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Section 11: Repeated Eigenvalues


  • First, you should watch the concepts videos below explaining the topics in the section. 
  • Second, you should attempt to solve the exercises and then watch the videos explaining the exercises. 
  • Last, 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.


  • Solving a system of equations where the coefficient matrix has a repeated eigenvalue
  • Finding a generalized eigenvector


Directions: You should attempt to solve the problems first and then watch the video to see the solution.
  1. Find the general solution of the system and the fundamental matrix. \[{\bf x}'=\begin{pmatrix} 3&-4\\1&-1\end{pmatrix}{\bf x}\]

    The general solution is \[\mathbf{x}(t)=C_1e^{t}\begin{pmatrix}2\\1\\\end{pmatrix}+C_2e^{t}\left[t\begin{pmatrix}2\\1\end{pmatrix}+\begin{pmatrix}1\\0\end{pmatrix}\right]\] and the fundamental matrix is \[\begin{pmatrix}2e^{t}&(2t+1)e^{t}\\e^{t}&te^{t}\end{pmatrix}\]
    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 V77b


Self-Assessment Questions

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. How do you find a second solution if an eigenvalue is repeated twice?
  2. Does a generalized eigenvector always exist in the repeated case?