Please help with Eigenvalue question

Question

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

{x'(t)=x(t)+2y(t)

{y'(t)=-x(t)+4y(t)}

Answer 1

We have matrix A = [(1 -1) (2 4)]. then det(A - λE) det[(1- λ - 1) (2 4 - λ)] = (1 - λ)(4 - λ) + 2 = λ² - 5λ + 6 = 0;

So (λ - 2)(λ - 3) = 0. Now igenvalues λ = 2 and λ = 3.

For λ = 2 we have system for igenvector (x, y): - x + 2 y = 0; - x + 2y = 0, x = 2y and (2, 1) corresponding igenvector. For λ = 3 we have system for igenvector (x, y): - 2x + 2y = 0 and - x + y = 0, so y = x and (1, 1) is corresponding second igenvector.

Now we can get general solution of the system: x(t) = 2C₁e^2t + C₂e^3t;

y(t) = C₁e^2t + C₂e^3t