The system of first order differential equations:

$\displaystyle \frac{dx}{dt} = 0x - 1y $

$\displaystyle \frac{dy}{dt} = 1x +2y $

where $\displaystyle x(0) = -4, y(0) = -5$ has the solution $\displaystyle x(t) = $ and $\displaystyle y(t) = $

When i solved for the eigenvalues, i got one double root where $\displaystyle \lambda = 1 $. I know for a fact that given a double eigenvalue of the same value, there can only be one associated eigenvector. Therefore i must find a generalized eigenvector for the other root.

The resulting eigenvectors are: $\displaystyle V_{1} = \left[\begin{array}{cc}-1\\1\end{array}\right], V_{2} = \left[\begin{array}{cc}1\\0\end{array}\right]$

I finally get to my final answer equal to: $\displaystyle \left[\begin{array}{cc}-5e^{t}+9te^{t}\\-4e^{t}-9te^{t}\end{array}\right] $ where i found the constants $\displaystyle C_{1} = -5 $ and $\displaystyle C_{2} = -9. $

I can't seem to get the correct answer! Anyone who contributes I thank them now.