.

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

\[y'= -4 \, x + 6 \, y\hspace{2em}y(0)=-5\]

Answer.

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

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

Example 17

S1: System of IVPs (ver. 17)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 18

S1: System of IVPs (ver. 18)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= 3 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-11 \, t\right)}\]

\[y= -e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}\]

Example 19

S1: System of IVPs (ver. 19)

Find the solution to the given system of IVPs.

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

\[y'= 4 \, x + 9 \, y\hspace{2em}y(0)=-3\]

Answer.

\[x= e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}\]

\[y= -4 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}\]

Example 20

S1: System of IVPs (ver. 20)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -e^{\left(7 \, t\right)} + 4 \, e^{\left(4 \, t\right)}\]

\[y= e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}\]

Example 21

S1: System of IVPs (ver. 21)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 22

S1: System of IVPs (ver. 22)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 23

S1: System of IVPs (ver. 23)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= 4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}\]

\[y= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}\]

Example 24

S1: System of IVPs (ver. 24)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -e^{\left(6 \, t\right)} - 4 \, e^{\left(3 \, t\right)}\]

\[y= e^{\left(6 \, t\right)} + e^{\left(3 \, t\right)}\]

Example 25

S1: System of IVPs (ver. 25)

Find the solution to the given system of IVPs.

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

\[y'= -2 \, x + 9 \, y\hspace{2em}y(0)=-3\]

Answer.

\[x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}\]

\[y= -2 \, e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}\]

Example 26

S1: System of IVPs (ver. 26)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= 2 \, e^{\left(10 \, t\right)} - 3 \, e^{\left(-3 \, t\right)}\]

\[y= 3 \, e^{\left(10 \, t\right)} + 2 \, e^{\left(-3 \, t\right)}\]

Example 27

S1: System of IVPs (ver. 27)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -e^{\left(6 \, t\right)} + e^{t}\]

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

Example 28

S1: System of IVPs (ver. 28)

Find the solution to the given system of IVPs.

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

\[y'= -x + 9 \, y\hspace{2em}y(0)=-2\]

Answer.

\[x= -e^{\left(8 \, t\right)} - 4 \, e^{\left(5 \, t\right)}\]

\[y= -e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}\]

Example 29

S1: System of IVPs (ver. 29)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -3 \, e^{\left(6 \, t\right)} - 2 \, e^{\left(-7 \, t\right)}\]

\[y= -2 \, e^{\left(6 \, t\right)} + 3 \, e^{\left(-7 \, t\right)}\]

Example 30

S1: System of IVPs (ver. 30)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 31

S1: System of IVPs (ver. 31)

Find the solution to the given system of IVPs.

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

\[y'= -x - y\hspace{2em}y(0)=2\]

Answer.

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

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

Example 32

S1: System of IVPs (ver. 32)

Find the solution to the given system of IVPs.

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

\[y'= -4 \, x + 9 \, y\hspace{2em}y(0)=5\]

Answer.

\[x= e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}\]

\[y= 4 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}\]

Example 33

S1: System of IVPs (ver. 33)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= e^{\left(6 \, t\right)} - e^{t}\]

\[y= 4 \, e^{\left(6 \, t\right)} + e^{t}\]

Example 34

S1: System of IVPs (ver. 34)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 35

S1: System of IVPs (ver. 35)

Find the solution to the given system of IVPs.

\[x'= 5 \, x + 4 \, y\hspace{2em}x(0)=-3\]

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

Answer.

\[x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}\]

\[y= e^{\left(9 \, t\right)} + e^{\left(4 \, t\right)}\]

Example 36

S1: System of IVPs (ver. 36)

Find the solution to the given system of IVPs.

