Find the general solution of the following matrix differenti

Find the general solution of the following matrix differential equation

and

Solution

y\' = [5 -1 ]

[ 3 -1 ]

finding first eigen values

(5-x) *(-1 -x) +3 =0

-5 -5x +x +x ^2 + 3 =0

x^2 -4x -2 =0

x = 2+sqrt (6), 2 -sqrt(6)

eigen vectors

matrix [x1] = [0]

[ x2] [0]

=> [1

sqrt6 + 3]

=>1

3 - sqrt 6

general solution is therefore

y(t) = c1 e^(eigen value) * eigen vector + c1 e^(eigen value) * eigen vector