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Fundamentals of differential equations PDF

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F u n d a m e n Fundamentals of Differential Equations t a l s Nagle Saff Snider : Eighth Edition D i f f e r e n t i a l E q u a t i o n s N a g l e e t a l . ISBN 978-1-29202-382-3 8 e 9 781292 023823 Fundamentals of Differential Equations Nagle Saff Snider Eighth Edition ISBN 10: 1-292-02382-1 ISBN 13: 978-1-292-02382-3 Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affi liation with or endorsement of this book by such owners. ISBN 10: 1-292-02382-1 ISBN 13: 978-1-292-02382-3 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America 12334556639553642988173935955937 P E A R S O N C U S T O M L I B R AR Y Table of Contents 1. Introduction R. Kent Nagle/Edward B. Saff/Arthur David Snider 1 2. First-Order Differential Equations R. Kent Nagle/Edward B. Saff/Arthur David Snider 37 3. Mathematical Models and Numerical Methods Involving First-Order Equations R. Kent Nagle/Edward B. Saff/Arthur David Snider 93 4. Linear Second-Order Equations R. Kent Nagle/Edward B. Saff/Arthur David Snider 159 5. Introduction to Systems and Phase Plane Analysis R. Kent Nagle/Edward B. Saff/Arthur David Snider 253 6. Theory of Higher-Order Linear Differential Equations R. Kent Nagle/Edward B. Saff/Arthur David Snider 335 7. Laplace Transforms R. Kent Nagle/Edward B. Saff/Arthur David Snider 369 8. Series Solutions of Differential Equations R. Kent Nagle/Edward B. Saff/Arthur David Snider 445 9. Matrix Methods for Linear Systems R. Kent Nagle/Edward B. Saff/Arthur David Snider 525 10. Partial Differential Equations R. Kent Nagle/Edward B. Saff/Arthur David Snider 599 Tables and Equations R. Kent Nagle/Edward B. Saff/Arthur David Snider 683 Index 687 I II Introduction 1 BACKGROUND In the sciences and engineering,mathematical models are developed to aid in the understanding of physical phenomena. These models often yield an equation that contains some derivatives of an unknown function. Such an equation is called a differential equation. Two examples of models developed in calculus are the free fall of a body and the decay of a radioactive substance. In the case of free fall, an object is released from a certain height above the ground and falls under the force of gravity.†Newton’s second law,which states that an object’s mass times its acceleration equals the total force acting on it, can be applied to the falling object. This leads to the equation (see Figure 1) d2h m (cid:3)(cid:2)mg , dt2 where mis the mass of the object,his the height above the ground,d2h/dt2 is its acceleration, gis the (constant) gravitational acceleration,and (cid:2)mgis the force due to gravity. This is a differ- ential equation containing the second derivative of the unknown height has a function of time. Fortunately,the above equation is easy to solve for h. All we have to do is divide by mand integrate twice with respect to t. That is, d2h (cid:3)(cid:2)g , dt2 so dh (cid:3)(cid:2)gt(cid:4)c dt 1 and (cid:2)gt2 h(cid:3)h t (cid:3) (cid:4) c t (cid:4)c . A B 2 1 2 −mg h Figure 1Apple in free fall †We are assuming here that gravity is the onlyforce acting on the object and that this force is constant. More general models would take into account other forces,such as air resistance. From Chapter 1 of Fundamentals of Differential Equations,Eighth Edition. R. Kent Nagle,Edward B. Saff and Arthur David Snider. Copyright ©2012 by Pearson Education,Inc. Publishing as Pearson Addison-Wesley. All rights reserved. 1 Introduction We will see that the constants of integration, c and c , are determined if we know the initial 1 2 height and the initial velocity of the object. We then have a formula for the height of the object at time t. In the case of radioactive decay (Figure 2), we begin from the premise that the rate of decay is proportional to the amount of radioactive substance present. This leads to the equation dA (cid:3)(cid:2)kA , k(cid:5)0 , dt where A (cid:5)0 is the unknown amount of radioactive substance present at time tand kis the pro- A B portionality constant. To solve this differential equation,we rewrite it in the form 1 dA(cid:3)(cid:2)k dt A and integrate to obtain (cid:2) 1 (cid:2) dA(cid:3) (cid:2)k dt A ln A(cid:4)C (cid:3)(cid:2)kt(cid:4)C . 