# General solution of a system

• Feb 8th 2010, 10:34 AM
CHAYNES
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
• Feb 8th 2010, 02:24 PM
Jester
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.