General solution of a system

Quote:

Originally Posted by CHAYNES

Hey, iv been stuck on this question for a few hours now and was wondering if anyone could give me some help.

Determine the general solution of the system:
x'= x-y y'= 2x+3y

its to be in complex form and im to then find the solution satisfying x(0)=0, y(0)=2 in real form

iv managed to get as far as finding the eigenvalues which i got to be 2+i and 2-i but iv not been able to calculate the eigenvectors. Any help you can give me would be great.

thanks Caitlin

So $ |\lambda \mathbb{I} - A| = \left|\begin{array}{cc} \lambda - 1 & 1\\ -2 & \lambda - 3 \end{array}\right| = 0 $ gives $\lambda = 2 \pm i$ as you've said. To find the eigenvectors you want to find $\bar{v}$ such that

$\left(\lambda \mathbb{I} - A\right) \vec{v} = \vec{0}$ . So for the first eigenvalue $\lambda = 2 + i$ then

$ \left(\begin{array}{cc} 1 + i & 1\\ -2 & -1 + i\end{array}\right) \left(\begin{array}{c} v_1\\ v_2\end{array}\right) = \left(\begin{array}{cc} 0\\ 0\end{array}\right) $ which gives $ \left(\begin{array}{c} v_1\\ v_2\end{array}\right) = \left(\begin{array}{c} -1\\ 1+i\end{array}\right) $ . The second just gives the comples conjugate.