Toronto Math Forum
MAT244-2013S => MAT244 Math--Tests => MidTerm => Topic started by: Victor Ivrii on March 06, 2013, 09:10:42 PM
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Solve the system of ordinary differential equations
\begin{equation*}
\left\{
\begin{aligned}
&x'_t=5x-3y,\\
&y'_t=6x-4y.
\end{aligned}
\right.\end{equation*}
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Heres my solution
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solution
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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 $Y_G=c_1e^{2t}\xi_1+c_2e^{-t}\xi_2$
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This is my solution