Differential Equations: Part I

227 Pages·1965·12.047 MB·English

by  L. W. F. Elen M.Sc. (auth.)

DIFFERENTIAL EQUATIONS
PART I
I8s.
Differential
Equations
Part I
L.W. F. ELEN,
M.Sc.
Principal Lecturer in Mathematics
West Ham College 0/ Technology
Macmillan Education
ISBN 978-0-333-09384-9 ISBN 978-1-349-86213-9 (eBook)
DOll 0.1 007/978-1-349-86213-9
Copyright © L. W. F. Elen 1965
Reprint of the original edition 1965
MACMILLAN AND COMPANY LIMITED
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also Bombay Calcutta Madras Melbourne
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Preface
This book aims to present the more elementary parts of the
subject and although no attempt is made to cover the syllabus
for any particular examination the work should be sufficient for
the differential equation requirements for H.N.C., H.N.D. and
Diploma in Technology in Engineering or Science and also for
London University B.Sc. (General) Parts land Hand B.Sc.
(Eng.) Parts land H.
In compiling this book consideration has been given to the
O.E.C.D. report on M athematics jor Physicists and Engineers
and also to the Mathematical Association syllabuses for the
Diploma in Mathematics (Technology).
The author wishes to thank the Senate of the University of
London for permission to make use of examination questions.
The answers to such questions are, of course, the responsibility
of the author and not the University.
L. W. F. ELEN
Contents
1 Introduction
Classification of differential equations. Formation of
differential equations. Arbitrary constants and the
order of a differential equation.
2 Ordinary first order differential equations 8
Equations with variables separable. Homogeneous
equations. Linear equations. Bernoulli's equation.
Exact equations. Equations of higher degree. Clai
raut's form. Singular solutions.
3 Applications of first order differential equations 41
Variables separable. Motion under a resistance. Elec-
trical applieations. Geometrical applications.
4 Ordinary differential equations of the second and 65
higher orders
Equations with constant coefficients. Complementary
function and particular integral. Determination of
particular integrals. The operator D. Equations of
higher order. Simultaneous equations. Homogeneous
equations. Miscellaneous methods. Equations of
higher degree.
5 The Laplace Transform 113
Transforms of basic functions. Inversion. Solution of
differential equations.
6 Applications of differential equations of the second 127
and higher orders
Simple harmonie motion. Damped oscillations. Forced
oscillations. Beams, struts and whirling shafts. Elec
trical circuits.
viii CONTENTS
7 Solution of second order differential equations by series 161
Simple power series. Frobenius' method. Coefficient
indeterminate. Roots of indicial equation equal.
Coefficients infinite.
S Partial differential equations 181
Formation. Part of solution known. Variables sepa
rable. Solution by Fourier series. Lagrange's linear
equation. Homogeneous equations of the second order.
Notes and formulae 205
Index 219
Introduction
§ 1.1 CLASSIFICATION OF DIFFERENTIAL EQUATIONS
Equations such as
dy = kx (1)
dx
d3y _ 2 dy + Y = eX (2)
dx3 dx
d2 (dY)2 ]
-y= [ 1 + - % (3)
d-r2 dx
02V 02V
ox2 + oy2 = 0 (+)
02y _ 202y
(5)
ot2 - ox2
C
involving derivatives are called differential equations.
Nos. (1), (2) and (3) are called ordinary differential equations
and they involve only one independent variable whilst Nos. (4)
and (5) are called partial differential equations and they involve
more than one independent variable.
The order of the equation is determined by the highest
derivative. No. (1) is first order, No. (2) is third order and Nos.
(3), (4), (5) are second order.
The degree of an equation is the degree of the highest deri
vative when the equation has been made rational and integral
as far as the derivatives and dependent variable are concemed.
All the above equations are of the first degree except (3) which
is of the second degree since it can be written
2 DIFFERENTIAL EQUATIONS
A linear differential equation is one which is linear in the
dependent variable and all its derivatives.
Nos. (1), (2), (4) and (5) are linear equations but (3) is non
linear.
d2 (d )2
y y
Ydx2+ dx =0
is another example of a non-linear equation.
§ 1.2 FORMATION OF DIFFERENTIAL EQUATIONS
In practice, differential equations arise naturally from prob
lems in mathematics, engineering and science. For example the
equation
d2x dx 2
dt2 + k dt + n x = 0
occurs in problems on damped oscillations while the equation
L ~:f + R ~; + ~ = f(t)
occurs when an inductance, resistance and capacitance are
connected in aseries circuit.
The solution of partial differential equations such as Lap
lace's Equation (fJ2Vjox2) + (02Vjoy2) = 0 which arises in
electrical problems and problems of fluid flow and the Wave
Equation (02yjot2) = C2(02yjoX2) arising from all forms of wave
motion-present much more difficult problems because each
equation possesses different types of solutions which depend on
the particular practical problem under consideration. Simple
solutions are discussed in Chapter 8 and a more extensive treat
ment is given in Volume H.
Many differential equations cannot be solved in exact terms
and we then have to make use of graphical and numerical
methods which are also discussed in Volume 11.
Although artificial from a practical point of view, it is helpful
initially to derive equations by eliminating constants since we
obtain some idea of the forms differential equations can take.
EXAMPLE 1 Form the differential equation by eliminating
the constants A and B from the expression
y = Ae2x + Be-3X•
INTRODUCTION 3
Since y = Ae2x + Be-3x (1)
then dy = 2Ae2~' _ 3Be-3x (2)
dx
d2
.2 = 4Ae2X + 9Be-3x (3)
dx2
We require three equations to eliminate two arbitrary constants
and consequently the resulting differential equation will be
second order.
Eliminating B from (1) and (2)
3y + dy = 5Ae2x (4)
dx
and eliminating B from (2) and (3)
3 dy + d2y = lOAe2x (5)
dx dx2
Hence eliminating A from (4) and (5) gives the differential
equation
d2y + dy _ 6y = 0
dx2 dx
Conversely it would seem that a second-order differential
equation requires two arbitrary constants in its solution.
The result can be expressed in general terms and is proved in
the next seetion.
EXAMPLE 2 Form the differential equation by eliminating the
constant from
y = Cx2 + C2.
dy
Different iat ing - = 2Cx
dx
Eliminating C gives
(~r + 2x3 ~ - 4x2y = 0
EXAMPLE 3 Find the differential equation which represents
(a) All circ1es of radius a
(b) All circ1es.