Table Of Contenta 's
n
K r i s h
TEXT BOOK on
Differential Equations
& Integral Transforms
(For B.A. and B.Sc. IInd year students of All Colleges affiliated to universities in Uttar Pradesh)
As per U.P. UNIFIED Syllabus
(w.e.f. 2012-2013)
By
A. R. Vasishtha
Retd. Head, Dep’t. of Mathematics
Meerut College, Meerut (U.P.)
KRISHNA Prakashan Media (P) Ltd.
KRISHNA HOUSE, 11, Shivaji Road, Meerut-250 001 (U.P.), India
Jai Shri Radhey Shyam
Dedicated
to
Lord
Krishna
Authors & Publishers
Preface
This book on DIFFERENTIAL EQUATIONS & INTEGRAL TRANSFORMS has
been specially written according to the latest Unified Syllabus to meet the requirements
of the B.A. and B.Sc. Part-II Students of all Universities in Uttar Pradesh.
The subject matter has been discussed in such a simple way that the students will find
no difficulty to understand it. The proofs of various theorems and examples have been
given with minute details. Each chapter of this book contains complete theory and a
fairly large number of solved examples. Sufficient problems have also been selected from
various university examination papers. At the end of each chapter an exercise containing
objective questions has been given.
We have tried our best to keep the book free from misprints. The authors shall be
grateful to the readers who point out errors and omissions which, inspite of all care, might
have been there.
The authors, in general, hope that the present book will be warmly received by the
students and teachers. We shall indeed be very thankful to our colleagues for their
recommending this book to their students.
The authors wish to express their thanks to Mr. S.K. Rastogi, Managing Director, Mr.
Sugam Rastogi, Executive Director, Mrs. Kanupriya Rastogi Director and entire team of
KRISHNA Prakashan Media (P) Ltd., Meerut for bringing out this book in the present
nice form.
The authors will feel amply rewarded if the book serves the purpose for which it is
meant. Suggestions for the improvement of the book are always welcome.
Preface to the Revised Edition
The authors feel great pleasure in presenting the thoroughly revised edition of the
book DIFFERENTIAL EQUATIONS & INTEGRAL TRANSFORMS and wish to
record thanks to the teachers and students for their warm reception to the previous
edition.
The present edition has been specially designed, made up-to-date and well organised
in a systematic order according to the latest syllabus.
The authors have always endeavoured to keep the text update in the best interests of
the students community- a gesture which the authors hope would be appreciated by the
students and teachers alike.
Suggestions for the improvement of the book will be thankfully received.
— Authors
Syllabus
Differential Equations & Integral Transforms
U.P. UNIFIED (w.e.f. 2012-13)
B.A./B.Sc. Paper-II M.M. : 33 / 65
Differential Equations
Unit-1: Formation of a differential equation (D.E.), Degree, order and solution of a D.E.,
Equations of first order and first degree : Separation of variables method, Solution of
homogeneous equations, linear equations and exact equations, Linear differential
equations with constant coefficients, Homogeneous linear differential equations,
Unit-2: Differential equations of the first order but not of the first degree, Clairaut's
equations and singular solutions, Orthogonal trajectories, Simultaneous linear
differential equations with constant coefficients, Linear differential equations of the
second order (including the method of variation of parameters),
Unit-3: Series solutions of second order differential equations, Legendre and Bessel
functions (P and J only) and their properties. Order, degree and formation of partial
n n
differential equations, Partial differential equations of the first order, Lagrange's
equations, Charpit's general method, Linear partial differential equations with constant
coefficients.
Unit-4(i): Partial differential equations of the second order, Monge's method.
Integral Transforms
Unit-4(ii): The concept of transform, Integral transforms and kernel, Linearity property
of transforms, Laplace transform, Inverse Laplace transform, Convolution theorem,
Applications of Laplace transform to solve ordinary differential equations.
Unit-5: Fourier transforms (finite and infinite), Fourier integral, Applications of Fourier
transform to boundary value problems, Fourier series.
