Toronto Math Forum
MAT2442018S => MAT244Tests => Quiz6 => Topic started by: Victor Ivrii on March 16, 2018, 08:17:05 PM

a. Express the general solution of the given system of equations in terms of realvalued functions.
b. Also draw a direction field, sketch a few of the trajectories, and describe the behavior of the solutions as $t\to \infty$.
$$\mathbf{x}' =\begin{pmatrix}
1 &1\\
5 &3
\end{pmatrix}\mathbf{x}$$

Setting $x = \xi t^r$ results in the algebraic equations
$$\begin{pmatrix}1r & 1\\\ 5 & 3r\end{pmatrix} \begin{pmatrix}\xi_1 \\ \xi_2 \end{pmatrix} = \begin{pmatrix}0 \\ 0 \end{pmatrix}$$
The characteristic equation is $r^2+2r+2=0$, with root $r = 1+i, r = 1i$. Substituting reduces $r = 1i$ the system of equations to $(2+i)\xi_{1}\xi_{2}=0$,The eigenvectors are $\xi^{(1)} = (1,2+i)^T$, $\xi^{(2)} = (1,2i)^T$.Hence one of the complexvalued solutions is given by $$x^{(1)} = \begin{pmatrix}1 \\\ 2+i \end{pmatrix}e^{(1+i)t} =\begin{pmatrix}1 \\\ 2+i \end{pmatrix}e^{t}(\cos t  i \sin t) $$
Therefore the general solution is $$x = c_1 e^{t}\begin{pmatrix}\cos t \\\ 2\cos t + \sin t \end{pmatrix} + c_2 e^{t}\begin{pmatrix}\sin t \\\ \cos t + 2\sin t \end{pmatrix}$$.
Attached is the graph.