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Math Help - General solution of a system

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by CHAYNES View Post
    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  <br />
|\lambda \mathbb{I} - A| = <br />
\left|\begin{array}{cc}<br />
\lambda - 1 & 1\\<br />
-2 & \lambda - 3<br />
\end{array}\right| = 0<br />
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

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