Toronto Math Forum
MAT2442013S => MAT244 MathTests => MidTerm => Topic started by: Victor Ivrii on March 06, 2013, 09:10:42 PM

Solve the system of ordinary differential equations
\begin{equation*}
\left\{
\begin{aligned}
&x'_t=5x3y,\\
&y'_t=6x4y.
\end{aligned}
\right.\end{equation*}

Heres my solution

solution

We begin by finding eigenvalues for the systems matrix. We solve $(5\lambda)(4\lambda)+18=\lambda^2\lambda2=0$. 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 $Y_G=c_1e^{2t}\xi_1+c_2e^{t}\xi_2$

This is my solution