Mathematica Tutorial: Differential Equation Solving With ...

Wolfram Mathematica? Tutorial Collection

DIFFERENTIAL EQUATION SOLVING WITH DSOLVE

For use with Wolfram Mathematica? 7.0 and later.

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Contents

Introduction to Differential Equation Solving with DSolve . . . . . . . . . . . . . . . . . . . . . . 1 Classification of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Ordinary Differential Equations (ODEs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Overview of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 First-Order ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Linear Second-Order ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Nonlinear Second-Order ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Higher-Order ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Lie Symmetry Methods for Solving Nonlinear ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Partial Differential Equations (PDEs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Introduction to PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 First-Order PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Second-Order PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Differential-Algebraic Equations (DAEs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Introduction to DAEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Examples of DAEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Initial and Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Introduction to Initial and Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Linear IVPs and BVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Nonlinear IVPs and BVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 IVPs with Piecewise Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Working with DSolve~A User's Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Setting Up the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Verification of the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Plotting the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 The GenerateParameters Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Symbolic Parameters and Inexact Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Is the Problem Well-Posed? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Introduction to Differential Equation Solving with DSolve

The Mathematica function DSolve finds symbolic solutions to differential equations. (The Mathe-

matica function NDSolve, on the other hand, is a general numerical differential equation

solver.) DSolve can handle the following types of equations:

Ordinary Differential Equations (ODEs), in which there is a single independent variable t and one or more dependent variables xiHtL. DSolve is equipped with a wide variety of techniques for solving single ODEs as well as systems of ODEs.

Partial Differential Equations (PDEs), in which there are two or more independent variables and one dependent variable. Finding exact symbolic solutions of PDEs is a difficult problem, but DSolve can solve most first-order PDEs and a limited number of the second-order PDEs found in standard reference books.

Differential-Algebraic Equations (DAEs), in which some members of the system are differential equations and the others are purely algebraic, having no derivatives in them. As with PDEs, it is difficult to find exact solutions to DAEs, but DSolve can solve many examples of such systems that occur in applications.

DSolve@eqn,y@xD,xD

solve a differential equation for y@xD

DSolve@8eqn1,eqn2,... ................
................

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