An Application to ODE's by Derrick Head (dhead@centenary.edu) Adapted from Chapter 7 of Elementary Differential Equations and Boundary Value Problems, Boyce and DiPrima, 8th edition restart:with(plots):with(LinearAlgebra): Warning, the name changecoords has been redefined Notes in Green are added for MAA Prep Workshop Participants and not part of the worksheet. This is a first draft of a notebook that I plan to use in my differential equations course next year. Most, if not all, students taking this course will have had a linear algebra, but they will need a quick review of matrix notation, eigenvalues, and eigenvectors.
<Text-field style="Heading 1" layout="Heading 1">Linear Algebra Review</Text-field> To be completed later --Should be skipped for now
<Text-field style="Heading 2" layout="Heading 2">Elementary Matrices</Text-field>
<Text-field style="Heading 2" layout="Heading 2">Eigenvalues</Text-field>
<Text-field style="Heading 2" layout="Heading 2">Eigenvectors</Text-field>
The next four sections present the material to the students. There are no Maple commands here.
<Text-field style="Heading 1" layout="Heading 1">Homogeneous Linear Systems with Constant Coefficients</Text-field> 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<Text-field style="Heading 1" layout="Heading 1">Equilibrium Solutions</Text-field> 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<Text-field style="Heading 1" layout="Heading 1">Phase Plane</Text-field> 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<Text-field style="Heading 1" layout="Heading 1">Solving Homogeneous Systems</Text-field> 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The next section is the first to utilize Maple.
<Text-field style="_cstyle14" layout="_pstyle28">Using Maple to compute eigenvalues and eigenvectors</Text-field> This is adapted from a notebook by Russell and Mike. Since a detailed discussed of eigenvalues and eigenvectors is left for a course in linear algebra, let's concentrate on how to use Maple to ease the computations. Let's start with a randomly generated 2 by 2 matrix called A. A := RandomMatrix(2,2,generator=rand(-10..10)); 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 We now use the Eigenvectors command in Maple to find the eigenvalues and their corresponding eigenvectors. (Remember that the LinearAlgebra package must be loaded for this command to work. Type ?Eigenvectors to learn more about this command. Eigenvectors(A); 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