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

\[y'= -9 \, x - 7 \, y\hspace{2em}y(0)=-8\]

Answer.

\[x= -e^{\left(2 \, t\right)} - 4 \, e^{\left(-11 \, t\right)}\]

\[y= e^{\left(2 \, t\right)} - 9 \, e^{\left(-11 \, t\right)}\]

Example 37

S1: System of IVPs (ver. 37)

Find the solution to the given system of IVPs.

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

\[y'= 9 \, x - y\hspace{2em}y(0)=8\]

Answer.

\[x= -e^{\left(8 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}\]

\[y= -e^{\left(8 \, t\right)} + 9 \, e^{\left(-5 \, t\right)}\]

Example 38

S1: System of IVPs (ver. 38)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 39

S1: System of IVPs (ver. 39)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= 3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}\]

\[y= -4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}\]

Example 40

S1: System of IVPs (ver. 40)

Find the solution to the given system of IVPs.

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

\[y'= -2 \, x - 9 \, y\hspace{2em}y(0)=-3\]

Answer.

\[x= 2 \, e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}\]

\[y= -e^{\left(-5 \, t\right)} - 2 \, e^{\left(-8 \, t\right)}\]

Example 41

S1: System of IVPs (ver. 41)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= 2 \, e^{\left(6 \, t\right)} - e^{t}\]

\[y= e^{\left(6 \, t\right)} + 2 \, e^{t}\]

Example 42

S1: System of IVPs (ver. 42)

Find the solution to the given system of IVPs.

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

\[y'= 2 \, x + 9 \, y\hspace{2em}y(0)=3\]

Answer.

\[x= -e^{\left(8 \, t\right)} - 2 \, e^{\left(5 \, t\right)}\]

\[y= 2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}\]

Example 43

S1: System of IVPs (ver. 43)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= e^{\left(8 \, t\right)} - 2 \, e^{\left(3 \, t\right)}\]

\[y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}\]

Example 44

S1: System of IVPs (ver. 44)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= 2 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}\]

\[y= -e^{\left(-t\right)} + 2 \, e^{\left(-6 \, t\right)}\]

Example 45

S1: System of IVPs (ver. 45)

Find the solution to the given system of IVPs.

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

\[y'= -12 \, x + 6 \, y\hspace{2em}y(0)=-1\]

Answer.

\[x= -e^{\left(10 \, t\right)} - 3 \, e^{\left(-3 \, t\right)}\]

\[y= 3 \, e^{\left(10 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}\]

Example 46

S1: System of IVPs (ver. 46)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 47

S1: System of IVPs (ver. 47)

Find the solution to the given system of IVPs.

\[x'= -4 \, x - 6 \, y\hspace{2em}x(0)=-1\]

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

Answer.

\[x= 2 \, e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}\]

\[y= -3 \, e^{\left(5 \, t\right)} - 2 \, e^{\left(-8 \, t\right)}\]

Example 48

S1: System of IVPs (ver. 48)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -e^{\left(6 \, t\right)} + 4 \, e^{t}\]

\[y= -e^{\left(6 \, t\right)} - e^{t}\]

Example 49

S1: System of IVPs (ver. 49)

Find the solution to the given system of IVPs.

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

\[y'= -4 \, x - 9 \, y\hspace{2em}y(0)=-3\]

Answer.

\[x= -e^{\left(-5 \, t\right)} + e^{\left(-8 \, t\right)}\]

\[y= e^{\left(-5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}\]

Example 50

S1: System of IVPs (ver. 50)

Find the solution to the given system of IVPs.

\[x'= -x - y\hspace{2em}x(0)=-2\]

\[y'= 4 \, x - 6 \, y\hspace{2em}y(0)=-5\]

Answer.

