Scroll back to the top

Virtual Math Learning Center Logo

Section 7.5 – Homogeneous Linear Equations with Constant Coefficients

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. 
  1. Find the general solution of the system and the fundamental matrix. Classify the type of the critical point (0,0), and determine whether it is stable or unstable. Sketch the phase portrait.\[{\bf x}'=\begin{pmatrix} 3&-2\\2&-2\end{pmatrix}{\bf x}\]

    The general solution is \[\mathbf{x}(t)=C_1e^{-t}\begin{pmatrix}1\\2\\\end{pmatrix}+C_2e^{2t}\begin{pmatrix}2\\1\end{pmatrix}\] and the fundamental matrix is \[\begin{pmatrix}e^{-t}&2e^{2t}\\2e^{-t}&e^{2t}\end{pmatrix}\]The critical point \((0,0)\) is a saddle point and unstable. Refer to the video for a sketch of the phase portrait. 

    To see the full video page and find related videos, click the following link.
    MATH 308 WIR22A V77a