Engineering mathematics II

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Copyright © 2009, New Age International (P) Ltd., Publishers
Published by New Age International (P) Ltd., Publishers
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Preface
This Textbook has been prepared as per the syllabus for the Engineering Mathematics Second
semester B.E classes of Visveswaraiah Technological University. The book contains eight chapters,
and each chapter corresponds to one unit of the syllabus. The topics covered are: Unit I and II—
Differential Calculus, Unit III and IV—Integral Calculus and Vector Integration, Unit V and VI—
Differential Equations and Unit VII and VIII—Laplace Transforms.

It gives us a great pleasure in presenting this book. In this edition, the modifications have
been dictated by the changes in the VTU syllabus. The main consideration in writing the book
was to present the considerable requirements of the syllabus in as simple manner as possible. This
will help students gain confidence in problem-solving.
Each unit treated in a systematic and logical presentation of solved examples is followed by
an exercise section and includes latest model question papers with answers from an integral part
of the text in which students will get enough questions for practice.
The book is designed as self-contained, comprehensive and friendly from students’ point of
view. Both theory and problems have been explained by using elegant diagrams wherever necessary.
We are grateful to New Age International (P) Limited, Publishers and the editorial department
for their commitment and encouragement in bringing out this book within a short span of period.
AUTHORS


Acknowledgement
It gives us a great pleasure to present this book ENGINEERING MATHEMATICS-II as per the
latest syllabus and question pattern of VTU effective from 2008-2009.
Let us take this opportunity to thank one and all who have actually given me all kinds of
support directly and indirectly for bringing up my textbook.
We whole heartedly thank our Chairman Mr. S. Narasaraju Garu, Executive Director, Mr.
S. Ramesh Raju Garu, Director Prof. Basavaraju, Principal Dr. T. Krishnan, HOD Dr. K. Mallikarjun,
Dr. P.V.K. Perumal, Dr. M. Surekha, The Oxford College of Engineering, Bangalore. We would like
to thank the other members of our Department, Prof. K. Bharathi, Prof. G. Padhmasudha,
Mr. Ravikumar, Mr. Sivashankar and other staffs of The Oxford College of Engineering, Bangalore
for the assistance they provided at all levels for bringing out this textbook successfully.
We must acknowledge Prof. M. Govindaiah, Principal, Prof. K.V. Narayana, Reader,
Department of Mathematics, Vivekananda First Grade Degree College, Bangalore are the ones
who truly made a difference in our life and inspired us a lot.
We must acknowledge HOD Prof. K. Rangasamy, Mr. C. Rangaraju, Dr. S. Murthy, Department
of Mathematics, Govt. Arts College (Men), Krishnagiri.

We are also grateful to Dr. A.V. Satyanarayana, Vice-Principal of R.L. Jalappa Institute of
Technology, Doddaballapur, Prof. A.S. Hariprasad, Sai Vidya Institute of Technology, Prof. V.K.
Ravi, Mr. T. Saravanan, Bangalore college of Engineering and Technology, Prof. L. Satish, Raja
Rajeshwari College of Engineering, Prof. M.R. Ramesh, S.S.E.T., Bangalore.
A very special thanks goes out to Mr. K.R. Venkataraj and Bros., our well wisher friend
Mr. N. Aswathanarayana Setty, Mr. D. Srinivas Murthy without whose motivation and
encouragement this could not have been completed.
We express our sincere gratitude to Managing Director, New Age International (P) Limited,
and Bangalore Division Marketing Manager Mr. Sudharshan for their suggestions and provisions
of the font materials evaluated in this study.
We would also like to thank our friends and students for exchanges of knowledge, skills
during our course of time writing this book.
AUTHORS


Dedicated to
my dear parents,
Shiridi Sai Baba,
my dear loving son Monish Sri Sai G
and my wife and best friend S. Mamatha
— A . Ganesh

&
my dear parents,
and my wife S. Geetha
— G. Balasubramanian


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QUESTION PAPER LAYOUT
Engineering Mathematics-II
O6MAT21
Units-1, 2, 3, 4

PART-A

4 Qns.

1 Qn. from each unit

PART-B

4 Qns.

Units-5, 6, 7, 8

1 Qn. from each unit

To answer fivefull questions choosing at leasttwo questions from each part

Time: 3 Hrs.