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

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

Example 51

S1: System of IVPs (ver. 51)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 52

S1: System of IVPs (ver. 52)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= e^{\left(7 \, t\right)} - 2 \, e^{\left(4 \, t\right)}\]

\[y= 2 \, e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}\]

Example 53

S1: System of IVPs (ver. 53)

Find the solution to the given system of IVPs.

\[x'= 4 \, x - 9 \, y\hspace{2em}x(0)=8\]

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

Answer.

\[x= 9 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}\]

\[y= -4 \, e^{\left(8 \, t\right)} - e^{\left(-5 \, t\right)}\]

Example 54

S1: System of IVPs (ver. 54)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= 4 \, e^{\left(12 \, t\right)} - 3 \, e^{\left(-t\right)}\]

\[y= -3 \, e^{\left(12 \, t\right)} - e^{\left(-t\right)}\]

Example 55

S1: System of IVPs (ver. 55)

Find the solution to the given system of IVPs.

\[x'= x - 9 \, y\hspace{2em}x(0)=10\]

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

Answer.

\[x= 9 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}\]

\[y= -4 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}\]

Example 56

S1: System of IVPs (ver. 56)

Find the solution to the given system of IVPs.

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

\[y'= -4 \, x + 3 \, y\hspace{2em}y(0)=-5\]

Answer.

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

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

Example 57

S1: System of IVPs (ver. 57)

Find the solution to the given system of IVPs.

\[x'= -x + 9 \, y\hspace{2em}x(0)=-8\]

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

Answer.

\[x= e^{\left(8 \, t\right)} - 9 \, e^{\left(-5 \, t\right)}\]

\[y= e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}\]

Example 58

S1: System of IVPs (ver. 58)

Find the solution to the given system of IVPs.

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

\[y'= 4 \, x - 8 \, y\hspace{2em}y(0)=5\]

Answer.

\[x= e^{\left(-4 \, t\right)} + e^{\left(-7 \, t\right)}\]

\[y= e^{\left(-4 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}\]

Example 59

S1: System of IVPs (ver. 59)

Find the solution to the given system of IVPs.

\[x'= -x - y\hspace{2em}x(0)=-2\]

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

Answer.

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

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

Example 60

S1: System of IVPs (ver. 60)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 61

S1: System of IVPs (ver. 61)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -e^{\left(8 \, t\right)} - 4 \, e^{\left(3 \, t\right)}\]

\[y= -e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}\]

Example 62

S1: System of IVPs (ver. 62)

Find the solution to the given system of IVPs.

\[x'= x - y\hspace{2em}x(0)=2\]

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

Answer.

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

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

Example 63

S1: System of IVPs (ver. 63)

Find the solution to the given system of IVPs.

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

\[y'= -9 \, x + 2 \, y\hspace{2em}y(0)=10\]

Answer.

\[x= -4 \, e^{\left(6 \, t\right)} + e^{\left(-7 \, t\right)}\]

\[y= 9 \, e^{\left(6 \, t\right)} + e^{\left(-7 \, t\right)}\]

Example 64

S1: System of IVPs (ver. 64)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -2 \, e^{\left(10 \, t\right)} + 3 \, e^{\left(-3 \, t\right)}\]

\[y= -3 \, e^{\left(10 \, t\right)} - 2 \, e^{\left(-3 \, t\right)}\]

Example 65

S1: System of IVPs (ver. 65)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= 3 \, e^{\left(2 \, t\right)} + 4 \, e^{\left(-11 \, t\right)}\]

\[y= e^{\left(2 \, t\right)} - 3 \, e^{\left(-11 \, t\right)}\]

Example 66

S1: System of IVPs (ver. 66)

Find the solution to the given system of IVPs.

\[x'= -x - y\hspace{2em}x(0)=0\]

\[y'= 4 \, x - 6 \, y\hspace{2em}y(0)=-3\]

Answer.

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

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

Example 67

S1: System of IVPs (ver. 67)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= 3 \, e^{\left(8 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}\]

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

Example 68

S1: System of IVPs (ver. 68)

Find the solution to the given system of IVPs.

