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Section 9: Complex Eigenvalues

Instructions

  • 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.

Concepts

  • Solving systems of linear, first-order differential equations with real and complex roots
  • Finding complex eigenvalues and the corresponding eigenvectors
 

Exercises


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. \[{\bf x}'=\begin{pmatrix} 1&-1\\5&-3\end{pmatrix}{\bf x}\]

    The general solution is \[\begin{align}\mathbf{x}(t)=&C_1e^{-t}\left[\begin{pmatrix}1\\2\\\end{pmatrix}\cos(t)+\begin{pmatrix}0\\1\end{pmatrix}\sin(t)\right]\\&+C_2e^{-t}\left[\begin{pmatrix}1\\2\end{pmatrix}\sin(t)+\begin{pmatrix}0\\-1\end{pmatrix}\cos(t)\right]\end{align}\] In a later section, we will discuss the fundamental matrix, and the fundamental matrix for this problem is\[\begin{pmatrix}e^{-t}\cos(t)&e^{-t}\sin(t)\\e^{-t}(2\cos(t)+\sin(t))&e^{-t}(2\sin(t)-\cos(t))\end{pmatrix}\]

    To see the full video page and find related videos, click the following link.
    MATH 308 WIR22A V77c


  2. Find the general solution of the system of equations \[\mathbf{x}'=\begin{pmatrix} 1 &-4 & -1\\ 3 & 2 & 3 \\ 1 & 1 & 2\\ \end{pmatrix} \mathbf{x}\]

    \[\begin{align}
    \mathbf{x} =
    &c_1e^{2t}\begin{pmatrix}
    1\\
    0\\
    -1
    \end{pmatrix}
    +c_2e^{2t}\left[
    \cos (3t)\begin{pmatrix}
    -5\\
    3\\
    2
    \end{pmatrix}
    -\sin(3t)
    \begin{pmatrix}
    3\\
    3\\
    0
    \end{pmatrix}
    \right]\\[8pt]
    &+c_3
    e^{2t}
    \left[\sin(3t)
    \begin{pmatrix}
    -5\\
    3\\
    2
    \end{pmatrix}
    +\cos(3t)
    \begin{pmatrix}
    3\\
    3\\
    0
    \end{pmatrix}

    \right]
    \end{align}\]
     


 

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. Are there any relations between eigenvectors of complex conjugate eigenvalues?
  2. Write a pair of vector-valued solutions for a system with an eigenvalue of \(\lambda+i\mu\) and eigenvector of \(\vec{a}+i\vec{b}\).
  3. When do the solutions converge to zero as \(t\rightarrow{\infty}\)? Can you express the condition in terms of the eigenvalues?