Unit/Qn. No.
1.

Max. Marks: 100

Topics
DIFFERENTIAL CALCULUS-I

Unit/Qn. No.
5.

Topics
DIFFERENTIAL EQUATIONS-I
Linear differential equation with
constant coefficients, Solution of
homogeneous and non homogeneous linear D.E., Inverse differential
operator and the Particular Integral
(P.I.)

Radius of Curvature: Cartesian
curve Parametric curve, Pedal
curve, Polar curve and some
fundamental theorems.

Method of undetermined coefficients.
2.

DIFFERENTIAL CALCULUS-II

6.

Taylor’s, Maclaurin’s Maxima and
Minima for a function of two
variables.

3.

INTEGRAL CALCULUS-II

Method of variation of parameters,
Solutions of Cauchy's homogeneous
linear equation and Legendre’s
linear equation, Solution of initial
and Boundary value problems.
7.

VECTOR INTEGRATION AND
ORTHOGONAL CURVILINEAR
COORDINATES

LAPLACE TRANSFORMS
Periodic function, Unit step function
(Heaviside function), Unit impulses
function.

Double and triple integral, Beta and
Gamma functions.
4.

DIFFERENTIAL EQUATIONS-II

8.

INVERSE LAPLACE TRANSFORMS

Applications of Laplace transforms.


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Contents
PREFACE

(v)

ACKNOWLEDGEMENT

(vi)

UNIT I Differential Calculus–I

1–60

1.1 Introduction
1.2 Radius of Curvature
1.2.1 Radius of Curvature in Cartesian Form
1.2.2 Radius of Curvature in Parametric Form
Worked Out Examples
Exercise 1.1
1.2.3 Radius of Curvature in Pedal Form
1.2.4 Radius of Curvature in Polar Form
Worked Out Examples

Exercise 1.2
1.3 Some Fundamental Theorem
1.3.1 Rolle’s Theorem
1.3.2 Lagrange’s Mean Value Theorem
1.3.3 Cauchy’s Mean Value Theorem
1.3.4 Taylor’s Theorem
Worked Out Examples
Exercise 1.3
Additional Problems (from Previous Years VTU Exams.)
Objective Questions

UNIT II Differential Calculus–II
2.1 Indeterminate Forms
2.1.1 Indeterminate Form

1
1
2
3
4
18
19
19
21
26
27
27
27
28
29

30
50
52
57

61–104
61

0
0

Worked Out Examples
Exercise 2.1
2.1.2 Indeterminate Forms ∞ – ∞ and 0 × ∞
Worked Out Examples
Exercises 2.2
2.1.3 Indeterminate Forms 00, 1∞, ∞0, 0∞
Worked Out Examples
Exercise 2.3
2.2 Taylor’s Theorem for Functions of Two Variables

61
62
68
69
69
74
74
74
77

78


(xii)

Worked Out Examples
Exercise 2.4
2.3 Maxima and Minima of Functions of Two Variables
2.3.1 Necessary and Sufficient Conditions for Maxima and Minima
Worked Out Examples
Exercise 2.5
2.4 Lagrange’s Method of Undetermined Multipliers
Working Rules
Worked Out Examples
Exercise 2.6
Additional Problems (from Previous Years VTU Exams.)
Objective Questions

UNIT III Integral Calculus

78
82
83
83
84
89
89
90
91
94

94
101

105–165

3.1 Introduction
3.2 Multiple Integrals
3.3 Double Integrals
Worked Out Examples
Exercise 3.1
3.3.1 Evaluation of a Double Integral by Changing the Order of Integration
3.3.2 Evaluation of a Double Integral by Change of Variables
3.3.3 Applications to Area and Volume
Worked Out Examples
Type 1. Evaluation over a given region
Type 2. Evaluation of a double integral by changing the order of integration
Type 3. Evaluation by changing into polars
Type 4. Applications of double and triple integrals
Exercise 3.2
3.4 Beta and Gamma Functions
3.4.1 Definitions
3.4.2 Properties of Beta and Gamma Functions
3.4.3 Relationship between Beta and Gamma functions
Worked Out Examples
Exercise 3.3
Additional Problems (From Previous Years VTU Exams.)
Objective Questions

