# Schaum's Easy Outline: Differential Equations (Schaum's Outline Series)

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<ul><li><p>SCHAUMS Easy OUTLINES</p><p>DIFFERENTIALEQUATIONS</p></li><li><p>Other Books in SchaumsEasy Outlines Series Include:</p><p>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 </p><p>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 </p><p>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</p></li><li><p>SCHAUMS Easy OUTLINES</p><p>DIFFERENTIALEQUATIONS</p><p>B a s e d o n S c h a u m s</p><p>Out l ine o f Theory and Problems o f </p><p>Di f ferent ia l Equat ions , Second Edi t ion</p><p>b y R i c h a r d B r o n s o n , Ph.D.</p><p>A b r i d g e m e n t E d i t o r</p><p>E r i n J . B r e d e n s t e i n e r , Ph.D.</p><p>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</p><p>New York Chicago San Francisco Lisbon London Madrid</p><p>Mexico City Milan New Delhi San Juan</p><p>Seoul Singapore Sydney Toronto</p></li><li><p>Copyright 2003 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of</p><p>America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be repro-</p><p>duced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior writ-</p><p>ten permission of the publisher. </p><p>0-07-142846-1</p><p>The material in this eBook also appears in the print version of this title: 0-07-140967-X </p><p>All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occur-</p><p>rence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner,</p><p>with no intention of infringement of the trademark. Where such designations appear in this book, they have been</p><p>printed with initial caps. </p><p>McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for</p><p>use in corporate training programs. For more information, please contact George Hoare, Special Sales, at</p><p>george_hoare@mcgraw-hill.com or (212) 904-4069. </p><p>TERMS OF USEThis is a copyrighted work and The McGraw-Hill Companies, Inc. (McGraw-Hill) and its licensors reserve all</p><p>rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act</p><p>of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engi-</p><p>neer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sub-</p><p>license the work or any part of it without McGraw-Hills prior consent. You may use the work for your own non-</p><p>commercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be</p><p>terminated if you fail to comply with these terms. </p><p>THE WORK IS PROVIDED AS IS. McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR</p><p>WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE</p><p>OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED</p><p>THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WAR-</p><p>RANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MER-</p><p>CHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant</p><p>or guarantee that the functions contained in the work will meet your requirements or that its operation will be unin-</p><p>terrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccu-</p><p>racy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has</p><p>no responsibility for the content of any information accessed through the work. Under no circumstances shall</p><p>McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar</p><p>damages that result from the use of or inability to use the work, even if any of them has been advised of the possi-</p><p>bility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim</p><p>or cause arises in contract, tort or otherwise.</p><p>DOI: 10.1036/0071428461</p></li><li><p>Want to learn more?</p><p>We hope you enjoy this McGraw-Hill eBook!If you d like more information about thisbook, its author, or related books and websites,please click here.</p><p>8,</p></li><li><p>Contents</p><p>v</p><p>Chapter 1 Basic Concepts and ClassifyingDifferential Equations 1</p><p>Chapter 2 Solutions of First-Order Differential Equations 8</p><p>Chapter 3 Applications of First-Order Differential Equations 20</p><p>Chapter 4 Linear Differential Equations: Theory of Solutions 29</p><p>Chapter 5 Solutions of Linear HomogeneousDifferential Equations with Constant Coefficients 33</p><p>Chapter 6 Solutions of Linear Nonhomogeneous Equations and Initial-Value Problems 39</p><p>Chapter 7 Applications of Second-Order Linear Differential Equations 47</p><p>Chapter 8 Laplace Transforms and InverseLaplace Transforms 55</p><p>Chapter 9 Solutions by Laplace Transforms 65Chapter 10 Matrices and the Matrix </p><p>Exponential 69Chapter 11 Solutions of Linear Differential</p><p>Equations with Constant Coefficients by Matrix Methods 78</p><p>For more information about this title, click here.