Topics in Differential Equations

Introductory Material
First Order Differential Equations
Applications of First Order Differential Equations
Numerical Methods for First Order Differential Equations
Linear Second Order Equations
Applications of Linear Second Order Differential Equations
Qualitative Solutions to DEs
Higher Order Linear Differential Equations
Laplace Transforms
Series Solutions of Differential Equations
Matrix Methods for Linear Systems
Applications of Systems

Notes on these worksheets

Introductory Material (tutorials) DE quick tour for Maple A first day demo of using dsolve, odeplot, DEplot1, etc. Using Maple for DE Example of using Maple to analyze two nonlinear DEs Intro to DEs on Maple Several DEs solved and checked w/ Maple. Maple V R.3 Maple V R.4 Using dsolve to find exact solutions of many differential equations or systems of equations, with or without initial conditions.Intro to DEs on Maple. Maple V R3: Text Version (Worksheet) Maple V R4: Text Version (Worksheet)
First Order Differential Equations (tutorials) A look at the separable DE dy x^2 -- = ------- dx 1-y^2 Solutions are found by finding the integral curves. Maple Code to determine whether or not a DE is separable. Solving y'=x^2+y^2 Three different methods of solving this: exactly, graphically, numerically. An Solving y'=sin(x) + sin(y) Example of a DE Maple cannot solve. DEplot is used to figure out what is happening. Using Phase Diagrams to look at first Order ODEs of the form dN/dt = F(N). Maple V R.3 Maple V R.4 Show how Maple can be used to investigate a first order linear differential equation, using both graphical and analytical methods. Maple V R3: Text Version (Worksheet) Maple V R4: Text Version (Worksheet)
Applications of First Order Differential Equations (tutorials) Flourescent Keychain Brightness of a flourescent light is modeled by the DE: y'=IC (1-y)-K y
Numerical Methods for First Order Differential Equations (tutorials) Euler's, Improved Euler, and Runge-Kutta Maple Code for the methods. Results are compared. Euler method from "Share" library is compared for varying step sizes. Maple V R.3 Maple V R.4 Euler's method Demo Maple commands to graphically demonstrate how step size affects accuracy of solution. (Denise Kirschner) Improved Euler's method Demo Maple commands to graphically compare Euler's method with the Improved Euler's method. (Denise Kirschner) Improved Euler method from "Share" library is compared for varying step sizes. Maple V R.3 Maple V R.4 Runge-Kutta method from "Share" library is compared for varying step sizes. Maple V R.3 Maple V R.4 Runge-Kutta method with hardware floating point is compared for varying step sizes. This is by the far the fastest of the numerical methods presented here. Maple V R.3 Maple V R.4 Higher Order Taylor's Series Methods Euler is extended using higher order Taylor approximations and Maple. Midterm Project: Solve a DE via the direction field w/ some looks at the error. Text version Maple R3 version
Linear Second Order Equations (tutorials) Homogeneous ODE's Several are solved and checked via an automated Maple procedure. Maple V R3 Maple V R4 Non-homogeneous ODEs via undetermined coeffiecents Maple's "subs" and "solve" commands used to find the "particular" solution of 2nd order non-homogeneous ODEs. Non-homogeneous ODEs via undetermined coefficents Maple code to demonstrate the complete "by hand" solution process. (Denise Kirschner) Non-homogeneous ODEs via undetermined coefficients A complete, no thought required, solution of these ODEs in Maple. A first problem Maple V R.3 Maple V R.4 A second problem Maple V R.3 Maple V R.4 Method of Undetermined Coefficients (Postscript, 4 pgs) Method is explained and illustrated using Maple. (by Maury Rahe) Variation of Parameters Maple code to demonstrate the "by hand" solution process (Denise Kirschner). Variation of Parameters method implemented using a single Maple procedure. Complete technique of Variation of Parameters is explained, illustrated, and checked in Maple. Maple V R.3 Maple V R.4 Variation of Parameters by Maury Rahe. (Postscript, 7 pgs) Method is completely explained and demonstrated in Maple Wronskian Maple is used to find and explore the meaning of the Wronskian Solutions to quiz problems. One uses the Wronskian. The other dsolve with type=numeric
Applications of Linear Second Order Differential Equations (tutorials) Springs Demonstrate 2nd order ODE using springs. Mechanical and Electrical Vibrations: A spring problem Maple V R3 Maple V R4 Forced Vibrations: A circuit problem is solved Maple V R3 Maple V R4
Qualitative Solutions to DEs (tutorials) Matlab demo by John Polking Phase Planes I Maple code to demonstrate the use of Phase portraits. (Denise Kirschner) Phase Planes II More Maple code to demonstrate the more use of Phase portraits. (Denise Kirschner) Phase Planes: Linear Systems. Elliptical and Spiral trajectory. Maple V R3 Maple V R4 Phase Planes: Nearly Linear Systems. Maple V R3 Maple V R4 Analyzing a "Competing Species" problem using Phase Planes. Maple V R3 Maple V R4 Analyzing a "Predator-Prey" problem using Phase Planes. Maple V R3 Maple V R4 Analyzing a 2nd "Predator-Prey" problem using Phase Planes. Maple V R3 Maple V R4 Systematically investigate the behavior of a 2-dim nonlinear autonomous system. Applied to a competing species DE Maple V R3: (Text Version) (Worksheet) Maple V R4: (Text Version) (Worksheet) Illustrate the construction of direction fields and numerical approximations to solution curves for first order DEs. Maple V R3: (Text Version) (Worksheet) Maple V R4: (Text Version) (Worksheet)
