Solving a System of First Order Differential Equations

The system of first order differential equations:

where has the solution and

When i solved for the eigenvalues, i got one double root where . 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:

I finally get to my final answer equal to: where i found the constants and

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