Solve the system with the initial value 7 3 2 2 SolutionFirs

Solve the system

with the initial value

$\"\\displaystyle$

$\"\\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\$	-7	3	$\"\\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\$
-2	-2

$\"x\"$

First let us determine the eigen values of matrix A

-7 3

-2 -2

A- mI =0 gives char polynomial as

(m^2+9m+20) =0

Hence Eigen values are -5, -4

Eigen vectors are

1 1

2/3, 1 respectively.

Hence general solution is

x(t) = c1e^-5t+c2e^-4t

x(0) = c1+c2 =

Thus two equations are c1+c2 =-7 and 2c1/3+c2=-6

Subtract to get c1/3 = -1 or c1 =-3

and c2 =-4

Hence soltuion is x(t) = -3e^-5t-4e^-4t