2500 Solved Problems in Differential Equations

452 Pages·057.311 MB·English

by  Richard Bronson

SCHAUMS
SOLVED PROBLEMS
SERIES
SOLVED PROBLEMS
2500 IN
DIFFERENTIAL
EQUATIONS
• A complete and expert source of problems with
solutions for college and university students.
• Solutions are worked out step-by-step, are easy
to follow, and teach the subject thoroughly.
• Usable with any textbook.
Digitized by the Internet Archive
2012
in
http://archive.org/details/2500solvedproble00rich
SCHAUM'S SOLVED
PROBLEMS
SERIES
SOLVED PROBLEMS
2500 IN
DIFFERENTIAL
EQUATIONS
by
Richard Bronson, Ph.D.
Fairleigh Dickinson University
SCHAUM'S OUTLINE SERIES
McGRAW-HILLPUBLISHINGCOMPANY
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# Richard Bronson, Ph.D., Professor ofMathematics and Computer
Science at Fairleigh Dickinson University.
Dr. Bronson,besidesteaching,editstwomathematicaljournalsand has
written numerous technical papers. Amongthebookshehaspublished
are Schaum's Outlines in the areas ofdifferential equations, operations
research, and matrix methods.
Other Contributors to This Volume
# Frank Ayres, Jr., Ph.D., Dickinson College
I James Crawford, B.S., Fairleigh Dickinson College
# Thomas M. Creese, Ph.D., University ofKansas
f Robert M. Harlick, Ph.D., University ofKansas
f Robert H. Martin, Jr., Ph.D., North Carolina State University
I George F. Simmons, Ph.D., Colorado College
I Murray R. Spiegel, Ph.D., Rensselaer Polytechnic Institute
I C. Ray Wylie, Ph.D., Furman University
Project supervision by The Total Book.
Library of Congress Cataloging-in-Publication Data
Bronson, Richard.
2500 solved problems in differential equations / by Richard
Bronson.
—
p. cm. (Schaum's solved problems series)
ISBN 0-07-007979-X
—
1. Differential equations Problems, exercises, etc. I. Title.
II. Series.
QA371.B83 1988
515.3'5'076—dc 19 88-17705
CIP
2 3 4 5 6 7 8 9 SHP/SHP 8 9 *
ISBN D-D7-DD7T7T-X
Copyright © 1989 McGraw-Hill, Inc. All rights reserved. Printed in the United
States of America. Except as permitted under the United States Copyright Act
of 1976, no part ofthis publication may be reproduced or distributed in any form
or by any means, or stored in a data base or retrieval system, without the prior
written permission of the publisher.
CONTENTS
Chapter 1 BASIC CONCEPTS 1
Classifications / Formulating proportionality problems / Problems involving
Newton's law of cooling / Problems involving Newton's second law of motion /
Spring problems / Electric circuit problems / Geometrical problems / Primitives /
SOLUTIONS
Chapter 2 19
Validating solutions / Primitives / Direction fields / Initial and boundary
conditions / Particular solutions / Simplifying solutions /
SEPARABLE FIRST-ORDER DIFFERENTIAL EQUATIONS
Chapter 3 37
Solutions with rational functions / Solutions with logarithms / Solutions with
transcendental functions / Homogeneous equations / Solutions of homogeneous
equations / Miscellaneous transformations / Initial-value problems /
Chapter 4 EXACT FIRST-ORDER DIFFERENTIAL EQUATIONS 66
Testing for exactness / Solutions of exact equations / Integrating factors /
Solution with integrating factors / Initial-value problems /
Chapter 5 LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 92
Homogeneous equations / Nonhomogeneous equations / Bernoulli equations /
Miscellaneous transformations / Initial-value problems /
Chapter 6 APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 110
Population growth problems Decay problems Compound-interest problems
/ / /
Cooling and heating problems / Flow problems / Electric circuit problems /
Mechanics problems Geometrical problems
/ /
Chapter 7 LINEAR DIFFERENTIAL EQUATIONS^THEORY OF SOLUTIONS 149
Wronskian / Linear independence / General solutions of homogeneous
equations / General solutions of nonhomogeneous equations /
Chapter 8 LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS WITH 166
CONSTANT COEFFICIENTS
Distinct real characteristic roots / Distinct complex characteristic roots /
Distinct real and complex characteristic roots / Repeated characteristic roots /
