Solving ODE's with Maple V

by Barrow,David

• ISBN13: 9780534345556
• ISBN10: 0534345557
• Format: Paperback
• Copyright: 1997-08-26
• Publisher: Brooks Cole

Summary

This comprehensive book helps students tap into the power of Maple?, thereby simplifying the computations and graphics that are often required in the practical use of mathematics. Numerous examples and exercises provide a thorough introduction to the basic Maple? commands that are needed to solve differential equations. Topics include: numerical algorithms, first order linear systems, homogeneous and nonhomogeneous equations, beats and resonance, Laplace Transforms, qualitative theory, nonlinear systems, and much more.

Table of Contents

Maple Review	1	(16)
Assignment Statements and Arithmetic	1	(1)
Expressions versus Functions	2	(2)
Plots	4	(4)
Solving Equations	8	(2)
Differentiation	10	(2)
Integration	12	(2)
Taylor Approximations	14	(1)
Exercises	15	(2)
Differential Equations and Maple	17	(12)
Maple Commands	17	(1)
Examples	18	(4)
dsolve	18	(1)
DEplot	19	(2)
dsolve/numeric	21	(1)
Euler's Method Approximation	22	(3)
Exercises	25	(4)
Applications	29	(14)
Art History	29	(3)
Rural Water Supplies	32	(4)
Growth of Microorganisms: Linear and Nonlinear	36	(7)
Higher Order Equations	43	(12)
dsolve, dsolve/numeric, and DEplot	43	(4)
Homogeneous equations	47	(2)
Nonhomogeneous Equations	49	(3)
Undetermined coefficients	49	(2)
Variation of Parameters	51	(1)
Exercises	52	(3)
Applications of Higher Order Equations	55	(8)
Beats and Resonance	55	(4)
Beats	55	(3)
Resonance	58	(1)
Damped Forced Vibrations	59	(2)
Exercises	61	(2)
Laplace Transforms	63	(20)
Definition of the Laplace Transform	64	(1)
Properties of the Laplace Transform	65	(4)
Derivatives	69	(1)
Initial Value Problems	69	(3)
Unit Step Function	72	(5)
Impulse Functions	77	(2)
The Convolution Integral	79	(1)
The Transfer Function	80	(1)
Exercises	80	(3)
Systems	83	(40)
dsolve, dsolve/numeric, and DEplot	83	(6)
Geometric Insight into Eigenvectors	89	(3)
Maple's Linear Algebra Package	92	(12)
Homogeneous First Order Linear Systems	104	(10)
Distinct eigenvalues	104	(3)
Repeated Eigenvalues with Insufficient Eigenvectors	107	(3)
Complex Eigenvalues	110	(4)
Nonhomogeneous First Order Linear Systems	114	(4)
Exercises	118	(5)
The Phase Plane	123	(22)
Linear Systems	123	(8)
Real Eigenvalues	123	(6)
Complex Eigenvalues	129	(2)
Autonomous Systems	131	(12)
Exercises	143	(2)
Numerical Solutions of Systems	145	(18)
The Order of Convergence for a Numerical Method	149	(1)
The Taylor Series Method	150	(2)
Finite Difference Method for Boundary Value Problem	152	(1)
Numerical Solution of Systems of First Order Equations	153	(4)
Two-Dimensional Motion of a Projectile with Drag	157	(2)
Exercises	159	(4)
Partial Differential Equations	163	(52)
The Heat Equation	163	(5)
Fourier Series	168	(9)
Cosine series	171	(2)
General Fourier Series	173	(4)
The Wave Equation	177	(5)
Laplace's Equation	182	(8)
Laplace Transforms	190	(5)
Numerical Methods	195	(17)
Heat Equation	196	(5)
Wave Equation	201	(6)
Laplace's Equation	207	(5)
Exercises	212	(3)
Projects	215	(14)
Heat-Seeking Particles	215	(2)
Autonomous Differential Equations	217	(3)
The Tennis Serve	220	(2)
Modeling Lake Pollution	222	(1)
Shooting a Projectile	222	(1)
Rocket Propulsion	223	(1)
The Laplace Transform and Square Waves	224	(1)
Taylor Series Methods for Differential Equations	225	(1)
Controlling the Motion of a Cart	226	(3)
Appendix.	229
Nuts and Bolts	229	(11)
Culling Eigenvectors	229	(4)
Systems and Matrices	233	(3)
Data Types and Sizes	236	(4)
A Tridiagonal Solver	240