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

\[y'= 4 \, x + 8 \, y\hspace{2em}y(0)=5\]

Answer.

\[x= e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}\]

\[y= 4 \, e^{\left(9 \, t\right)} + e^{\left(4 \, t\right)}\]

Example 69

S1: System of IVPs (ver. 69)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 70

S1: System of IVPs (ver. 70)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}\]

\[y= -3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}\]

Example 71

S1: System of IVPs (ver. 71)

Find the solution to the given system of IVPs.

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

\[y'= -9 \, x - 6 \, y\hspace{2em}y(0)=-10\]

Answer.

\[x= e^{\left(3 \, t\right)} - 4 \, e^{\left(-10 \, t\right)}\]

\[y= -e^{\left(3 \, t\right)} - 9 \, e^{\left(-10 \, t\right)}\]

Example 72

S1: System of IVPs (ver. 72)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -3 \, e^{\left(7 \, t\right)} + 2 \, e^{\left(-6 \, t\right)}\]

\[y= 2 \, e^{\left(7 \, t\right)} + 3 \, e^{\left(-6 \, t\right)}\]

Example 73

S1: System of IVPs (ver. 73)

Find the solution to the given system of IVPs.

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

\[y'= 9 \, x + 7 \, y\hspace{2em}y(0)=-8\]

Answer.

\[x= -4 \, e^{\left(11 \, t\right)} - e^{\left(-2 \, t\right)}\]

\[y= -9 \, e^{\left(11 \, t\right)} + e^{\left(-2 \, t\right)}\]

Example 74

S1: System of IVPs (ver. 74)

Find the solution to the given system of IVPs.

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

\[y'= -x - y\hspace{2em}y(0)=0\]

Answer.

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

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

Example 75

S1: System of IVPs (ver. 75)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 76

S1: System of IVPs (ver. 76)

Find the solution to the given system of IVPs.

\[x'= 4 \, x + 12 \, y\hspace{2em}x(0)=7\]

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

Answer.

\[x= 3 \, e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}\]

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

Example 77

S1: System of IVPs (ver. 77)

Find the solution to the given system of IVPs.

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

\[y'= -9 \, x - y\hspace{2em}y(0)=8\]

Answer.

\[x= e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}\]

\[y= -e^{\left(8 \, t\right)} + 9 \, e^{\left(-5 \, t\right)}\]

Example 78

S1: System of IVPs (ver. 78)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= e^{\left(6 \, t\right)} - e^{t}\]

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

Example 79

S1: System of IVPs (ver. 79)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -2 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}\]

\[y= e^{\left(-t\right)} + 2 \, e^{\left(-6 \, t\right)}\]

Example 80

S1: System of IVPs (ver. 80)

Find the solution to the given system of IVPs.

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

\[y'= 6 \, x - 4 \, y\hspace{2em}y(0)=5\]

Answer.

\[x= 3 \, e^{\left(5 \, t\right)} - 2 \, e^{\left(-8 \, t\right)}\]

\[y= 2 \, e^{\left(5 \, t\right)} + 3 \, e^{\left(-8 \, t\right)}\]

Example 81

S1: System of IVPs (ver. 81)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}\]

\[y= 4 \, e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}\]

Example 82

S1: System of IVPs (ver. 82)

Find the solution to the given system of IVPs.

\[x'= -4 \, x - 12 \, y\hspace{2em}x(0)=7\]

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

Answer.

\[x= 4 \, e^{\left(5 \, t\right)} + 3 \, e^{\left(-8 \, t\right)}\]

\[y= -3 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}\]

Example 83

S1: System of IVPs (ver. 83)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}\]

\[y= e^{\left(-t\right)} + 4 \, e^{\left(-6 \, t\right)}\]

Example 84

S1: System of IVPs (ver. 84)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -3 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}\]

\[y= -4 \, e^{\left(2 \, t\right)} - 3 \, e^{\left(-11 \, t\right)}\]

Example 85

S1: System of IVPs (ver. 85)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -2 \, e^{\left(-3 \, t\right)} + e^{\left(-6 \, t\right)}\]

\[y= e^{\left(-3 \, t\right)} - 2 \, e^{\left(-6 \, t\right)}\]

Example 86

S1: System of IVPs (ver. 86)

Find the solution to the given system of IVPs.