Characteristic roots of various types / Euler's equation /
Chapter 9 THE METHOD OF UNDETERMINED COEFFICIENTS 191
Equations with exponential right side / Equations with constant right-hand side /
Equations with polynomial right side / Equations whose right side is the product
of a polynomial and an exponential / Equations whose right side contains sines
and cosines / Equations whose right side contains a product involving sines and
cosines / Modifications of trial particular solutions / Equations whose right side
contains a combination of terms
/
Chapter 10 VARIATION OF PARAMETERS 232
Formulas / First-order differential equations / Second-order differential
equations / Higher-order differential equations /
Chapter 11 APPLICATIONS OF SECOND-ORDER LINEAR DIFFERENTIAL 255
EQUATIONS
Spring problems Mechanics problems Horizontal-beam problems Buoyancy
/ / /
problems / Electric circuit problems /
iii
CONTENTS
iv
LAPLACE TRANSFORMS
Chapter 12 283
Transforms of elementary functions / Transforms involving gamma functions /
Linearity / Functions multiplied by a power of the independent variable /
Translations / Transforms of periodic functions /
Chapter 13 INVERSE LAPLACE TRANSFORMS AND THEIR USE IN SOLVING 306
DIFFERENTIAL EQUATIONS
Inverse Laplace transforms by inspection / Linearity / Completing the square and
translations / Partial-fraction decompositions / Convolutions / Solutions using
Laplace transforms
/
MATRIX METHODS
Chapter 14 337
Finding eAt / Matrix differential equations / Solutions /
Chapter 15 INFINITE-SERIES SOLUTIONS 354
Analytic functions / Ordinary and singular points Recursion formulas /
Solutions to homogeneous differential equations about an ordinary point
/
Solutions to nonhomogeneous differential equations about an ordinary point
/
Initial-value problems / The method of Frobenius / Bessel functions /
Chapter 16 EIGENFUNCTION EXPANSIONS 415
Sturm-Liouville problems / Fourier series / Parseval's identity / Even and odd
functions / Sine and cosine series /
To the Student
This collection of solved problems covers analytical techniques for solving differential equations. It is meant to
be used as both a supplement for traditional courses in differential equations and a reference book for engineers
and scientists interested in particular applications. The only prerequisite for understanding the material in this
book is calculus.
The material within each chapter and the ordering ofchapters are standard. The book begins with methods
for solving first-order differential equations and continues through linear differential equations. In this latter
category we include the methods of variation of parameters and undetermined coefficients, Laplace transforms,
matrix methods, and boundary-value problems. Much of the emphasis is on second-order equations, but
extensions to higher-order equations are also demonstrated.
Two chapters are devoted exclusively to applications, so readers interested in a particular type can go directly
to the appropriate section. Problems in these chapters are cross-referenced to solution procedures in previous
chapters. By utilizing this referencing system, readers can limit themselves to just those techniques that have
value within a particular application.

CHAPTER
1
Basic Concepts
CLASSIFICATIONS
1.1 Determine which ofthe following are ordinary differential equations and which are partial differential equations:
^+3^
<«) + 2y =
dx2 dx
—dz —dz
(b) = z + x
ox oy
I Equation (a) is an ordinary differential equation because it contains only ordinary (nonpartial) derivatives;
{b) is a partial differential equation because it contains partial derivatives.
1.2 Determine which of the following are ordinary differential equations and which are partial differential equations:
(a) xy' + v = 3
(b) /" + 2(y")2 + y = cos x
(/Cx) d2z + d2z = X2 +y
Ix-2 8?
I Equations (a) and (b) are ordinary differential equations because they contain only ordinary derivatives; (c) is
a partial differential equation because it contains at least one partial derivative.