1 2 Solving for Ayields A(cid:3)A t (cid:3)eln A(cid:3)e(cid:2)kteC2(cid:2)C1 (cid:3)Ce(cid:2)kt , A B where C is the combination of integration constants eC2(cid:2)C1. The value of C, as we will see later,is determined if the initial amount of radioactive substance is given. We then have a for- mula for the amount of radioactive substance at any future time t. Even though the above examples were easily solved by methods learned in calculus,they do give us some insight into the study of differential equations in general. First,notice that the solu- tion of a differential equation is a function,like h t or A t ,not merely a number. Second,inte- A B A B gration is an important tool in solving differential equations (not surprisingly!). Third,we cannot expect to get a unique solution to a differential equation,since there will be arbitrary “constants of integration.”The second derivative d2h/dt2in the free-fall equation gave rise to two constants, c and c ,and the first derivative in the decay equation gave rise,ultimately,to one constant,C. 1 2 Whenever a mathematical model involves the rate of changeof one variable with respect to another, a differential equation is apt to appear. Unfortunately, in contrast to the examples for free fall and radioactive decay,the differential equation may be very complicated and diffi- cult to analyze. A Figure 2Radioactive decay 2 Introduction R L + emf C − Figure 3Schematic for a series RLC circuit Differential equations arise in a variety of subject areas,including not only the physical sci- ences but also such diverse fields as economics,medicine,psychology,and operations research. We now list a few specific examples. 1. In banking practice,if P t is the number of dollars in a savings bank account that A B pays a yearly interest rate of r% compounded continuously,then Psatisfies the dif- ferential equation dP r (1) (cid:3) P , t in years. dt 100 2. A classic application of differential equations is found in the study of an electric cir- cuit consisting of a resistor,an inductor,and a capacitor driven by an electromotive force (see Figure 3). Here an application of Kirchhoff’s laws leads to the equation d2q dq 1 (2) L (cid:4) R (cid:4) q(cid:3)E t , dt2 dt C A B where Lis the inductance,Ris the resistance,Cis the capacitance,E t is the electro- A B motive force,q t is the charge on the capacitor,and tis the time. A B 3. In psychology,one model of the learning of a task involves the equation dy/dt 2p (3) (cid:3) . y3/2 1(cid:2)y 3/2 1n A B Here the variable yrepresents the learner’s skill level as a function of time t. The constants pand ndepend on the individual learner and the nature of the task. 4. In the study of vibrating strings and the propagation of waves,we find the partial differential equation 02u 02u (4) (cid:2)c2 (cid:3)0 ,†† 0t2 0x2 where trepresents time,xthe location along the string,cthe wave speed,and uthe displacement of the string,which is a function of time and location. ††Historical Footnote:This partial differential equation was first discovered by Jean le Rond d’Alembert (1717–1783) in 1747. 3 Introduction To begin our study of differential equations, we need some common terminology. If an equation involves the derivative of one variable with respect to another, then the former is called a dependent variable and the latter an independent variable. Thus,in the equation d2x dx (5) (cid:4)a (cid:4)kx(cid:3)0 , dt2 dt tis the independent variable and xis the dependent variable. We refer to aand kas coefficients in equation (5). In the equation 0u 0u (6) (cid:2) (cid:3)x(cid:2)2y , 0x 0y xand yare independent variables and uis the dependent variable. A differential equation involving only ordinary derivatives with respect to a single indepen- dent variable is called an ordinary differential equation. A differential equation involving partial derivatives with respect to more than one independent variable is a partial differential equation. Equation (5) is an ordinary differential equation,and equation (6) is a partial differential equation. The orderof a differential equation is the order of the highest-order derivatives present in the equation. Equation (5) is a second-order equation because d2x/dt2 is the highest-order derivative present. Equation (6) is a first-order equation because only first-order partial deriva- tives occur. It will be useful to classify ordinary differential equations as being either linear or nonlin- ear. Remember that lines (in two dimensions) and planes (in three dimensions) are especially easy to visualize,when compared to nonlinear objects such as cubic curves or quadric surfaces. For example, all