A TEXTBOOK OF ENGINEERING MATHEMATICS

A TEXTBOOK OF

ENGINEERING MATHEMATICS For B.E./B.TECH. (Semester–II) ANNA UNIVERSITY, CHENNAI (Strictly According to the Latest Revised Syllabus)

By

N.P. BALI

DR. MANISH GOYAL

Former Principal S.B. College, Gurgaon Haryana

M.Sc. (Mathematics), Ph.D., CSIR–NET Associate Professor Department of Mathematics Institute of Applied Sciences & Humanities G.L.A. University, Mathura U.P.

UNIVERSITY SCIENCE PRESS (An Imprint of Laxmi Publications (P) Ltd.) BANGALORE l CHENNAI JALANDHAR

l

KOLKATA

l

COCHIN

l

GUWAHATI

l

HYDERABAD

l

LUCKNOW

l

MUMBAI

l

RANCHI

NEW DELHI

l

BOSTON, USA

Copyright © 2014 by Laxmi Publications Pvt. Ltd. All rights reserved with the Publishers. 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 the prior written permission of the publisher.

Published by:

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First Edition: 2005, Second Edition: 2007, Third Edition: 2009, Fourth Edition: 2010, Fifth Edition: 2011, Reprint: 2012, 2013, Sixth Edition: 2014 OFFICES

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CONTENTS Preface

...

(vii)

Syllabus

...

(viii)

Standard Results

...

(ix)–(xiii)

Symbols

...

(xiv)

Units

Pages

1.

Vector Calculus

...

1

2.

Ordinary Differential Equations

...

98

3.

Laplace Transform

...

197

4.

Analytic Functions

...

287

5.

Complex Integration

...

360

Appendices

...

(i)–(ix)

PREFACE TO THE SIXTH EDITION

vkfnR;kukega fo".kqT;ksfZ r"kka jfoja'kqeku~A ejhfpeZ#rkefLe u{k=kk.kkega 'k'khAA It gives us immense pleasure to bring before all, the Sixth Edition of a textbook of Engineering Mathematics–II for all students of B.E./B.Tech. I Year II Semester studying under Anna University. The book has been made up-to-date by incorporating all the questions, asked recently, with their to the point solutions/answers at the appropriate places. It will also help the readers in developing their necessary basic mathematical skills that are imperative for effective understanding of the subject. We are really thankful to Dr. Hari Kishan, Associate Professor and Head, Mathematics Dept., K.R. (P.G.) College, Mathura, for his valuable suggestions and necessary instructions. However, any more suggestions with a view to enhance the quality of the text are always welcome.

— Authors

(vii)

SYLLABUS ENGINEERING MATHEMATICS–II ANNA UNIVERSITY MA6251

LTPC 3 1 0 4

OBJECTIVES • • •

•

To make the student acquire sound knowledge of techniques in solving ordinary differential equations that model engineering problems. To acquaint the student with the concepts of vector calculus, needed for problems in all engineering disciplines. To develop an understanding of the standard techniques of complex variable theory so as to enable the student to apply them with confidence, in application areas such as heat conduction, elasticity, fluid dynamics and flow of the electric current. To make the student appreciate the purpose of using transforms to create a new domain in which it is easier to handle the problem that is being investigated.

UNIT I: VECTOR CALCULUS

9+3

Gradient, divergence and curl—Directional derivative—Irrotational and solenoidal vector fields— Vector integration— Green’s theorem in a plane, Gauss divergence theorem and Stokes’ theorem (excluding proofs)—Simple applications involving cubes and rectangular parallelopipeds. UNIT II: ORDINARY DIFFERENTIAL EQUATIONS

9+3

Higher order linear differential equations with constant coefficients—Method of variation of parameters—Cauchy’s and Legendre’s linear equations—Simultaneous first order linear equations with constant coefficients. UNIT III: LAPLACE TRANSFORM

9+3

Laplace transform—Sufficient condition for existence—Transform of elementary functions—Basic properties—Transforms of derivatives and integrals of functions—Derivatives and integrals of transforms—Transforms of unit step function and impulse functions—Transform of periodic functions. Inverse Laplace transform—Statement of Convolution theorem—Initial and final value theorems—Solution of linear ODE of second order with constant coefficients using Laplace transformation techniques. UNIT IV: ANALYTIC FUNCTIONS