B C
rief ontents
Dedication.........................................................................(v)
Preface ...........................................................................(vi)
Syllabus ........................................................................(vii)
Brief Contents ...............................................................(viii)
Section-A: Differential Equation .............................D-01—D-398
1. Differential Equations of First Order and First Degree...................................D-03—D-46
2. Differential Equations of the First Order but not of the First Degree...........D-47—D-72
3. Orthogonal Trajectories...........................................................................................D-73—D-82
4. Linear Differential Equations with Constant Coefficients............................D-83—D-118
5. Homogeneous Linear Differential Equations..................................................D-119—D-130
6. Ordinary Simultaneous Differential Equations..............................................D-131—D-152
7. Linear Equations of Second Order with Variable Coefficients.................D-153—D-194
8. Partial Differential Equations of the First Order..........................................D-195—D-246
9. Linear Partial Differential Equations of Second and Higher Order ..................................
with Constant Coefficients................................................................................D-247—D-286
10. Partial Differential Equations of Second Order with Variable ...........................................
Coefficients...........................................................................................................D-287—D-306
11. Monge's Method...................................................................................................D-307—D-324
12. Series Solutions of Differential Equations....................................................D-325—D-348
13. Legendre's Functions.........................................................................................D-349—D-376
14. Bessel’s Functions..............................................................................................D-377—D-398
Section-B: Integral Transforms ...............................T-01—T-192
1. The Laplace Transform...............................................................................................T-03—T-42
2. The Inverse Laplace Transform...............................................................................T-43—T-76
3. Applications of Laplace Transform.......................................................................T-77—T-104
4. Fourier Transforms.................................................................................................T-105—T-134
5. Finite Fourier Transforms......................................................................................T-135—T-148
6. Applications of Fourier Transforms in Initial and Boundary .............................................
Value Problems.........................................................................................................T-149—T-168
7. Fourier Series............................................................................................................T-169—T-192
(cid:83)(cid:69)(cid:67)(cid:84)(cid:73)(cid:79)(cid:78)
(cid:65)
