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Section 8: Homogeneous Linear Systems with Constant Coefficients

Instructions

  • You should start with the example video below. 
  • Next, you should attempt to answer the self-assessment questions to determine how well you learned the material.
  • When you have finished the material below, you can start on the next section or return to the main differential equations page.

Concepts

  • Solving a homogeneous first-degree, linear system of differential quations with constant coefficients
 

There is no conceptual video for this topic.

 

Exercises


Directions: Since there is no conceptual video above, you should watch the video first to learn how to solve the problem.
  1. Find the general solution of the system\[{\bf x}'=\begin{pmatrix} 3&-2\\2&-2\end{pmatrix}{\bf x}\]

    The general solution is \[\mathbf{x}(t)=C_1e^{-t}\begin{pmatrix}1\\2\\\end{pmatrix}+C_2e^{2t}\begin{pmatrix}2\\1\end{pmatrix}\] In a later section, we will discuss the fundamental matrix, and the fundamental matrix for this problem is \[\begin{pmatrix}e^{-t}&2e^{2t}\\2e^{-t}&e^{2t}\end{pmatrix}\]
    Note: you can skip the last part of the problem in the video that says "Classify the type of the critical point (0,0), and determine whether it is stable or unstable. Sketch the phase portrait." You will not be required to learn that material.


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


 
Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
  1. Explain how the eigenvalue problem can help with solving a linear system of ODEs with constant coefficients?
  2. What is the asymptotic behavior of the solutions as \(t\rightarrow\infty\) when the eigenvalues are negative?