Differential equation system with complex eigenvalues I need to solve this equation 
$$ x'=x+y,     y'=-2x+3y$$
I get the matrix rigor and the eigenvectors complex $2-i$ and $2+i$.  When I try to apply the eigenvectors associated the solution for $x$ I simply can't find ( I do find the $y$ though).
The solution on the book for $x=e^{2t}(a\cos(t)+b\sin(t))$
Any help would be appreciated.

Let's rewrite your equations as follows,assuming that $Dx=x',~Dy=y'$:
$$ \left\{
        \begin{array}{ll}
            (D-1)x-y=0 \\
            2x+(D-3)y=0
        \end{array}
    \right.
$$
By solving this homogenous system of equations, we have:
$$ \left\{
        \begin{array}{ll}
            (D-1)(D-3)x+2x=0 \\
            (D-1)(D-3)y+2y=0
        \end{array}
    \right.
$$ which is equivalent to the following system:
$$ \left\{
        \begin{array}{ll}
            x''(t)-4x'(t)+5x(t)=0 \\
            y''(t)-4y'(t)+5y(t)=0
        \end{array}
    \right.
$$ Now, I think you can do the rest.
