Differential Equations with Maple

Preface  iii 1  Introduction 1 1.1  Guiding Philosophy  1 1.2  Student s Guide  3 1.3  Instructor s Guide  4 1.3.1  Maple  4 1.3.2  ODE Chapters  5 1.3.3  Computer Problem Sets  5 1.4  A Word about Software Versions  6 2  Getting Started with Maple  7 2.1  Platforms and Versions  7 2.2  Instrallation  7 2.3  Starting Maple  8 2.4 Maple Input  8 2.5  Online Help  9 2.6  Ending a Session  9 3  Doing Mathematics with Maple  11 3.1  Arithmetic  11 3.2  Symbolic Computation  12 3.3  Assignments  13 3.4  Working with Output  14 3.5  Recovering from Problems  15 3.5.1  Errors in Input  15 3.5.2  Aborting Calculations  15 3.6  Suppressing Output  16 3.7  The Restart Command  16 3.8  Equations  16 3.9  Solving Equations  17 3.10  Functions and Expressions  19 3.10.1  Built in Functions  19 3.10.2  User defined Functions  20 3.10.3  Expressions  21 3.11  Substitution  22 3.12  Sequences, Sets, and Lists  23 3.13  Packages  24 3.14  Graphics  25 3.14.1 Plotting Functions and Expressions  25 3.14.2  Plotting Multiple Curves  26 3.14.3  Plotting Points  27 3.14.4  Adding Text to a Plot  27 3.14.5  Parametric Plots  29 3.14.6  Implicit Plots  30 3.14.7  Contour Plots  30 3.15  Calculus  31 3.16  More on Sequences, Lists, and Sets  34 4.17  Procedures  37 3.18  Some Tips and Reminders  39 4  Using Maple Documents  41 4.1  The Maple Window  41 4.2  Organization of a Document  42 4.3  Document Blocks  42 4.4  Graphics  43 4.5  Preparing Homework Solutions  44 Problem Set A: Practice with Maple  47 5  Solutions of Differential Equations  51 5.1  Finding Symbolic Solutions  51 5.2  Existence and Uniqueness  53 5.3  Stability of Differential Equations  55 5.4  Different Types of Symbolic Solutions  59 6  A Qualitative Approach to Differential Equations  65 6.1  Direction Field for a First Order Linear Equation  65 6.2  Direction Filed for a Non Linear Equation  67 6.3  Autonomous Equations  68 6.3.1  Examples of Autonomous Equations  70 Problem Set B: First Order Equations  73 7  Numerical Methods  83 7.1  Numerical Solutions Using Maple  84 7.2  Some Numerical Methods  86 7.2.1  The Euler Method  86 7.2.2  The Improved Euler Method  89 7.2.3  The Rung Kutta Method  90 7.2.4  Inside dsolve(...,numeric)  91 7.2.5  Round off Error  91 7.3 Controlling the Error in dsolve(...,numeric)  92 7.4  Reliability of Numerical Methods  92 8  Features of Maple  97 8.1  Names and Values  97 8.2  Clearing Values  98 8.3  Vectors and Matrices  98 8.3.1  Solving Linear Systems  100 8.3.2  Calculating Eigenvalues and Eigenvectors  101 8.4  Plots for ODEs  102 8.4.1  Commands for Plotting Direction Fields  102 8.4.2  Plotting Families of Numerical Solutions of ODEs  102 8.4.3  More about D Eplot  103 8.5  Stopping Conditions  105 8.6  Numerical Solutions of Higher Order Differential Equations  106 8.7  Troubleshooting  107 8.7.1  The Common Mistakes  108 8.7.2  Error and Warning Messages  108 Problem Set C: Numerical Solutions  111 9  Solving and Analyzing Second Order Linear Equations  119 9.1  Second Order Equations with Maple  121 9.2  Comparison Methods  124 9.2.1  The Interlacing of Zeros  126 9.2.2  Proof of the Sturm Comparison Theorem  127 9.3  A Geometric Method  127 9.3.1  The Constant Coefficient Case  128 9.3.2  The Variable Coefficient Case  130 9.3.3  Airy s Equation  130 9.3.4  Bessel s Equation  131 9.3.5  Other Equations  132 Problem Set D: Second Order Equations  135 10  Series Solutions  149 10.1  Series Solutions  150 10.2  Singular Points  152 11  Laplace Transforms  155 11.1  Differential Equations and Laplace Transforms  157 11.2  Discontinuous Functions  160 11.3  Differential Equations with Discontinuous Forcing  162   Problem Set E: Series Solutions and Laplace Transforms  165 12  Higher Order Equations and Systems of First Order Equations  177 12.1  Higher Order Linear Equations  178 12.2  Systems of First Order Equations  179 12.2.1  Linear First Order Systems  179 12.2.2. Using Maple to Find Eigenpairs  182 12.3  Phase Portraits  186 12.3.1  Plotting a Single Trajectory  187 12.3.2  Plotting Several Trajectories  187 12.3.3  Numerical Solutions of First Order Systems  189 13  Qualitative Theory for Systems of Differential Equations  201 Problem Set F: Systems of Differential Equations  215 Glossary  215 Sample Solutions  233 Index  243