Higher Order Linear Differential Equations (tutorials) Complete technique of Variation of Parameters is explained, illustrated, and checked in Maple. Third Order ODE. Maple V R.3 Maple V R.4 Forth Order ODE. Maple V R.3 Maple V R.4 Maple project to solve a 4 spring/3 mass system. Text version Maple R3 version
Laplace Transforms (tutorials) Laplace transforms Concept is introduced along with a couple Maple functions. An example of a damped oscillator with a sawtooth forcing function. Solved via Laplace transforms. Laplace Transforms by Maury Rahe. (Postscript, 9pgs.) Concepts and techniques are discussed. Maple is used to illustrate the techniques. Definition and solution IVP using Laplace Transforms An extensive worksheet on the use of Laplace Transforms for solving IVP - expecially with discontinuous forcing functions. Maple V R3 Maple V R4 Use of Heaviside functions is explored to study jumps in otherwise continuous functions. These types of functions are used in rhs of IVP. Maple V R3 Maple V R4 Dirac distribution is introduced and used in IVP. Several "interesting" graphs are produced. Maple V R3 Maple V R4 Convolution Integrals introduced and used as rhs of DEs. Several "interesting" graphs are produced. Maple V R3 Maple V R4
Series Solutions of Differential Equations (tutorials) Worksheet on series solutions. Series solutions to DEs are explored with Maple. Legendre polynomials are introduced. Series soln of DEs by Maury Rahe. (Postscript, 5 pgs.) Series techniques are explored and explained using Maple to illustrate and graph results. A tutorial/demo on infinite series and solving ODE's using infinite series, text file (by Don allen) Review of Power Series Maple V R3 Maple V R4 Series Solutions near an Ordinary Point Problem 1a Maple V R3 Maple V R4 Problem 1b Graphs of solutions of different orders. Maple V R3 Maple V R4 Problem 2 dsolve(..., series) is used. Radius of convergence is discussed and analyzed. Maple V R3 Maple V R4 A Series Solution near a Regular Singular Point Part 1 Maple V R3 Maple V R4 Part 2 Maple V R3 Maple V R4 A Series Solution of Euler's Equation Maple V R3 Maple V R4 A Fourier series example: Calculate the coefficients in Fourier series and plot partial sums. Maple V R3: Text Version Worksheet Two examples of Fourier series including plots of partial sums and the limiting function. Maple V R3: Text Version Worksheet Series solution to heat conduction PDE. Maple V R3: Text Version Worksheet
Matrix Methods for Linear Systems (tutorials) Linear Algebra functions are introduced and used to solve DEs. Intro. to matrices with Maple and applications to DEs by Don Allen. Elementary operations on matrices are introduced. Eigenvalues and explained and explored by Maury Rahe (Postscript, 4 pgs.) Complex Eigenvalues are explored by Maury Rahe (Postscript, 2 pgs.) Eigenvalues: Some examples of finding the eigenvales and eigenvectors of a matrix. Maple V R3: Text Version (Worksheet ) Maple V R4: Text Version (Worksheet ) Decomposing a matrix A into A = T D T^{-1} Review of matrices, linear independence, eigenvalues, and eigenvectors Requires use of "vec_calc" package which is available on TAMU - math and calclab machines and is available for downloading. Maple V R3 Maple V R4 Basic theory of 1st order systems and Homogeneous linear systems with constant coefficients. Requires use of "vec_calc" package which is available on TAMU - math and calclab machines and is available for downloading. Maple V R3 Maple V R4 Complex Eigenvalues of systems are explored Requires use of "vec_calc" package which is available on TAMU - math and calclab machines and is available for downloading. Maple V R3 Maple V R4 Repeated Complex Eigenvalues of systems are explored Requires use of "vec_calc" package which is available on TAMU - math and calclab machines and is available for downloading. Maple V R3 Maple V R4 Fundamental Matrices for 1st order systems Requires use of "vec_calc" package which is available on TAMU - math and calclab machines and is available for downloading. Maple V R3 Maple V R4 Nonhomogeneous Linear Systems Requires use of "vec_calc" package which is available on TAMU - math and calclab machines and is available for downloading. Maple V R3 Maple V R4
Applications of Systems (tutorials) Panthers and Deer problem. (text) Maple Demo (text). Maple is used to construct an animation of an orbiting comet. Graphic of comet orbit. Postscript An example of coupled springs and a pendulum. (text) Student solutions to applied problems. (text) Predator-Prey Problem (by John Armstrong) Predator-Prey Problem ( by Trisha L. Hames and ViVi Darmansyah ) Field Goals ( by Robert Duane Edmonds and Thavy Chumni Un ) Systematically investigate the behavior of a 2-dim nonlinear autonomous system. Applied to a competing species DE Maple V R3: (Text Version) (Worksheet) Maple V R4: (Text Version) (Worksheet)

