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Section 5: Systems of Ordinary Differential Equations

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

  • Systems of normal, first-order differential equations
  • Writing systems of first-order linear differential equations in matrix form
 

Exercises


Directions: You should attempt to solve the problems first and then watch the video to see the solution. 
  1. Convert the given system of ordinary differential equations into matrix form.\begin{align}\dfrac{dx}{dt}&=tx-t^2y+z,\quad x(0)=1\\[8pt]\dfrac{dy}{dt}&=z+\cos(t)y,\quad y(0)=-1\\[8pt]\dfrac{dz}{dt}&=3z+x+\tan(t),\quad z(0)=3\\[8pt]\end{align}

    \[\begin{pmatrix}
    \dfrac{dx}{dt}\\[8pt]
    \dfrac{dy}{dt}\\[8pt]
    \dfrac{dz}{dt}\\[8pt]
    \end{pmatrix}
    =
    \begin{pmatrix}
    t & -t^2 & 1\\[8pt]
    0 & \cos(t) & 1\\[8pt]
    1 & 0 & 3\\[8pt]
    \end{pmatrix}
    \begin{pmatrix}
    x\\[8pt]
    y\\[8pt]
    z\\[8pt]
    \end{pmatrix}
    +
    \begin{pmatrix}
    0\\[8pt]
    0\\[8pt]
    \tan(t)\\[8pt]
    \end{pmatrix}
    ,\quad 
    \begin{pmatrix}
    x(0)\\[8pt]
    y(0)\\[8pt]
    z(0)\\[8pt]
    \end{pmatrix}
    =
    \begin{pmatrix}
    1\\[8pt]
    -1\\[8pt]
    3\\[8pt]
    \end{pmatrix}
    \]
    or it can be rewritten in the form \(\bf{X}'=\bf{AX}+\bf{b}\) as
    \[
    \bf{X}'
    =
    \begin{pmatrix}
    t & -t^2 & 1\\[8pt]
    0 & \cos(t) & 1\\[8pt]
    1 & 0 & 3\\[8pt]
    \end{pmatrix}
    \bf{X}
    +
    \begin{pmatrix}
    0\\[8pt]
    0\\[8pt]
    \tan(t)\\[8pt]
    \end{pmatrix}
    ,\quad 
    \bf{X}(0)
    =
    \begin{pmatrix}
    1\\[8pt]
    -1\\[8pt]
    3\\[8pt]
    \end{pmatrix}
    \]


 

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 many independent variables are there in any system of ODEs?
  2. How many initial conditions should be coupled with a system of ODEs to form an IVP?
  3. Give an example of a linear, nonhomogeneous and non-autonomous system of ODEs. Then write it in matrix form.