UNIT IV Vector Integration and Orthogonal Curvilinear Coordinates
4.1 Introduction

4.2 Vector Integration
4.2.1 Vector Line Integral
Worked Out Examples
Exercise 4.1

105
105
105
106
112
113
113
113
114
114
119
122
124
128
129
129
129
133
135
156
159
162

166–213
166

166
166
166
172


(xiii)

4.3 Integral Theorem
4.3.1. Green’s Theorem in a Plane
4.3.2. Surface Integral and Volume Integral
4.3.3. Stoke’s Theorem
4.3.4. Gauss Divergence Theorem
Worked Out Examples
Exercise 4.2
4.4 Orthogonal Curvilinear Coordinates
4.4.1 Definition
4.4.2 Unit Tangent and Unit Normal Vectors
4.4.3. The Differential Operators
Worked Out Examples
Exercise 4.3
4.4.4. Divergence of a Vector
Worked out Examples
Exercise 4.4
4.4.5. Curl of a Vector
Worked Out Examples
Exercise 4.5
4.4.6. Expression for Laplacian ∇2ψ
4.4.7. Particular Coordinate System
Worked Out Examples

Exercise 4.6
Additional Problems (From Previous Years VTU Exams.)
Objective Questions

UNIT V Differential Equations–I
5.1 Introduction
5.2 Linear Differential Equations of Second and Higher Order with
Constant Coefficients
5.3 Solution of a Homogeneous Second Order Linear Differential Equation
Worked Out Examples
Exercise 5.1
5.4 Inverse Differential Operator and Particular Integral
5.5 Special Forms of x
Worked Out Examples
Exercise 5.2
Exercise 5.3
Exercise 5.4
5.6 Method of Undetermined Coefficients
Worked Out Examples
Exercise 5.5

173
173
173
174
174
174
188
189
189

189
191
192
194
194
195
196
196
197
198
199
199
203
208
208
210

214–279
214
214
215
215
219
220
221
224
236
241
251
251

252
262


(xiv)

5.7 Solution of Simultaneous Differential equations
Worked Out Examples
Exercise 5.6
Additional Problems (From Previous Years VTU Exams.)
Objective Questions

UNIT VI Differential Equations–I

280—320

6.1 Method of Variation of Parameters
Worked Out Examples
Exercise 6.1
6.2 Solution of Cauchy’s Homogeneous Linear Equation and Lengendre’s
Linear Equation
Worked Out Examples
Exercise 6.2
6.3 Solution of Initial and Boundary Value Problems
Worked Out Examples
Exercise 6.3
Additional Problems (From Previous Years VTU Exams.)
Objective Questions

UNIT VII Laplace Transforms

280
281
291
292
294
306
308
308
310
310
318

321—368

7.1 Introduction
7.2 Definition
7.3 Properties of Laplace Transforms
7.3.1 Laplace Transforms of Some Standard Functions
Worked out Examples
Exercise 7.1
7.3.2 Laplace Transforms of the form eat f (t)
Worked Out Examples
Exercise 7.2
7.3.3 Laplace Transforms of the form tn f (t) where n is a positive integer
7.3.4 Laplace Transforms of

264
265
267

268
277

=B

f t
t

Worked out Examples
Exercise 7.3
7.4 Laplace Transforms of Periodic Functions
Worked Out Examples
Exercise 7.3
7.5 Laplace Transforms of Unit Step Function and Unit Impulse Function
Unit Step Function (Heaviside Function)
7.5.1 Properties Associated with the Unit Step Function

321
321
321
322
325
330
331
332
335
336
337
337
345

346
347
351
352
352
352


(xv)

7.5.2 Laplace Transform of the Unit Impulse Function
Exercise 7.4
Additional Problems (From Previous Years VTU Exams.)
Objective Questions

354
359
360
365

UNIT VIII Inverse Laplace Transforms
8.1 Introduction
8.2 Inverse Laplace Transforms of Some Standard Functions
Worked Out Examples
8.3 Inverse Laplace Transforms using Partial Fractions
Exercise 8.1
8.4 Inverse Laplace Transforms of the Functions of the Form