</p><p>Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.</p></li><li><p>Chapter 12 Power Series Solutions 85Chapter 13 Gamma and Bessel Functions 98Chapter 14 Numerical Methods 104Chapter 15 Boundary-Value Problems </p><p>and Fourier Series 115Appendix Laplace Transforms 124Index 133</p><p>vi DIFFERENTIAL EQUATIONS</p></li><li><p>Chapter 1</p><p>Basic Conceptsand Classifying</p><p>DifferentialEquations</p><p>In This Chapter:</p><p> Differential Equations Notation Solutions Initial-Value and Boundary-Value</p><p>Problems Standard and Differential Forms Linear Equations Bernoulli Equations Homogeneous Equations Separable Equations Exact Equations</p><p>1Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.</p></li><li><p>Differential Equations</p><p>A differential equation is an equation involving an unknown function andits derivatives.</p><p>Example 1.1: The following are differential equations involving the un-known function y.</p><p>(1.1)</p><p>(1.2)</p><p>(1.3)</p><p>(1.4)</p><p>(1.5)</p><p>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.</p><p>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.</p><p>=2</p><p>2</p><p>2</p><p>24 0y</p><p>t</p><p>y</p><p>x</p><p>d y</p><p>dxy</p><p>dy</p><p>dxy</p><p>dy</p><p>dxx</p><p>2</p><p>2</p><p>3 73</p><p>2</p><p>3 5</p><p>+ +</p><p> =</p><p>4 5 03</p><p>3</p><p>2</p><p>2d y</p><p>dxx</p><p>d y</p><p>dxxy+ + =(sin )</p><p>ed y</p><p>dx</p><p>dy</p><p>dxy</p><p>2</p><p>2</p><p>2</p><p>2 1+ =</p><p>dy</p><p>dxx= +5 3</p><p>2 DIFFERENTIAL EQUATIONS</p></li><li><p>Note!</p><p>The order of a differential equation is the order ofthe highest derivative appearing in the equation.</p><p>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.</p><p>Notation</p><p>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-</p><p>sents if the independent variable is x, but represents </p><p>if the independent variable is p. Observe that parenthesis are used in y(n)</p><p>to distinguish it from the nth power, yn. If the independent variable istime, usually denoted by t, primes are often replaced by dots. Thus,</p><p>represent, respectively.</p><p>Solutions</p><p>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 .</p><p>Example 1.4: Is , where c1 and c2 are arbi-trary constants, a solution of ?</p><p>Differentiating y, we find y = 2c1cos2x 2c2 sin2x and y =. Hence, c x c x4 2 4 21 2sin cos</p><p> + =y y4 0y x c x c x( ) sin cos= +1 22 2</p><p>dy dt d y dt d y dt/ , / , / ,2 2 3 3and, , y y yand</p><p>d y dp2 2/d y dx2 2/</p><p>y</p><p> y y y y y n, , , ,...,( ) ( )4</p><p>CHAPTER 1: Basic Concepts and Classification 3</p></li><li><p>Thus, satisfies the differential equation for allvalues of x and is a solution on the interval .</p><p>Example 1.5: Determine whether is a solution of .</p><p>Note that the left side of the differential equation must be nonnegative forevery real function y(x) and any x, since it is the sum of terms raised tothe second and fourth powers, while the right side of the equation is neg-ative. Since no function y(x) will satisfy this equation, the given differ-ential equation has no solutions.</p><p>We see that some differential equations have infinitely many solu-tions (Example 1.4), whereas other differential equations have no solu-tions (Example 1.5). It is also possible that a differential equation has ex-actly one solution. Consider , which for reasons identicalto those given in Example 1.5 has only one solution .</p><p>You Need to Know </p><p>A particular solution of a differential equation is anyone solution. The general solution of a differentialequation is the set of all solutions.</p><p>Example 1.6: The general solution to the differential equation in Ex-ample 1.4 can be shown to be (see Chapters Four and Five) </p><p>. That is, every particular solution of the differentialequation has this general form. A few particular solutions are: (a) </p><p>(choose and ), (b) (chooseand ), and (c) (choose ).c c1 2 0= =y 0c2 0=c1 1=</p><p>y x= sin 2c2 3= c1 5=x x5 2 3 2sin cosy =</p><p>c x c x+1 22 2sin cosy =</p><p>y 0( ) + =y y4 2 0</p><p>= 1( ) +y y4 2y x= 2 1</p><p>( , ) y c x c x= +1 22 2sin cos</p><p> + = + += + + +=</p><p>y y c x c x c x c x</p><p>c c x c c x</p><p>4 4 2 4 2 4 2 2</p><p>4 4 2 4 4 2</p><p>0</p><p>1 2 1 2</p><p>1 1 2 2</p><p>( sin cos ) ( sin cos )</p><p>( )sin ( )cos</p><p>4 DIFFERENTIAL EQUATIONS</p></li><li><p>The general solution of a differential equation cannot always be ex-pressed by a single formula. As an example consider the differential equa-</p><p>tion , which has two particular solutions and .