9+3

Functions of a complex variable—Analytic functions: Necessary conditions—Cauchy-Riemann equations and sufficient conditions (excluding proofs)—Harmonic and orthogonal properties of analytic function—Harmonic conjugate—Construction of analytic functions—Conformal mapping: w = z + k, kz, 1/z, z2, ez and bilinear transformation. UNIT V: COMPLEX INTEGRATION

9+3

Complex integration—Statement and applications of Cauchy’s integral theorem and Cauchy’s integral formula—Taylor’s and Laurent’s series expansions —Singular points—Residues — Cauchy’s residue theorem—Evaluation of real definite integrals as contour integrals around unit circle and semi-circle (excluding poles on the real axis).

TOTAL : 60 PERIODS (viii)

STANDARD RESULTS 1.

d (xn) = nxn–1 dx

2.

d (ax) = ax loge a dx

3.

d (ex) = ex dx

4.

d 1 (loge x) = dx x

5.

d 1 (log10 x) = log10 e dx x

6.

d (sin x) = cos x dx

7.

d (cos x) = – sin x dx

8.

d (tan x) = sec2 x dx

9.

d (cosec x) = – cosec x cot x dx

10.

d (cot x) = – cosec2 x dx

11.

d (sec x) = sec x tan x dx

12.

d (sin–1 x) = dx

13.

d (cos–1 x) = dx

14.

1 d (tan–1 x) = 1 + x2 dx

15.

1 d (sec–1 x) = dx x x2 − 1

16.

−1 d (cot–1 x) = dx 1 + x2

17.

1 d (cosec–1 x) = – dx x x2 − 1

18. sinh x =

19. cosh x =

−1 1− x

2

e x + e− x 2

20. tanh x =

1 1 − x2

e x − e− x 2

e x − e− x e x + e− x

21. cosh2 x – sinh2 x = 1, sech2 x + tanh2 x = 1, coth2 x = 1 + cosech2 x 22. cosh2 x + sinh2 x = cosh 2x

23. sinh–1 x = log (x +

x2 + 1 )

cosh–1 x = log ( x +

x2 − 1 )

24.

d (sinh x) = cosh x dx

25.

d (cosh x) = sinh x dx

26.

d (tanh x) = sech2 x dx

27.

d (coth x) = – cosech2 x dx

28.

d (sech x) = – sech x tanh x dx

29.

d (cosech x) = – cosech x coth x dx

d dv du 30. Product rule : (uv) = u +v dx dx dx

d 31. Quotient rule : dx

( ix )

FG u IJ H vK

=

v

du dv −u dx dx v2

(x) 32.

dy dy dt . = dx dt dx

if y = f1(t) and x = f2(t)

33. sin–1 x + cos–1 x = 34. tan–1

35. tan–1

π π π , tan–1 x + cot–1 x = , sec–1 x + cosec–1 x = 2 2 2

FG a − b IJ = tan a – tan b, tan F a + b I = tan GH 1 − ab JK H 1 + ab K F 2x I = sin F 2x I = 2 tan x GH 1 + x JK GH 1 − x JK –1

2

–1

–1

2

–1

–1

a + tan–1 b

–1

36. sin 3x = 3 sin x – 4 sin3 x, cos 3x = 4 cos3 x – 3 cos x, tan 3x = sin 2x = 2 sin x cos x ; tan 2x = cos2 x – sin2 x =

2 tan x 1 − tan 2 x

3 tan x − tan3 x 1 − 3 tan 2 x

, cos 2x = 2 cos2 x – 1 = 1 – 2 sin2 x

1 − tan 2 x 1 + tan 2 x

x7 x3 x5 + – + ... 7! 3! 5! x4 x2 x6 cos x = 1 – + – + ... 4! 2! 6! x2 x3 ex = 1 + x + + + ... 2! 3!

37. sin x = x –

(1 – x)–1 = 1 + x + x2 + x3 + ... ; | x | < 1 (1 + x)–1 = 1 – x + x2 – x3 + ... (1 – x)–2 = 1 + 2x + 3x2 + 4x3 + ... (1 + x)–2 = 1 – 2x + 3x2 – 4x3 + ...