8.5

8.6
8.7

8.8

Worked Out Example
Exercise 8.2
Convolution Theorem
Worked Out Examples
Exercise 8.3
Laplace Transforms of the Derivatives
Solution of Linear Differential Equations
Worked Out Examples
Solution of Simultaneous Differential Equations
Exercise 8.4
Applications of Laplace Transforms
Worked Out Examples
Exercise 8.5
Additional Problems (From Previous Years VTU Exams.)
Objective Questions

369–426

=B

Fs
s

369
369

372
376
382
384
384
387
388
389
396
397
398
398
406
410
411
412
416
417
422

Model Question Paper–I

427–440

Model Question Paper–II

441–447


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UNIT

1

Differential Calculus—I
1.1 INTRODUCTION
In many practical situations engineers and scientists come across problems which involve quantities
of varying nature. Calculus in general, and differential calculus in particular, provide the analyst with
several mathematical tools and techniques in studying how the functions involved in the problem
behave. The student may recall at this stage that the derivative, obtained through the basic operation
of calculus, called differentiation, measures the rate of change of the functions (dependent variable)
with respect to the independent variable. In this chapter we examine how the concept of the derivative
can be adopted in the study of curvedness or bending of curves.

1.2 RADIUS OF CURVATURE
Let P be any point on the curve C. Draw the tangent at P to the
circle. The circle having the same curvature as the curve at P
touching the curve at P, is called the circle of curvature. It is also
called the osculating circle. The centre of the circle of the curvature is called the centre of curvature. The radius of the circle
of curvature is called the radius of curvature and is denoted
by ‘ρ’.

Y

C
O

P

Note : 1. If k (> 0) is the curvature of a curve at P, then the radius

1
. This follows from the definition
k
of radius of curvature and the result that the curvature of a circle is the
reciprocal of its radius.

O

of curvature of the curve of ρ is

Note : 2. If for an arc of a curve, ψ decreases as s increases, then
But the radius of a circle is non-negative. So to take ρ =
i.e., k =

dψ
.
ds
1

X

Fig. 1.1

dψ
is negative, i.e., k is negative.
ds

1
ds
=
some authors regard k also as non-negative
dψ
k


2

ENGINEERING MATHEMATICS—II

dψ
indicates the convexity and concavity of the curve in the neighbourhood of
ds
ds
the point. Many authors take ρ =
and discard negative sign if computed value is negative.
dψ
The sign of

∴ Radius of curvature ρ =

1
·
k

1.2.1 Radius of Curvature in Cartesian Form
Suppose the Cartesian equation of the curve C is given by y = f (x) and A be a fixed point on it. Let

P(x, y) be a given point on C such that arc AP = s.
Then we know that

dy
= tan ψ
dx
where ψ is the angle made by the tangent to the curve C at P with the x-axis and
ds
=
dx

R|1 + FG dy IJ
S| H dx K
T

2

Differentiating (1) w.r.t x, we get

U|
V|
W

...(1)

1
2

...(2)

dψ
d2y
sec 2 ψ ⋅
2 =
dx
dx

d

2
= 1 + tan ψ

i ddsψ ⋅ dxds

LM1 + FG dy IJ OP 1 LM1 + FG dy IJ OP
MN H dx K PQ ρ MN H dx K PQ
1 R| F dy I U|
S1 + G J V
ρ |T H dx K |W
R|1 + FG dy IJ U|
S| H dx K V|
T
W

1
2 2

2

=

2

=

2

Therefore,

where y1 =

ρ=

dy
d2y
and y2 =
.
dx
dx 2

d2y
dx 2

[By using the (1) and (2)]

3
2

3
2

...(3)


3

DIFFERENTIAL CALCULUS—I

Equation (3) becomes,
ρ=

3
2 2
1

o1 + y t
y2

This is the Cartesian form of the radius of curvature of the curve y = f (x) at P (x, y) on it.