(cid:68)(cid:68)(cid:73)(cid:73)(cid:70)(cid:70)(cid:70)(cid:70)(cid:69)(cid:69)(cid:82)(cid:82)(cid:69)(cid:69)(cid:78)(cid:78)(cid:84)(cid:84)(cid:73)(cid:73)(cid:65)(cid:65)(cid:76)(cid:76)(cid:32)(cid:32)(cid:69)(cid:69)(cid:81)(cid:81)(cid:85)(cid:85)(cid:65)(cid:65)(cid:84)(cid:84)(cid:73)(cid:73)(cid:79)(cid:79)(cid:78)(cid:78)(cid:83)(cid:83)
(cid:67)
(cid:104)(cid:97)(cid:112)(cid:116)(cid:101)(cid:114)(cid:115)
(cid:68)(cid:105)(cid:102)(cid:102)(cid:101)(cid:114)(cid:101)(cid:110)(cid:116)(cid:105)(cid:97)(cid:108)(cid:32)(cid:69)(cid:113)(cid:117)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:115)(cid:32)(cid:111)(cid:102)(cid:32)(cid:70)(cid:105)(cid:114)(cid:115)(cid:116)(cid:32)(cid:79)(cid:114)(cid:100)(cid:101)(cid:114)
(cid:49)(cid:49)(cid:46)(cid:46)
(cid:65)(cid:110)(cid:100)(cid:32)(cid:70)(cid:105)(cid:114)(cid:115)(cid:116)(cid:32)(cid:68)(cid:101)(cid:103)(cid:114)(cid:101)(cid:101)
(cid:68)(cid:105)(cid:102)(cid:102)(cid:101)(cid:114)(cid:101)(cid:110)(cid:116)(cid:105)(cid:97)(cid:108)(cid:32)(cid:69)(cid:113)(cid:117)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:115)(cid:32)(cid:111)(cid:102)(cid:32)(cid:84)(cid:104)(cid:101)(cid:32)(cid:70)(cid:105)(cid:114)(cid:115)(cid:116)(cid:32)
(cid:50)(cid:46)
(cid:79)(cid:114)(cid:100)(cid:101)(cid:114)(cid:32)(cid:66)(cid:117)(cid:116)(cid:32)(cid:78)(cid:111)(cid:116)(cid:32)(cid:111)(cid:102)(cid:32)(cid:84)(cid:104)(cid:101)(cid:32)(cid:70)(cid:105)(cid:114)(cid:115)(cid:116)(cid:32)(cid:68)(cid:101)(cid:103)(cid:114)(cid:101)(cid:101)
(cid:49)(cid:49)(cid:46)(cid:46)
(cid:51)(cid:46) (cid:79)(cid:114)(cid:116)(cid:104)(cid:111)(cid:103)(cid:111)(cid:110)(cid:97)(cid:108)(cid:32)(cid:84)(cid:114)(cid:97)(cid:106)(cid:101)(cid:99)(cid:116)(cid:111)(cid:114)(cid:105)(cid:101)(cid:115)
(cid:76)(cid:105)(cid:110)(cid:101)(cid:97)(cid:114)(cid:32)(cid:68)(cid:105)(cid:102)(cid:102)(cid:101)(cid:114)(cid:101)(cid:110)(cid:116)(cid:105)(cid:97)(cid:108)(cid:32)(cid:69)(cid:113)(cid:117)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:115)(cid:32)(cid:87)(cid:105)(cid:116)(cid:104)
(cid:52)(cid:46)
(cid:67)(cid:111)(cid:110)(cid:115)(cid:116)(cid:97)(cid:110)(cid:116)(cid:32)(cid:67)(cid:111)(cid:101)(cid:102)(cid:102)(cid:105)(cid:99)(cid:105)(cid:101)(cid:110)(cid:116)(cid:115)
(cid:72)(cid:111)(cid:109)(cid:111)(cid:103)(cid:101)(cid:110)(cid:101)(cid:111)(cid:117)(cid:115)(cid:32)(cid:76)(cid:105)(cid:110)(cid:101)(cid:97)(cid:114)(cid:32)(cid:68)(cid:105)(cid:102)(cid:102)(cid:101)(cid:114)(cid:101)(cid:110)(cid:116)(cid:105)(cid:97)(cid:108)
(cid:53)(cid:46)
(cid:69)(cid:113)(cid:117)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:115)
(cid:79)(cid:114)(cid:100)(cid:105)(cid:110)(cid:97)(cid:114)(cid:121)(cid:32)(cid:83)(cid:105)(cid:109)(cid:117)(cid:108)(cid:116)(cid:97)(cid:110)(cid:101)(cid:111)(cid:117)(cid:115)(cid:32)(cid:68)(cid:105)(cid:102)(cid:102)(cid:101)(cid:114)(cid:101)(cid:110)(cid:116)(cid:105)(cid:97)(cid:108)
(cid:54)(cid:46)
(cid:69)(cid:113)(cid:117)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:115)
(cid:76)(cid:105)(cid:110)(cid:101)(cid:97)(cid:114)(cid:32)(cid:69)(cid:113)(cid:117)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:115)(cid:32)(cid:111)(cid:102)(cid:32)(cid:83)(cid:101)(cid:99)(cid:111)(cid:110)(cid:100)(cid:32)(cid:79)(cid:114)(cid:100)(cid:101)(cid:114)
(cid:55)(cid:46)
