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.

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

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