Someone please help with this question!

Question

x'=-x+2y

y'=-2x+3y

What is the genral solution with C₁ and C₂?

λ₁=λ₂=1

Eigenvector=

[2]

[2]

Answer 1

Denote h₁=[2 2]^T the eigenvector for λ=1, then f₁(t)=h₁e^t is a particular solution. Another(linearly independent) solution can be found as the following: find vector h₂ that is a solution of linear algebraic system of equations (A-1*Id)h₂=h₁. Here A is the matrix of your system, Id is the identity matrix. h₂=[0 1]^T is a such vector. Then f₂(t)=(h₂+th₁)e^t is the second linearly independent solution, and the general solution is C₁f₁(t) + C₂ f₂(t), i. e.

x(t)=(2C₁+2C₂t)e^t

y(t)=(2C₁+C₂+2C₂t)e^t