1.2.2 Radius of Curvature in Parametric Form
Let x = f (t) and y = g (t) be the Parametric equations of a curve C and P (x, y) be a given point
on it.
Then

dy dt
dy
= d x dt
dx

...(4)

RS
T

UV
W

d2y
d dy / dt dt
⋅
2 =
dt dx / dt dx
dx

and

dx d 2 y dy d 2 x
⋅
−
⋅
dt dt 2
dt dt 2 ⋅ 1
=
2
dx
dx
dt
dt

FG IJ
H K

dx d 2 y dy d 2 x
–
⋅
⋅
d2y
dt dt 2 dt dt 2
=
3
dx 2
dx
dt

FG IJ
H K

Substituting the values of

...(5)

dy
d2y
and
in the Cartesian form of the radius of curvature of the
dx
dx 2

curve y = f (x) [Eqn. (3)]

∴

ρ=

=

R|1 + FG dy IJ
o1 + y t = S|T H dx K
3
2 2
1

d2y
dx 2

y2

|RS1 + FG dy / dt IJ
|T H dx / dt K
2

2

2

3
2

3

2

3

2

R|FG dx IJ + FG dy IJ U|
S|H dt K H dt K V|
T
W
2

ρ=

U|
V|
W

|UV
|W
RS dx ⋅ d y – dy ⋅ d x UV / FG dx IJ
T dt dt dt dt W H dt K
2

∴

2

2

3
2

dx d 2 y dy d 2 x
⋅
−
⋅
dt dt 2
dt dt 2

...(6)


4

ENGINEERING MATHEMATICS—II
2
2
dx
dy
, y′ =
, x″ = d x , y″ = d y
2
dt
dt
dt
dt 2

where x′ =

ρ=

o

x′ 2 + y′ 2

t

3
2

x ′ y ″ – y′ x ″

This is the cartesian form of the radius of curvature in parametric form.

WORKED OUT EXAMPLES
1. Find the radius of curvature at any point on the curve y = a log sec
Solution
Radius of curvature ρ =
Here,

o

t

1 + y12

3
2

y2

y = a log sec

FG x IJ
H aK

1

y1 = a ×

sec

FG x IJ
H aK

⋅ sec

FG x IJ tan FG x IJ ⋅ 1
H aK H aK a

FG x IJ
H aK
F xI 1
= sec G J ⋅
H aK a

y 1 = tan
y2

2

RS1 + tan FG x IJ UV
H aKW
T
ρ =
1
F xI
sec G J
H aK
a
RSsec FG x IJ UV
T H aKW =
=
1
F xI
sec G J
H aK
a
3
2

2

Hence

2

32

2

2

∴

Radius of curvature = a sec

FG x IJ
H aK

FG x IJ
H aK
F xI
sec G J
H aK
F xI
= a sec G J
H aK
a sec 3
2

FG x IJ .
H aK


5

DIFFERENTIAL CALCULUS—I

FG x IJ , show that ρ =
H cK
d1 + y i
ρ =

2. For the curve y = c cos h

y2
·
c

3
2 2
1

Solution

y2

FG x IJ
H cK
F xI 1
F xI
y = c sin h G J × = sin h G J
H cK c
H cK
F xI 1
y = cos h G J ×
H cK c
RS1 + sin h FG x IJ UV c FG cos h x IJ

H cKW = H cK
T
ρ =
x
1
F xI
cos h
cos h G J
H cK
c
c
F xII
F xI 1 F
= c cos h G J = G c cos h G J J
H cKK
H cK c H
y = c cos h

Here,

1

and

2

3
2

2

2

3
2

2

2

=

1 2
⋅y
c

y2
ρ =
·
c

∴

Hence proved.

3. Find the radius of curvature at (1, –1) on the curve y = x2 – 3x + 1.

d1 + y i
ρ =

3
2 2
1

Solution. Where

y2

Here,
Now,

( y1)(1,
( y2)(1,

∴

ρ(1,

at (1, – 1)

y = x2 – 3x + 1
y 1 = 2x – 3, y2 = 2
–1) = – 1
–1) = 2

–1)

b1 + 1g
=

3
2

2

=

=

2 2
2

2

4. Find the radius of curvature at (a, 0) on y = x3 (x – a).

Solution. We have

ρ =

d

1 + y12
y2

i

3
2

at (a, 0)


6

ENGINEERING MATHEMATICS—II

y = x3(x – a) = x4 – x3a
y 1 = 4x3 – 3ax2
y 2 = 12x2 – 6ax
( y1)(a, 0) = 4a3 – 3a3 = a3
( y2)(a, 0) = 12a2 – 6a2 = 6a2