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

\[y'= x + y\hspace{2em}y(0)=2\]

Answer.

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

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

Example 87

S1: System of IVPs (ver. 87)

Find the solution to the given system of IVPs.

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

\[y'= x + 9 \, y\hspace{2em}y(0)=0\]

Answer.

\[x= -e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}\]

\[y= e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}\]

Example 88

S1: System of IVPs (ver. 88)

Find the solution to the given system of IVPs.

\[x'= -3 \, x + 9 \, y\hspace{2em}x(0)=-8\]

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

Answer.

\[x= e^{\left(6 \, t\right)} - 9 \, e^{\left(-7 \, t\right)}\]

\[y= e^{\left(6 \, t\right)} + 4 \, e^{\left(-7 \, t\right)}\]

Example 89

S1: System of IVPs (ver. 89)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -e^{\left(7 \, t\right)} - 3 \, e^{\left(-6 \, t\right)}\]

\[y= -3 \, e^{\left(7 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}\]

Example 90

S1: System of IVPs (ver. 90)

Find the solution to the given system of IVPs.

\[x'= x - y\hspace{2em}x(0)=0\]

\[y'= 4 \, x + 6 \, y\hspace{2em}y(0)=-3\]

Answer.

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

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

Example 91

S1: System of IVPs (ver. 91)

Find the solution to the given system of IVPs.

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

\[y'= 3 \, x - 7 \, y\hspace{2em}y(0)=4\]

Answer.

\[x= 3 \, e^{\left(2 \, t\right)} - 4 \, e^{\left(-11 \, t\right)}\]

\[y= e^{\left(2 \, t\right)} + 3 \, e^{\left(-11 \, t\right)}\]

Example 92

S1: System of IVPs (ver. 92)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}\]

\[y= -2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}\]

Example 93

S1: System of IVPs (ver. 93)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 94

S1: System of IVPs (ver. 94)

Find the solution to the given system of IVPs.

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

\[y'= 9 \, x + 3 \, y\hspace{2em}y(0)=8\]

Answer.

\[x= 4 \, e^{\left(7 \, t\right)} + e^{\left(-6 \, t\right)}\]

\[y= 9 \, e^{\left(7 \, t\right)} - e^{\left(-6 \, t\right)}\]

Example 95

S1: System of IVPs (ver. 95)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= 2 \, e^{\left(6 \, t\right)} + e^{t}\]

\[y= -e^{\left(6 \, t\right)} + 2 \, e^{t}\]

Example 96

S1: System of IVPs (ver. 96)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= e^{\left(11 \, t\right)} + 3 \, e^{\left(-2 \, t\right)}\]

\[y= 3 \, e^{\left(11 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}\]

Example 97

S1: System of IVPs (ver. 97)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= -e^{\left(6 \, t\right)} - 2 \, e^{t}\]

\[y= -2 \, e^{\left(6 \, t\right)} + e^{t}\]

Example 98

S1: System of IVPs (ver. 98)

Find the solution to the given system of IVPs.

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

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

Answer.

\[x= 2 \, e^{\left(-t\right)} - e^{\left(-6 \, t\right)}\]

\[y= -e^{\left(-t\right)} - 2 \, e^{\left(-6 \, t\right)}\]

Example 99

S1: System of IVPs (ver. 99)

Find the solution to the given system of IVPs.

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

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

Answer.

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

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

Example 100

S1: System of IVPs (ver. 100)

Find the solution to the given system of IVPs.

\[x'= x + y\hspace{2em}x(0)=0\]

\[y'= -4 \, x + 6 \, y\hspace{2em}y(0)=-3\]

Answer.

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

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