(cid:87)(cid:105)(cid:116)(cid:104)(cid:32)(cid:86)(cid:97)(cid:114)(cid:105)(cid:97)(cid:98)(cid:108)(cid:101)(cid:32)(cid:67)(cid:111)(cid:101)(cid:102)(cid:102)(cid:105)(cid:99)(cid:105)(cid:101)(cid:110)(cid:116)(cid:115)
(cid:80)(cid:97)(cid:114)(cid:116)(cid:105)(cid:97)(cid:108)(cid:32)(cid:68)(cid:105)(cid:102)(cid:102)(cid:101)(cid:114)(cid:101)(cid:110)(cid:116)(cid:105)(cid:97)(cid:108)(cid:32)(cid:69)(cid:113)(cid:117)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:115)(cid:32)(cid:111)(cid:102)
(cid:56)(cid:46)
(cid:84)(cid:104)(cid:101)(cid:32)(cid:70)(cid:105)(cid:114)(cid:115)(cid:116)(cid:32)(cid:79)(cid:114)(cid:100)(cid:101)(cid:114)
(cid:76)(cid:105)(cid:110)(cid:101)(cid:97)(cid:114)(cid:32)(cid:80)(cid:97)(cid:114)(cid:116)(cid:105)(cid:97)(cid:108)(cid:32)(cid:68)(cid:105)(cid:102)(cid:102)(cid:101)(cid:114)(cid:101)(cid:110)(cid:116)(cid:105)(cid:97)(cid:108)(cid:32)(cid:69)(cid:113)(cid:117)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:32)(cid:111)(cid:102)
(cid:57)(cid:46) (cid:83)(cid:101)(cid:99)(cid:111)(cid:110)(cid:100)(cid:32)(cid:97)(cid:110)(cid:100)(cid:32)(cid:72)(cid:105)(cid:103)(cid:104)(cid:101)(cid:114)(cid:32)(cid:79)(cid:114)(cid:100)(cid:101)(cid:114)(cid:32)(cid:119)(cid:105)(cid:116)(cid:104)
(cid:67)(cid:111)(cid:110)(cid:115)(cid:116)(cid:97)(cid:110)(cid:116)(cid:32)(cid:67)(cid:111)(cid:101)(cid:102)(cid:102)(cid:105)(cid:99)(cid:105)(cid:101)(cid:110)(cid:116)(cid:115)
(cid:80)(cid:97)(cid:114)(cid:116)(cid:105)(cid:97)(cid:108)(cid:32)(cid:68)(cid:105)(cid:102)(cid:102)(cid:101)(cid:114)(cid:101)(cid:110)(cid:116)(cid:105)(cid:97)(cid:108)(cid:32)(cid:69)(cid:113)(cid:117)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:115)(cid:32)(cid:111)(cid:102)(cid:32)(cid:83)(cid:101)(cid:99)(cid:111)(cid:110)(cid:100)
(cid:49)(cid:48)(cid:46)
(cid:79)(cid:114)(cid:100)(cid:101)(cid:114)(cid:32)(cid:119)(cid:105)(cid:116)(cid:104)(cid:32)(cid:86)(cid:97)(cid:114)(cid:105)(cid:97)(cid:98)(cid:108)(cid:101)(cid:32)(cid:67)(cid:111)(cid:101)(cid:102)(cid:102)(cid:105)(cid:99)(cid:105)(cid:101)(cid:110)(cid:116)(cid:115)
(cid:49)(cid:49)(cid:46) (cid:77)(cid:111)(cid:110)(cid:103)(cid:101)(cid:39)(cid:115)(cid:32)(cid:77)(cid:101)(cid:116)(cid:104)(cid:111)(cid:100)
(cid:49)(cid:50)(cid:46) (cid:83)(cid:101)(cid:114)(cid:105)(cid:101)(cid:115)(cid:32)(cid:83)(cid:111)(cid:108)(cid:117)(cid:116)(cid:105)(cid:111)(cid:110)(cid:115)(cid:32)(cid:111)(cid:102)(cid:32)(cid:68)(cid:105)(cid:102)(cid:102)(cid:101)(cid:114)(cid:101)(cid:110)(cid:116)(cid:105)(cid:97)(cid:108)(cid:32)(cid:69)(cid:113)(cid:117)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:115)
(cid:49)(cid:51)(cid:46) (cid:76)(cid:101)(cid:103)(cid:101)(cid:110)(cid:100)(cid:114)(cid:101)(cid:39)(cid:115)(cid:32)(cid:70)(cid:117)(cid:110)(cid:99)(cid:116)(cid:105)(cid:111)(cid:110)(cid:115)
(cid:49)(cid:52)(cid:46) (cid:66)(cid:101)(cid:115)(cid:115)(cid:101)(cid:108)(cid:39)(cid:115)(cid:32)(cid:70)(cid:117)(cid:110)(cid:99)(cid:116)(cid:105)(cid:111)(cid:110)(cid:115)
D-3
1
D E
ifferential quations of
F O F D
irst rder and irst egree
1.1 Definitions
A differential equation is an equation containing the dependent and independent
variables and different derivatives of the dependent variables w.r.t. one or more
independent variables.
The order of a differential equation is the order of the highest derivative (or
differential coefficient) occurring in the equation.
