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In some past exam papers for the Maths course that I attend,I found this example and I would really appreciate if someone looked at my solution. It goes like this: Find general solution to $$ y_1' = 2 y_1 - y_2 + (x+1) e^{3x}, \\ y_2' = y_1 + 4 y_2 + 2 x e^{3x}. $$

First of all, I set up the fundamental matrix and found its eigenvalues. I got that the matrix has repeated roots namely $r=3$. As I could not find two linearly independent eigenvectors (I had just one), I used this method I had found online to obtain another eigenvector given just one (not convinced that the method is $100 \%$ legitimate though). My eigenvectors are thus $v_1 = (1,-1)$ and $v_2 = (0,-1)$. Therefore my homogeneous solution matrix is $$ y = c_1 e^{3x} v_1 + c_2 (x e^{3x} v_1 + e^{3x} v_2). $$

Then, I searched for the inverse of this matrix and got $$ \left( \array{(1+x) e^{-3x} &x e^{-3x} \\ -e^{-3x} &-e^{-3x}} \right) $$ Multiplying this matrix with the inhomogeneous matrix $( (x+1) e^{3x} , 2 x e^{3x})$ I obtained $(3 x^2x+1, -3x-1)$, which upon integrating, I got $(x^3 +x^2+x, -\frac{3}{2} x-x)$. Adding this matrix and the homogeneous matrix, I obtained the general solution.

Is this correct?

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What you call the 'other eigenvector' is a generalized eigenvector. I don't feel like reading your answer, but for some examples on this kind of problem see this, this and this. –  Git Gud Jun 2 at 10:38
    
Please use MathJax to typeset formulas. As it's currently written, your question is far from being easy to understand. –  TZakrevskiy Jun 2 at 10:40

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