Differential Equations with Maple V®-

1st Edition - September 8, 1994
Authors: Martha L Abell, James P. Braselton
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 6 6 5 7 - 2

Differential Equations with Maple V provides an introduction and discussion of topics typically covered in an undergraduate course in ordinary differential equations as well as… Read more

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Differential Equations with Maple V provides an introduction and discussion of topics typically covered in an undergraduate course in ordinary differential equations as well as some supplementary topics such as Laplace transforms, Fourier series, and partial differential equations. It also illustrates how Maple V is used to enhance the study of differential equations not only by eliminating the computational difficulties, but also by overcoming the visual limitations associated with the solutions of differential equations. The book contains chapters that present differential equations and illustrate how Maple V can be used to solve some typical problems. The text covers topics on differential equations such as first-order ordinary differential equations, higher order differential equations, power series solutions of ordinary differential equations, the Laplace Transform, systems of ordinary differential equations, and Fourier Series and applications to partial differential equations. Applications of these topics are also provided. Engineers, computer scientists, physical scientists, mathematicians, business professionals, and students will find the book useful.

Preface

1 Introduction to Differential Equations

1.1 Ρurpose

1.2 Definitions and Concepts

1.3 Solutions of Differential Equations

1.4 Initial-and Boundary-Value Problems

1.5 Direction Fields

2 First-Order Ordinary Differential Equations

2.1 Separation of Variables

2.2 Homogeneous Equations

2.3 Exact Equations

Solving the Exact Differential Equation M(x, y)dx + N(x, y) dy = 0

2.4 Linear Equations

2.5 Some Special Differential Equations

Bernoulli Equations

Clairaut Equations

Lagrange Equations

Ricatti Equations

2.6 Theory of First-Order Equations

2.7 Numerical Approximation of First-Order Equations

Built-In Methods

Euler's Method

Improved Euler's Method

The Runge-Kutta Method

3 Applications of First-Order Ordinary Differential Equations

3.1 Orthogonal Trajectories

3.2 Population Growth and Decay

The Malthus Model

Solution of the Malthus Model

The Logistic Equation

Solution of the Logistic Equation

3.3 Newton's Law of Cooling

Newton's Law of Cooling

Solution of the Equation

3.4 Free-Falling Bodies

Newton's Second Law of Motion

4 Higher-Order Differential Equations

4.1 Preliminary Definitions and Notation

The nth-Order Ordinary Linear Differential Equation

Fundamental Set of Solutions

Existence of a Fundamental Set of Solutions

4.2 Solutions of Homogeneous Equations with Constant Coefficients

General Solution

Finding a General Solution for a Homogeneous Equation with Constant Coefficients

Rules for Determining the General Solution of a Higher-Order Equation

4.3 Nonhomogeneous Equations with Constant Coefficients: The Annihilator Method

General Solution of a Nonhomogeneous Equation

Operator Notation

Using the Annihilator Method

Solving Initial-Value Problems Involving Nonhomogeneous Equations

4.4 Nonhomogeneous Equations with Constant Coefficients: The Method of Undetennined Coefficients

Outline of the Method of Undetermined Coefficients

Determining the Form of ypix) (Step 2):

4.5 Nonhomogeneous Equations with Constant Coefficients: Variation of Parameters

Second-Order Equations

Higher-Order Nonhomogeneous Equations

5 Applications of Higher-Order Differential Equations

5.1 Simple Harmonic Motion

5.2 Damped Motion

5.3 Forced Motion

5.4 Other Applications

L-R-C Circuits

Deflection of a Beam

5.5 The Pendulum Problem

6 Ordinary Differential Equations with Nonconstant Coefficients

6.1 Cauchy-Euler Equations

Second-Order Cauchy-Euler Equations

Higher-Order Cauchy-Euler Equations

Variation of Parameters

6.2 Power Series Review

Basic Definitions and Theorems

Reindexing a Power Series

6.3 Power Series Solutions about Ordinary Points

Power Series Solution Method about an Ordinary Point

6.4 Power Series Solutions about Regular Singular Points

Regular and Irregular Singular Points

Method of Frobenius

Indicial Roots that Differ by an Integer

Equal Indicial Roots

6.5 Some Special Equations

Legendre's Equation

The Gamma Function

Bessel's Equation

7 Introduction to the Laplace Transform

7.1 The Laplace Transform: Preliminary Definitions and Notation

Exponential Order, Jump Discontinuities, and Piecewise Continuous Functions

Properties of the Laplace Transform

7.2 The Inverse Laplace Transform

Linear Factors (Nonrepeated)

