Use the Eigenvalue Method to solve the following system of ODES. Please help with this question!!!

Question

Use the Eigenvalue Method to solve the following system of ODES:

 

 

Hint: The corresponding coefficient matrix has a pair of complex eigenvalues.

Answer 1

M = [-2 1;-4 -2]

find eigen values

det (M-I2*2) = r^2+4r+8

eigenvalues λ=-2+2i,:λ=-2-2i

a =-2

b= 2

plug in values and rref (M-λI|0)

rewrite solution in terms of free var parametric form

[-i;2] [i;2]

thus solution

[x(t); y(t)] = [c1 * e^a*t * cos(b*t) + c2 * e^a*t * *i*sin(b*t) ] * [ λ=-2+2i]