Differential Equations with Maple

Preface. 1 Introduction.
1.1 Guiding Philosophy.
1.2 Student's Guide.
1.3 Instructor's Guide.
1.3.1 Maple.
1.3.2 ODE Chapters.
1.3.3 Computer Problem Sets.
1.4 A Word about Software Versions.

2 Getting Started with Maple.
2.1 Platforms and Versions.
2.2 Instrallation.
2.3 Starting Maple.
2.4 Maple Input.
2.5 Online Help.
2.6 Ending a Session.

3 Doing Mathematics with Maple.
3.1 Arithmetic.
3.2 Symbolic Computation.
3.3 Assignments.
3.4 Working with Output.
3.5 Recovering from Problems.
3.5.1 Errors in Input.
3.5.2 Aborting Calculations.
3.6 Suppressing Output.
3.7 The Restart Command.
3.8 Equations.
3.9 Solving Equations.
3.10 Functions and Expressions.
3.10.1 Built-in Functions.
3.10.2 User-defined Functions.
3.10.3 Expressions.
3.11 Substitution.
3.12 Sequences, Sets, and Lists.
3.13 Packages.
3.14 Graphics.
3.14.1 Plotting Functions and Expressions.
3.14.2 Plotting Multiple Curves.
3.14.3 Plotting Points.
3.14.4 Adding Text to a Plot.
3.14.5 Parametric Plots.
3.14.6 Implicit Plots.
3.14.7 Contour Plots.
3.15 Calculus.
3.16 More on Sequences, Lists, and Sets.
3.17 Procedures.
3.18 Some Tips and Reminders.

4 Using Maple Documents.
4.1 The Maple Window.
4.2 Organization of a Document.
4.3 Document Blocks.
4.4 Graphics.
4.5 Preparing Homework Solutions.

Problem Set A: Practice with Maple.

5 Solutions of Differential Equations.
5.1 Finding Symbolic Solutions.
5.2 Existence and Uniqueness.
5.3 Stability of Differential Equations.
5.4 Different Types of Symbolic Solutions.

6 A Qualitative Approach to Differential Equations.
6.1 Direction Field for a First Order Linear Equation.
6.2 Direction Filed for a Non-Linear Equation.
6.3 Autonomous Equations.
6.3.1 Examples of Autonomous Equations.

Problem Set B: First Order Equations.

7 Numerical Methods.
7.1 Numerical Solutions Using Maple.
7.2 Some Numerical Methods.
7.2.1 The Euler Method.
7.2.2 The Improved Euler Method.
7.2.3 The Rung-Kutta Method.
7.2.4 Inside dsolve(...,numeric).
7.2.5 Round-off Error.
7.3 Controlling the Error in dsolve(...,numeric).
7.4 Reliability of Numerical Methods.

8 Features of Maple.
8.1 Names and Values.
8.2 Clearing Values.
8.3 Vectors and Matrices.
8.3.1 Solving Linear Systems.
8.3.2 Calculating Eigenvalues and Eigenvectors.
8.4 Plots for ODEs.
8.4.1 Commands for Plotting Direction Fields.
8.4.2 Plotting Families of Numerical Solutions of ODEs.
8.4.3 More about D Eplot.
8.5 Stopping Conditions.
8.6 Numerical Solutions of Higher Order Differential Equations.
8.7 Troubleshooting.
8.7.1 The Common Mistakes.
8.7.2 Error and Warning Messages.

Problem Set C: Numerical Solutions.

9 Solving and Analyzing Second Order Linear Equations.
9.1 Second Order Equations with Maple.
9.2 Comparison Methods.
9.2.1 The Interlacing of Zeros.
9.2.2 Proof of the Sturm Comparison Theorem.
9.3 A Geometric Method.
9.3.1 The Constant Coefficient Case.
9.3.2 The Variable Coefficient Case.
9.3.3 Airy's Equation.
9.3.4 Bessel's Equation.
9.3.5 Other Equations.

Problem Set D: Second Order Equations.

10 Series Solutions.
10.1 Series Solutions.
10.2 Singular Points.

11 Laplace Transforms.
11.1 Differential Equations and Laplace Transforms.
11.2 Discontinuous Functions.
11.3 Differential Equations with Discontinuous Forcing.

Problem Set E: Series Solutions and Laplace Transforms.

12 Higher Order Equations and Systems of First Order Equations.
12.1 Higher Order Linear Equations.
12.2 Systems of First Order Equations.
12.2.1 Linear First Order Systems.
12.2.2. Using Maple to Find Eigenpairs.
12.3 Phase Portraits.
12.3.1 Plotting a Single Trajectory.
12.3.2 Plotting Several Trajectories.
12.3.3 Numerical Solutions of First Order Systems.

13 Qualitative Theory for Systems of Differential Equations.

Problem Set F: Systems of Differential Equations.

Glossary.

Sample Solutions.

Index.