38. sin C + sin D = 2 sin

C+D C−D C+D C−D cos , sin C – sin D = 2 cos sin 2 2 2 2

cos C + cos D = 2 cos

C+D C−D C+D D−C cos , cos C – cos D = 2 sin sin 2 2 2 2

39. 2 cos A cos B = cos (A + B) + cos (A – B), 2 sin A sin B = cos (A – B) – cos (A + B) 2 sin A cos B = sin (A + B) + sin (A – B), 2 cos A sin B = sin (A + B) – sin (A – B) 40. sin (A + B) = sin A cos B + cos A sin B, sin (A – B) = sin A cos B – cos A sin B cos (A + B) = cos A cos B – sin A sin B, cos (A – B) = cos A cos B + sin A sin B 41.

d (sinh–1 x) = dx

1 1+ x

2

,

d (cosh–1 x) = dx

1 d (tanh–1 x) = where | x | < 1, dx 1 − x2

1 2

x −1 1 d (coth–1 x) = 2 where | x | > 1 dx x −1

d 1 d 1 (sech–1 x) = – , (cosech–1 x) = – dx 2 dx x 1− x x x2 + 1

42. (cos θ + i sin θ)n = cos nθ + i sin nθ, (cos θ + i sin θ)–n = cos nθ – i sin nθ

( xi ) 43. sin2 θ + cos2 θ = 1, sec2 θ – tan2 θ = 1, 1 + cot2 θ = cosec2 θ 44. θ

0°

30°

45°

60°

sin θ

0

1/2

1/ 2

cos θ

1

3 /2

tan θ

0 90° – θ cos θ sin θ cot θ

45. θ sin θ cos θ tan θ

46. sine formula :

90°

180°

270°

360°

3 /2

1

0

–1

0

1/ 2

1/2

0

–1

0

1

1/ 3

1

3

∞

0

∞

0

90° + θ cos θ – sin θ – cot θ

π–θ sin θ – cos θ – tan θ

π+θ – sin θ – cos θ tan θ

a c b = = sin A sin C sin B

cosine formula : cos A = 47. Area of triangle ∆ = 48. 49.

50.

nC r

z z z z z z z z z z z

=

b2 + c2 − a2 2bc

s(s − a) (s − b) (s − c) where s =

a+b+c 2

n! r!n− r!

x n dx =

x n+ 1 +c;n≠–1 n+1

1 dx = log x + c ; e x

z

e x dx = ex + c ;

sin x dx = – cos x + c ;

z

tan x dx = log sec x + c ;

z

ax +c log e a

a x dx =

cos x dx = sin x + c

z

cot x dx = log sin x + c

sec x dx = log (sec x + tan x) + c = log tan

FG π + x IJ + c H 4 2K

cosec x dx = log (cosec x – cot x) + c = log tan sec x tan x dx = sec x + c ;

z z

x +c 2

cosec x cot x dx = – cosec x + c

− dx FG x IJ + c ; F xI = cos G J + c H H aK K a a −x a −x 1 1 dx − dx F xI x = tan FG IJ + c ; = cot G J + c H aK a a H aK a +x a +x dx 1 F a + x IJ + c ; dx = 1 log FG x − a IJ + c = log G 2a 2a a −x x −a H a − xK H x + aK dx 1 1 − dx F xI F xI = sec G J + c ; = cosec G J + c H aK H K a a a x x −a x x −a dx

2

2

2

2

2

2

2

= sin–1

–1

2

–1

–1

z

2

2

2

2

z z

2

–1

2

–1

2

2

51.

52.