</p><p>Initial-Value and Boundary-Value Problems</p><p>A differential equation along with subsidiaryconditions on the unknown function and its de-rivatives, all given at the same value of the in-dependent variable, constitutes an initial-valueproblem. The subsidiary conditions are initialconditions. If the subsidiary conditions are giv-en at more than one value of the independentvariable, the problem is a boundary-value prob-lem and the conditions are boundary conditions.</p><p>Example 1.7: The problem is an initialvalue problem, because the two subsidiary conditions are both given at</p><p>. The problem is a boundary-valueproblem, because the two subsidiary conditions are given at x = 0 and x = 1.</p><p>A solution to an initial-value or boundary-value problem is a func-tion y(x) that both solves the differential equation and satisfies all givensubsidiary conditions.</p><p>Standard and Differential Forms</p><p>Standard form for a first-order differential equation in the unknown func-tion y(x) is</p><p>(1.6)</p><p>where the derivative appears only on the left side of 1.6. Many, butnot all, first-order differential equations can be written in standard formby algebraically solving for and then setting f(x,y) equal to the rightside of the resulting equation.</p><p>y</p><p>y</p><p> =y f x y( , )</p><p> + = = =y y e y yx2 0 1 1 1; ( ) , ( )x = p</p><p> + = = =y y e y yx2 1 2; ( ) , ( )p p</p><p>y 0y x= 1 / + =y y2 0</p><p>CHAPTER 1: Basic Concepts and Classification 5</p></li><li><p>The right side of 1.6 can always be written as a quotient of two oth-er functions M(x,y) and N(x,y). Then 1.6 becomes</p><p>which is equivalent to the differential form </p><p>(1.7)</p><p>Linear Equations</p><p>Consider a differential equation in standard form 1.6. If f(x,y) can be writ-ten as (that is, as a function of x times y, plus an-other function of x), the differential equation is linear. First-order lineardifferential equations can always be expressed as</p><p>(1.8)</p><p>Linear equations are solved in Chapter Two.</p><p>Bernoulli Equations</p><p>A Bernoulli differential equation is an equation of the form</p><p>(1.9)</p><p>where n denotes a real number. When n = 1 or n = 0, a Bernoulli equationreduces to a linear equation. Bernoulli equations are solved in ChapterTwo.</p><p>Homogeneous Equations</p><p>A differential equation in standard form (1.6) is homogeneous if</p><p>(1.10)</p><p>for every real number t. Homogeneous equations are solved in ChapterTwo.</p><p>f tx ty f x y( , ) ( , )=</p><p> + =y p x y q x yn( ) ( )</p><p> + =y p x y q x( ) ( )</p><p>f x y p x y q x( , ) ( ) ( )= +</p><p>M x y dx N x y dy( , ) ( , )+ = 0</p><p>N x y( , ),dy dx M x y/ ( , ) /=</p><p>6 DIFFERENTIAL EQUATIONS</p></li><li><p>Note!</p><p>In the general framework of differential equations,the word homogeneous has an entirely differentmeaning (see Chapter Four). Only in the context of first-order differential equations does homoge-neous have the meaning defined above.</p><p>Separable Equations</p><p>Consider a differential equation in differential form (1.7). If M(x,y) = A(x)(a function only of x) and N(x,y) = B(y) (a function only of y), the differ-ential equation is separable, or has its variables separated. Separableequations are solved in Chapter Two.</p><p>Exact Equations</p><p>A differential equation in differential form (1.7) is exact if</p><p>(1.11)</p><p>Exact equations are solved in Chapter Two (where a more precise defini-tion of exactness is given).</p><p>= </p><p>M x y</p><p>y</p><p>N x y</p><p>x</p><p>( , ) ( , )</p><p>CHAPTER 1: Basic Concepts and Classification 7</p></li><li><p>Chapter 2 </p><p>Solutions ofFirst-Order</p><p>DifferentialEquations</p><p>In This Chapter:</p><p> Separable Equations Homogeneous Equations Exact Equations Linear Equations Bernoulli Equations Solved Problems</p><p>Separable Equations</p><p>General Solution</p><p>The solution to the first-order separable differential equation (see Chap-ter One). </p><p>(2.1)A x dx B y dy( ) ( )+ = 0</p><p>8Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.</p></li><li><p>is</p><p>(2.2)</p><p>where c represents an arbitrary constant.(See Problem 2.1)The integrals obtained in Equation 2.2 may be, for all practical pur-</p><p>poses, impossible to evaluate. In such case, numerical techniques (seeChapter 14) are used to obtain an approximate solution. Even if the indi-cated integrations in...</p></li></ul>