1.3 Determine which of the following are ordinary differential equations and which are partial differential equations:
dx
(a) -dfx- = 5x + 3
4-d34v + (sinxd)2-y4+5xv =
(c)
dxi dx1
I All three equations are ordinary differential equations because each contains only ordinary derivatives.
1.4 Determine which of the following are ordinary differential equations and which are partial differential equations:
d2y\3 (dy\ Jdy\,2
„
d2y d2y
(c) xy2 + 3xy — 2x3y = 1
I Equation (a) is an ordinary differential equation, while (b) is a partial differential equation. Equation (c) is
neither, since it contains no derivatives, it is not a differential equation of any type. It is an algebraic equation
in x and y.
1.5 Determine which of the following are ordinary differential equations and which are partial differential equations:
(a) (sinx)y2 + 2y = 3x3 — 5
(b) exy - 2x + 3y2 =
(c) (2x-5y)2 = 6
f None of these equations is a differential equation, either ordinary or partial, because none of them involves
derivatives.
.
CHAPTER
2 1
1.6 Determine which ofthe following are ordinary differential equations and which are partial differential equations:
dy
ax
(b) (y")2 + (y')3 + 3y = x2
I Both are ordinary differential equations because each contains only ordinary derivatives.
1.7 Define order for an ordinary differential equation.
# The order ofa differential equation is the order ofthe highest derivative appearing in the equation.
1.8 Define degree for an ordinary differential equation.
I If an ordinary differential equation can be written as a polynomial in the unknown function and its
derivatives, then its degree is the power to which the highest-order derivative is raised.
1.9 Define linearity for an ordinary differential equation.
I An nth-order ordinary differential equation in the unknown function y and the independent variable x is
linear if it has the form
bJLx)^d"v + b„-,(x) jd"~^xv + + 6,(x) -dv£ + b (x)y = g(x)
The functions bj(x) (j — 0, 1,2, ... ,n) and g(x) are presumed known and depend only on the variable x.
Differentiil equations that cannot be put into this form are nonlinear.
1.10 Determine the order, degree, linearity, unknown function, and independent variable of the ordinary differential
equation y" — 5xy' — ex + 1.
I Second order: the highest derivative is the second. The unknown function is y, and the independent variable
is x. First degree: the equation is written as a polynomial in the unknown function y and its derivatives,
with the highest derivative (here the second) raised to the first power. Linear: in the notation of Problem 1.9,
b2(x)=l, b1(x)=-5x, bo(x) = 0, and g(x) = e*+l.
1.11 Determine the order, degree, linearity, unknown function, and independent variable of the ordinary differential
equation y'" — 5xy' — ex + 1.
I Third order: the highest derivative is the third. The unknown function is y, and the independent variable is
x. First degree: the equation is a polynomial in the unknown function y and its derivatives, with its highest
derivative (here the third) raised to the first power. Linear: in the notation of Problem 1.9. b3(x) = 1,
6,(x) = -5x, b2(x) = b (x) = 0, and g(x) = e* + 1.
1.12 Determine the order, degree, linearity, unknown function, and independent variable of the differential equation
y - 5xy' = ex + 1
I First order: the highest derivative is the first. The unknown function is y. and the independent variable is
x. First degree: the equation is a polynomial in the unknown function y and its derivative, with its highest
derivative (here the first) raised to the first power. Linear: in the notation of Problem 1.9, b^x) = —5x.
b (x) = 1, and #(x) = ex + 1.
1.13 Determine the order, degree, linearity, unknown function, and independent variable of the differential equation
y-5x(y')2 = e*+l.
t First order: the highest derivative is the first. The unknown function is y, and the independent variable is x.
Second degree: the equation is a polynomial in the unknown function y and its derivative, with its highest
derivative (here the first) raised to the second power. Nonlinear: the derivative of the unknown function is
raised to a power other than the first.
1.14 Determine the order, degree, linearity, unknown function, and independent variable of the differential equation
y - 5x(y')4 = ex + 1.
I First order: the highest derivative is the first. The unknown function is y. and the independent variable is x.
Fourth degree: the equation is a polynomial in the unknown function y and its derivative, with its highest