Due to a recent change of textbook most of the chapter, page, and problem numbers refer to the old text by Boyce and DiPrima rather than the new text by Nagle and Saff. This will hopefully be rectified in the future.

Using text files:

Many of the files listed here are in Maple Text format. This means that you can view the file and if you choose can either save the file from your Web browser and load them into Maple using

the "File" menu's "Import Text" option on the X Windows (Unix) version, or
the "File" menu's "Open" and setting the file type to "Maple Text" at the bottom of the "Open" window on the Windows (Dos) version.

Share Library

Some of these worksheets require Maple's Share Library. For further information see Share Library Info.

Warning! Most of these worksheets are not intended to be "Howto" sheets in differential equations. Rather, they provide a framework to solve DE's using Maple once the underlying mathematical principles have been learned.

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Please send comments, questions, or suggestions regarding this page to Arlen Strader at "strader@calclab.math.tamu.edu" .
Last updated Tue Jun 18 1996

Topics in Differential Equations

Contents

Notes on these worksheets

Introductory Material (tutorials)

First Order Differential Equations (tutorials)

Applications of First Order Differential Equations (tutorials)

Numerical Methods for First Order Differential Equations (tutorials)

Linear Second Order Equations (tutorials)

Applications of Linear Second Order Differential Equations (tutorials)

Qualitative Solutions to DEs (tutorials)

Higher Order Linear Differential Equations (tutorials)

Laplace Transforms (tutorials)

Series Solutions of Differential Equations (tutorials)

Matrix Methods for Linear Systems (tutorials)

Applications of Systems (tutorials)

Notes.