Differential Equations with Maple V®
- 1st Edition - September 8, 1994
- Imprint: Academic Press
- Authors: Martha L Abell, James P. Braselton
- Language: English
- Paperback ISBN:9 7 8 - 0 - 1 2 - 0 4 1 5 4 8 - 9
- 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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Request a sales quoteDifferential 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
- Edition: 1
- Published: September 8, 1994
- No. of pages (eBook): 698
- Imprint: Academic Press
- Language: English
- Paperback ISBN: 9780120415489
- eBook ISBN: 9781483266572
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