Here,
and
Now

RS1 + da i UV
T
W
=
2

3

∴

ρ(a,

0)

6a 2

o

1 + a6

=

6a

2

t

3
2

d

1 + y12

Solution. We have

ρ =

Here

y = a sec
y1

y1

and

y2

i

·
πa
on y = a sec
4

5. Find the radius of curvature at x =

∴

3
2

3
2

at x =

y2

πa
4

FG x IJ
H aK
F xI F xI 1
= a sec G J ⋅ tan G J ×
H aK H aK a
F xI F xI
= sec G J tan G J
H aK H aK
x 1
F xI F xI 1
× + sec G J ⋅ tan G J ⋅
= sec
H aK H aK a
a a
1L
F xI F xI F xIO
= Msec G J + sec G J tan G J P
H aK H aK H aKQ
aN
3

2

3

and y2 =

2

πa

π
π
, y = sec ⋅ tan = 2
1
4
4
4

At x =

e

j

1
3 2
2 2+ 2 =
a
a

RS1 + e 2 j UV
T
W
2

∴

ρ

x=

πa
4

FG x IJ .
H aK

=

=

3 2
a

3
a.
2

3
2

=

3 3
3 2

⋅a


7

DIFFERENTIAL CALCULUS—I

FG x IJ .
H 2K

π
on y = 2 log sin
3

6. Find ρ at x =

Solution. We have ρ =

d1 + y i

3
2 2
1

y2

y = 2 log sin

The curve is

at x =

FG x IJ
H 2K

1

× cos

sin

y2

At

x=

π
,y
3 1

and

2

y2 =

–1
π
cosec 2 = – 2
2
6

=

RS1 + e 3j UV
T
W

=

b1 + 3g

2

∴

ρ

x=

π
3

Solution. We have ρ =

3
2

–2
3
2

–2

=

o

1 + y12

t

y2

4×2
= – 4.
–2

FG 3a , 3a IJ
H 2 2K

7. Find the radius of curvature at

x3

FG x IJ × 1
H 2K 2

FG x IJ
H 2K
F xI
= cot G J
H 2K

F xI 1
= – cosec G J ×
H 2K 2
F πI
= cot G J = 3
H 6K

y1 = 2 ⋅

and

π
3

3
2

at

on x3 + y3 = 3axy.

FG 3a , 3a IJ .
H2 2K

y3

+
= 3axy
Here,
Differentiating with respect to x

3x2 + 3y2 y1 = 3a (xy1 + y)
3 ( y2 – ax) y1 = 3 (ay – x2)
⇒

y1 =

ay – x 2
y 2 − ax

...(1)


8

ENGINEERING MATHEMATICS—II

Again differentiating w.r.t x.
⇒

y2 =

Now, from (1), at

dy

2

ib

g d

− ax ⋅ ay1 − 2 x − ay − x 2

FG 3a , 3a IJ
H 2 2K
F 3a I F 3a I
aG J −G J
H 2K H 2K
y =
FG 3a IJ − a FG 3a IJ
H 2K H 2K

dy

2

− ax

i

i b2 yy

1

−a

g

2

2

2

1

=

6a 2 − 9a 2
9a 2 − 6a 2

d

– 9a 2 − 6a 2
=

From (2), at

FG 3a , 3a IJ
H 2 2K
y2 =

d9 a

F 9a
GH 2

2

− 6a

−

2

3a 2
2

i

i

= –1

I b– a − 3ag − F 3a
JK
GH 2
F 9a − 3a I
GH 4 2 JK

2

2

–

3 2
3a 2
a × 4a –
× 4a

4
4

F 3a I
GH 4 JK

=

=

2

2

2

– 6a 3 – 32
=
3a
9a 4
16

Using these

ρ F 3a

GH 2 , 32a IJK

{1 + b– 1g }
FG − 32 IJ

H 3a K

3
2 2

=

= –
∴ Radius of curvature at

2 2 × 3a − 3a
=
32
8 2

FG 3a , 3a IJ
H 2 2K

is

3a
·
8 2

2

2

−

9a 2
4

I b– 3a − ag
JK

...(2)