schaum's easy outlines of differential equations books in schaum’s easy outlines series include:...
Post on 15-Feb-2018
Embed Size (px)
SCHAUMS Easy OUTLINES
Other Books in SchaumsEasy Outlines Series Include:
Schaums Easy Outline: CalculusSchaums Easy Outline: College AlgebraSchaums Easy Outline: College MathematicsSchaums Easy Outline: Discrete MathematicsSchaums Easy Outline: Elementary AlgebraSchaums Easy Outline: GeometrySchaums Easy Outline: Linear AlgebraSchaums Easy Outline: Mathematical Handbook
of Formulas and TablesSchaums Easy Outline: PrecalculusSchaums Easy Outline: Probability and StatisticsSchaums Easy Outline: StatisticsSchaums Easy Outline: TrigonometrySchaums Easy Outline: Business StatisticsSchaums Easy Outline: Principles of AccountingSchaums Easy Outline: Principles of EconomicsSchaums Easy Outline: BiologySchaums Easy Outline: BiochemistrySchaums Easy Outline: Molecular and Cell BiologySchaums Easy Outline: College ChemistrySchaums Easy Outline: GeneticsSchaums Easy Outline: Human Anatomy
and PhysiologySchaums Easy Outline: Organic ChemistrySchaums Easy Outline: PhysicsSchaums Easy Outline: Applied PhysicsSchaums Easy Outline: Programming with C++Schaums Easy Outline: Programming with JavaSchaums Easy Outline: Basic ElectricitySchaums Easy Outline: ElectromagneticsSchaums Easy Outline: Introduction to PsychologySchaums Easy Outline: FrenchSchaums Easy Outline: German Schaums Easy Outline: SpanishSchaums Easy Outline: Writing and Grammar
SCHAUMS Easy OUTLINES
B a s e d o n S c h a u m s
Out l ine o f Theory and Problems o f
Di f ferent ia l Equat ions , Second Edi t ion
b y R i c h a r d B r o n s o n , Ph.D.
A b r i d g e m e n t E d i t o r
E r i n J . B r e d e n s t e i n e r , Ph.D.
S C H A U M S O U T L I N E S E R I E SM c G R AW - H I L L
New York Chicago San Francisco Lisbon London Madrid
Mexico City Milan New Delhi San Juan
Seoul Singapore Sydney Toronto
Copyright 2003 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of
America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be repro-
duced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior writ-
ten permission of the publisher.
The material in this eBook also appears in the print version of this title: 0-07-140967-X
All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occur-
rence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner,
with no intention of infringement of the trademark. Where such designations appear in this book, they have been
printed with initial caps.
McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for
use in corporate training programs. For more information, please contact George Hoare, Special Sales, at
firstname.lastname@example.org or (212) 904-4069.
rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act
of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engi-
neer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sub-
license the work or any part of it without McGraw-Hills prior consent. You may use the work for your own non-
commercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be
terminated if you fail to comply with these terms.
THE WORK IS PROVIDED AS IS. McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR
WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE
OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED
THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WAR-
RANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MER-
CHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant
or guarantee that the functions contained in the work will meet your requirements or that its operation will be unin-
terrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccu-
racy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has
no responsibility for the content of any information accessed through the work. Under no circumstances shall
McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar
damages that result from the use of or inability to use the work, even if any of them has been advised of the possi-
bility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim
or cause arises in contract, tort or otherwise.
Chapter 1 Basic Concepts and ClassifyingDifferential Equations 1
Chapter 2 Solutions of First-Order Differential Equations 8
Chapter 3 Applications of First-Order Differential Equations 20
Chapter 4 Linear Differential Equations: Theory of Solutions 29
Chapter 5 Solutions of Linear HomogeneousDifferential Equations with Constant Coefficients 33
Chapter 6 Solutions of Linear Nonhomogeneous Equations and Initial-Value Problems 39
Chapter 7 Applications of Second-Order Linear Differential Equations 47
Chapter 8 Laplace Transforms and InverseLaplace Transforms 55
Chapter 9 Solutions by Laplace Transforms 65Chapter 10 Matrices and the Matrix
Exponential 69Chapter 11 Solutions of Linear Differential
Equations with Constant Coefficients by Matrix Methods 78
For more information about this title, click here.
Chapter 12 Power Series Solutions 85Chapter 13 Gamma and Bessel Functions 98Chapter 14 Numerical Methods 104Chapter 15 Boundary-Value Problems
and Fourier Series 115Appendix Laplace Transforms 124Index 133
vi DIFFERENTIAL EQUATIONS
Basic Conceptsand Classifying
In This Chapter:
Differential Equations Notation Solutions Initial-Value and Boundary-Value
Problems Standard and Differential Forms Linear Equations Bernoulli Equations Homogeneous Equations Separable Equations Exact Equations
A differential equation is an equation involving an unknown function andits derivatives.
Example 1.1: The following are differential equations involving the un-known function y.
Adifferential equation is an ordinary differential equation if the unknownfunction depends on only one independent variable. If the unknown func-tion depends on two or more independent variables, the differential equa-tion is a partial differential equation. In this book we will be concernedsolely with ordinary differential equations.
Example 1.2: Equations 1.1 through 1.4 are examples of ordinary differ-ential equations, since the unknown function y depends solely on the vari-able x. Equation 1.5 is a partial differential equation, since y depends onboth the independent variables t and x.
4 5 03
dxxy+ + =(sin )
2 1+ =
dxx= +5 3
2 DIFFERENTIAL EQUATIONS
The order of a differential equation is the order ofthe highest derivative appearing in the equation.
Example 1.3: Equation 1.1 is a first-order differential equation; 1.2, 1.4,and 1.5 are second-order differential equations. (Note in 1.4 that the or-der of the highest derivative appearing in the equation is two.) Equation1.3 is a third-order differential equation.
The expressions are often used to represent, re-spectively, the first, second, third, fourth, . . ., nth derivatives of y with re-spect to the independent variable under consideration. Thus, repre-
sents if the independent variable is x, but represents
if the independent variable is p. Observe that parenthesis are used in y(n)
to distinguish it from the nth power, yn. If the independent variable istime, usually denoted by t, primes are often replaced by dots. Thus,
A solution of a differential equation in the unknown function y and theindependent variable x on the interval is a function y(x) that satisfiesthe differential equation identically for all x in .
Example 1.4: Is , where c1 and c2 are arbi-trary constants, a solution of ?
Differentiating y, we find y = 2c1cos2x 2c2 sin2x and y =. Hence, c x c x4 2 4 21 2sin cos
+ =y y4 0y x c x c x( ) sin cos= +1 22 2
dy dt d y dt d y dt/ , / , / ,2 2 3 3and, , y y yand
d y dp2 2/d y dx2 2/
y y y y y n, , , ,...,( ) ( )4
CHAPTER 1: Basic Concepts and Classification 3