Using Eigenvalues and Eigenvectors, Find the general solution of the following coupled differential equations. x'=x+y and y'=-x+3y. Consider the matrix $A=\begin{bmatrix} 1 & 1 \\ -1 & 3 \end{bmatrix}$
I found the eigenvalue $\lambda=2$ with multiplicity $2$. However, the general solution I found degrees with the answer provided by the text.
I found the solution
$$\begin{bmatrix} x \\ y \end{bmatrix} = A\begin{bmatrix} 1 \\ 1 \end{bmatrix} e^{2t} + B\begin{bmatrix} t \\ t-1 \end{bmatrix}e^{2t}$$ 
whereas the book finds
$$\begin{bmatrix} x \\ y \end{bmatrix} = A\begin{bmatrix} 1 \\ 1 \end{bmatrix}e^{2t} + B\begin{bmatrix} t-1 \\ t \end{bmatrix} e^{2t}$$
What have I done wrong?
 A: Given:
$$\tag 1 x'(t) = A x(t) = \begin{bmatrix} 1 & 1 \\ -1 & 3 \end{bmatrix}x(t)$$
Find the solution to the system in $(1)$.
For the matrix A, we get the eigenvalues / eigenvectors:


*

*$\lambda_1 = 2, v_1 = (1,1)$

*$\lambda_2 = 2, v_2 = (-1,0)$ (A generalized eigenvector)


From this, we can write the solution:
$$X(t) = \begin{bmatrix}x(t)\\ y(t)\end{bmatrix} = e^{2t}\left(c_1 \begin{bmatrix}1\\1\end{bmatrix} + c_2\left(\begin{bmatrix}-1\\0\end{bmatrix} + t\begin{bmatrix}1\\1\end{bmatrix}\right)\right) = e^{2t}\begin{bmatrix}c_1 + c_2(t-1)\\c_1 + c_2 t\end{bmatrix}$$
This result matches your book.
Do you see where you went astray? Be careful when writing that second solution because we have a repeated eigenvalue.
Update
We need to find a generalized eigenvector, so we have $[A - 2I]v_2 = v_1$, and when we do RREF, we end up with:
$$\begin{bmatrix}1 & -1 \\0 & 0 \end{bmatrix}v_2 = \begin{bmatrix}-1\\0\end{bmatrix}$$
From this, we have:
$$a - b = -1 \rightarrow a = -1 + b \rightarrow ~\mbox{let}~ b = 0,~\mbox{so}~ a = -1$$
This results in the eigenvector:
$$v_2 = (-1,0)$$
Do you see your issue?
Update 2
To verify the result:


*

*Calculate $x'(t)$ from your result and see if it equals $x(t) + y(t)$

*Calculate $y'(t)$ from your result and see if it equals $-x(t) + 3y(t)$


If both of those match, it is a correct result.
A: Check out this website:
http://www.egwald.ca/linearalgebra/lineardifferentialequations.php
Look at example 5. I think (and I hope) it resembles your question really well!