the points on a line can be found if we know just two of them. Correspond- ingly,lineardifferential equations are more amenable to solution than nonlinear ones. Now the equations for lines ax(cid:4)by(cid:3)cand planes ax(cid:4)by(cid:4)cz(cid:3)dhave the feature that the variables appear in additive combinations of their first powers only. By analogy a linear differential equationis one in which the dependent variable yand its derivatives appear in additive combi- nations of their first powers. More precisely,a differential equation is linearif it has the format dny dn(cid:3)1y p dy (7) anAxB dxn (cid:2) an(cid:2)1AxB dxn(cid:3)1 (cid:2) (cid:2)a1AxB dx (cid:2)a0AxBy(cid:4) FAxB , where anAxB,an(cid:2)1AxB,. . . ,a0AxB and FAxB depend only on the independent variable x. The addi- tive combinations are permitted to have multipliers (coefficients) that depend on x; no restric- tions are made on the nature of this x-dependence. If an ordinary differential equation is not linear,then we call it nonlinear. For example, d2y (cid:4) y3(cid:3)0 dx2 is a nonlinear second-order ordinary differential equation because of the y3term,whereas dx t3 (cid:3)t3(cid:4) x dt is linear (despite the t3terms). The equation d2y dy (cid:2)y (cid:3)cos x dx2 dx is nonlinear because of the y dy/dx term. 4 Introduction Although the majority of equations one is likely to encounter in practice fall into the nonlin- ear category,knowing how to deal with the simpler linear equations is an important first step (just as tangent lines help our understanding of complicated curves by providing local approximations). 1 EXERCISES In Problems 1–12,a differential equation is given along dp 8. (cid:3)kp P(cid:2)p ,where kand Pare constants with the field or problem area in which it arises. Classify dt A B each as an ordinary differential equation (ODE) or a (logistic curve,epidemiology,economics) partial differential equation (PDE), give the order, and d4y indicate the independent and dependent variables. If the 9. 8 (cid:3)x 1(cid:2)x dx4 A B equation is an ordinary differential equation, indicate (deflection of beams) whether the equation is linear or nonlinear. d2y dy d2y dy 10. x (cid:4) (cid:4)xy(cid:3)0 1. (cid:2) 2x (cid:4) 2y (cid:3) 0 dx2 dx dx2 dx (aerodynamics,stress analysis) (Hermite’s equation,quantum-mechanical harmonic 0N 02N 1 0N 11. (cid:3) (cid:4) (cid:4)kN, where kis a constant oscillator) 0t 0r2 r 0r d2x dx (nuclear fission) 2. 5 (cid:4) 4 (cid:4) 9x(cid:3)2 cos 3t dt2 dt d2y dy 12. (cid:2)0.1 1(cid:2)y2 (cid:4)9y(cid:3)0 (mechanical vibrations,electrical circuits,seismology) dx2 A B dx (van der Pol’s equation,triode vacuum tube) 02u 02u 3. (cid:4) (cid:3)0 0x2 0y2 In Problems 13–16,write a differential equation that fits the physical description. (Laplace’s equation,potential theory,electricity,heat, aerodynamics) 13. The rate of change of the population pof bacteria at time tis proportional to the population at time t. dy y 2(cid:2)3x 4. (cid:3) A B 14. The velocity at time t of a particle moving along a dx x 1(cid:2)3y straight line is proportional to the fourth power of its A B position x. (competition between two species,ecology) 15. The rate of change in the temperature Tof coffee at dx 5. (cid:3)k 4(cid:2)x 1(cid:2)x ,where kis a constant time t is proportional to the difference between the dt A BA B temperature M of the air at time t and the tempera- (chemical reaction rates) ture of the coffee at time t. dy 2 16. The rate of change of the mass Aof salt at time tis 6. y 1(cid:4) (cid:3)C,where Cis a constant proportional to the square of the mass of salt present c adxb d at time t. (brachistochrone problem,†calculus of variations) 17. Drag Race. Two drivers,Alison and Kevin,are par- d2y dy ticipating in a drag race. Beginning from a standing 7. 11 (cid:2) y dx2 (cid:4) 2x dx (cid:3) 0 start,they each proceed with a constant acceleration. Alison covers the last 1/4 of the distance in 3 sec- (Kidder’s equation, flow of gases through a porous onds, whereas Kevin covers the last 1/3 of the dis- medium) tance in 4 seconds. Who wins and by how much time? †Historical Footnote:In 1630 Galileo formulated the brachistochrone problem bra´x´isto(cid:6)(cid:3)shortest,xro´no(cid:6)(cid:3)time,that is,to determine a A B path down which a particle will fall from one given point to another in the shortest time. It was reproposed by John Bernoulli in 1696 and solved by him the following year. 5 5

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