(Lucknow 2007; Meerut 09B, 10B; Bundelkhand 10)
The degree of a differential equation is the degree of the highest derivative (or diff.
coeff.) which occurs in it, after the differential equation has been rationalized (i.e.,
made free from radicals and fractions so far as derivatives are concerned). A
differential equation is called ordinary, if the unknown function depends on only one
argument (independent variable).
(Lucknow 2007; Meerut 09B, 10B; Bundelkhand 10)
A differential equation is said to be partial if there are two or more independent
variables.
A differential equation is said to be linear if the dependent variable, say, ‘y ' and all its
derivatives occur in the first degree, otherwise it is non-linear.
D-4
A function y = f (x) is called a Solution (or the primitive) of a differential equation if,
when substituted into the equation, it reduces the equation to an identity and the
process of finding all the solutions is called integrating (or solving) the differential
equation.
General solution: (Lucknow 2007; Gorakhpur 09)
A solution of a differential equation, containing independent arbitrary constants
equal in number to the order of the differential equation is called its general solution.
Particular solution: (Lucknow 2007)
A solution obtained by giving particular values to the arbitrary constants in the
general solution is called a particular solution or particular integral.
Arbitrary Constants: The solution of a differential equation may contain as many
arbitrary constants as is the order of the differential equation i.e., the solution of an nth
order differential equation may contain n arbitrary constants.
Ex am ple 1: Find the diff er ent ial equa tion of the fami ly of curves y = Aex +(B/ex), for dif fer ent
val ues of A and B.
So lu tion: We have y = Aex + Be-x. …(1)
To obtain the required differential equation the constants A and B are to be
eliminated with the help of the given equation (1) and the two equations obtained by
differentiating (1) once and twice. Thus differentiating (1), we get
dy
= Aex - Be-x. …(2)
dx
Now differentiating (2), we get
d2 y
= Aex + Be-x. …(3)
dx2
d2y
Eliminating A and B between (1), (2) and (3), we obtain = y, which is the
dx2
required differential equation.
Ex am ple 2: Find the dif fer ent ial equa tion of all cir cles of ra dius a, or By the elim i nat ion of the
con stants h and k, find the diff er ent ial equa tion of which (x- h)2 +(y - k)2 = a2, is a solution.
So lu tion: The equa tion of all cir cles of rad ius a is given by
(x- h)2 +(y - k)2 = a2, …(1)
h and k being parameters (i.e., arbitrary constants).
Differentiating (1), we get
D-5
dy
2(x- h)+2(y - k) =0. …(2)
dx
Differentiating (2), we get
d2y (cid:230)dy(cid:246)2
1+(y - k) + (cid:231) (cid:247) =0. …(3)
dx2 Łdxł
From (2) and (3), we obtain
[1+(dy/dx)2]
(x- h)= -(y - k)(dy/dx) and (y - k)= - .
d2 y/dx2
Substituting these values in (1) and simplifying, we obtain
غŒŒ1+ (cid:230)Ł(cid:231)ddyx(cid:246)ł(cid:247)2øßœœ3 = a2(cid:230)Ł(cid:231)(cid:231)dd2x2y(cid:246)ł(cid:247)(cid:247)2,
which is the required differential equation.
C E
omprehensive xercise 1
1. Find the differential equation of the family of curves
y = Ae2x + Be-2x,
for different values of A and B. (Bundelkhand 2003)
2. Find the differential equation corresponding to
y = ae2x + be-3x + cex,
where a,b,c are arbitrary constants.
3. Find the differential equation of the family of curves
y = ex (Acos x+ Bsin x).
where A and B are arbitrary constants.
4. By eliminating the constants a and b obtain the differential equation of which
xy = aex + be-x + x2 is a solution.
(Purvanchal 2014)
5. Find the differential equation corresponding to the family of curves
y = c(x- c)2, where c is an arbitrary constant.
6. Show that Ax2 + By2 =1 is the solution of
Ø d2y (cid:230)dy(cid:246)2ø dy
xŒy + (cid:231) (cid:247) œ - y =0.
ºŒ dx2 Łdxł ßœ dx
7. Show that v=(a/r)+ B is a solution of
d2v 2dv
+ =0.
dr2 r dr
