We begin by finding eigenvalues for the systems matrix. We solve $(5-\lambda)(-4-\lambda)+18=\lambda^2-\lambda-2=0$. This yields $\lambda_1=2, \lambda_2=-1$. We now search for eigenvectors.
For $\lambda_1=2$, the eigenvector is $\xi_1=(1,1)$
For $\lambda_2=-1$, the eigenvector is $\xi_2=(1,2)$.
General solution for the system is $Y_G=c_1e^{2t}\xi_1+c_2e^{-t}\xi_2$
