Find the general solution?

Question

Find the general solution of x'=(1, 1, 2, 1, 2, 1, 2, 1, 1)x. (this is a 3x3 matrix, 1, 1, 2 on the left, 1, 2, 1 in the middle, 2, 1, 1, on the right, please show your work step by step.)

Answer 1

Since you didn't give initial values, you are probably supposed to do this problem with eigenvalues.

 

Let A be your 3x3 matrix. From the characteristic equation det(A-λI)=0 you find the three eigenvalues of A to be λ = -1, 1, and 4. When you substitute these into the equation A-λI=0, you get, up to overall constants the three corresponding eigenvectors x₁=[-1,0,1], x₂=[1,-2,1], and x₃=[1,1,1].

 

By the theorem from your book, the general solution is therefore

 

x=c₁x₁e^-t + c₂x₂e^t + c₃x₃e^4t.