# Thread: General solution of a system

1. ## General solution of a system

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

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

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

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