Repeated Linear Factors

Irreducible Quadratic Factors

Laplace Transform of an Integral

7.3 Solving Initial-Value Problems with the Laplace Transform

7.4 Laplace Transforms of Several Important Functions

Piecewise Defined Functions: The Unit Step Function

Solving Initial-Value Problems

Periodic Functions

Impulse Functions: The Delta Function

7.5 The Convolution Theorem

The Convolution Theorem

Integral and Integrodifferential Equations

8 Applications of Laplace Transforms

8.1 Spring-Mass Systems Revisited

8.2 L-R-C Circuits Revisited

8.3 Population Problems Revisited

9 Systems of Ordinary Differential Equations

9.1 Systems of Equations: The Operator Method

Operator Notation

Solution Method with Operator Notation

9.2 Review of Matrix Algebra and Calculus

Basic Operations

Determinants and Inverses

Eigenvalues and Eigenvectors

Matrix Calculus

9.3 Preliminary Definitions and Notation

9.4 Homogeneous Linear Systems with Constant Coefficients

Distinct Real Eigenvalues

Complex Conjugate Eigenvalues

Repeated Eigenvalues

9.5 Variation of Parameters

9.6 Laplace Transforms

9.7 Nonlinear Systems, Linearization, and Classification of Equilibrium Points

Real Distinct Eigenvalues

Repeated Eigenvalues

Complex Conjugate Eigenvalues

Nonlinear Systems

9.8 Numerical Methods

Built-In Methods

Euler's Method

Runge-Kutta Method

10 Applications of Systems of Ordinary Differential Equations

10.1 L-R-C Circuits with Loops

L-R-C Circuit with One Loop

L-R-C Circuit with Two Loops

L-R-C Circuit with Three Loops

10.2 Diffusion Problems

Diffusion through a Membrane

Diffusion through a Double-Walled Membrane

10.3 Spring-Mass Systems

10.4 Population Problems

10.5 Applications Using Laplace Transforms

Coupled Spring-Mass Systems

The Double Pendulum

10.6 Special Nonlinear Equations and Systems of Equations

Biological Systems: Predator-Prey Interaction

Physical Systems: Variable Damping

11 Eigenvalue Problems and Fourier Series

11.1 Boundary-Value, Eigenvalue, and Sturm-Liouville Problems

Boundary-Value Problems

Eigenvalue Problems

Sturm-Liouville Problems

11.2 Fourier Sine Series and Cosine Series

Fourier Sine Series

Fourier Cosine Series

11.3 Fourier Series

11.4 Generalized Fourier Series: Bessel-Fourier Series

12 Partial Differential Equations

12.1 Introduction to Partial Differential Equations and Separation of Variables

12.2 The One-Dimensional Heat Equation

The Heat Equation with Homogeneous Boundary Conditions

Nonhomogeneous Boundary Conditions

Insulated Boundary

12.3 The One-Dimensional Wave Equation

D'Alembert's Solution

12.4 Problems in Two Dimensions: Laplace's Equation

12.5 Two-Dimensional Problems in a Circular Region

Laplace's Equation in a Circular Region

The Wave Equation in a Circular Region

Appendix Getting Help from Maple V

A Note Regarding Different Versions of Maple

Getting Started with Maple V

Getting Help from Maple V

Additional Ways of Obtaining Help from Maple V

The Maple V Tutorial

Loading Miscellaneous Library Functions

Loading Packages

Glossary

Selected References

Index