53.

z z z z z z z z z z

sech2 x dx = tanh x + c, sinh x dx = cosh x + c,

z

z

( xii ) cosech2 x dx = – coth x + c cosh x dx = sinh x + c

sech x tanh x dx = – sech x + c, a2 − x2 dx =

1 x 2

a2 − x2 +

a2 + x 2 dx =

1 x 2

a2 + x2

x2 − a2 dx =

1 x 2

x2 − a2 –

dx 2

a +x b

2

f ( x) dx =

a

a

−a

z

b

a

R| S| T R f ( x) dx = |S2 |T

z z

a

0

z

b

a

cosech x coth x dx = – cosech x + c

1 2 x a sin–1 + c 2 a 1 2 + a log (x + a2 + x2 ) + c 2

z

FG x IJ + c ; H aK

f ( y) dy ;

f ( x) dx = 2

2a

0

= sinh–1

z

1 2 a log (x + 2 dx 2

x −a

f ( x) dx = –

z

a

b

x2 − a2 ) + c

FG x IJ + c H aK

= cosh–1

2

f ( x) dx ,

z

a

0

f ( x) dx =

U| V if f ( x) is odd function |W U if f (2a − x) = f ( x) | V if f (2a − x) = − f ( x) |W

z

a

0

f (a − x) dx

f ( x) dx , if f ( x) is even function 0

a

0

,

f ( x) dx , 0

,

54. Leibnitz rule for differentiation under the integral sign

z

d dx

55. 56. 57. 58.

ψ (α )

φ (α )

f ( x, α) dx =

z

ψ (α )

φ (α )

∂ dψ (α) dφ(α) { f ( x, α)} dx + f{ψ(α), α} – f{φ(α), α} dα dα ∂α →

→

→

If r = xi
+ yj
+ zk
then | r | =

2

2

x +y +z

2

r

and r
=

→

|r|

–→

–→

–→

AB = position vector of B-position vector of A = OB – OA →

→ →

→

a . b = | a | | b | cos θ ; work done =

→

→

→

→

z

c

→

→

F . dr

a × b = | a | | b | sin θ n
→

→

→

→

59. Area of parallelogram = a × b , Moment of force = r × F 60.

=

a1 a2 →→→ → → a . ( b × c ) = [ a b c ] = b1 b2 c1 c2

→

→

a3 b3 c3

→

→

→

→

where a = Σ a1 i
, b = Σ b1 i
and c = Σc1 i
→

→

→

→

→

→

→

= (a × b) . c

If a . ( b × c ) = 0 then a , b , c are coplanar.

xi
+ yj
+ zk
x 2 + y2 + z 2

( xiii ) 61.

→

→

→ →

→

→

→

→

→

a × (b × c ) = (a . c) b – (a . b) c →

→

→

→

→

→ →

→ →

b. c

b.d

→ 62. ( a × b ) . ( c × d ) = a . c a . d → → → → →

→→→

→

→

→→→

→

63. ( a × b ) × ( c × d ) = [ a b d ] c – [ a b c ] d 64. A (Adj. A) = | A | I 65. AA–1 = I = A–1 A 66. AI = A = IA 67. (ABC)′ = C′B′A′ 68. (AB) C = A(BC) ; A(B + C) = AB + AC 69. A + B = B + A ; A + (B + C) = (A + B) + C 70. (AB)–1 = B–1A–1 71. Walli’s formula

z z z

π/2

0

72.

n

sin θ dθ =

z

π/2

0

eax

eax sin bx dx =

a2 + b2 eax

eax cos bx dx =

73. Γ(1/ 2) =

cos

n

2

a + b2

R| n − 1 . n − 3 . n − 5 ...... 3 . 1 . π n−2 n−4 4 2 2 θ dθ = S n n−1 n−3 n−5 || n . n − 2 . n − 4 ...... 45 . 23 T (a sin bx – b cos bx) + c (a cos bx + b sin bx) + c

π , Γ(− 1/ 2) = – 2 π

74. log (1 + x) = x –

x2 x3 x4 x5 x6 + – + – + ...... 2 3 4 5 6

log (1 – x) = – x –

x2 x3 x4 x5 x6 – – – – – ...... 2 3 4 5 6

75. x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)

if n is even if n is odd

SYMBOLS Greek Alphabets A B Γ ∆ E Z H Θ

α β γ δ ε ζ η θ ∃

Alpha Beta Gamma Delta Epsilon Zeta Eta Theta there exists

I K Λ M N Ξ O Π

ι κ λ µ ν ξ ο π V

( xiv )

Iota Kappa Lambda Mu Nu Xi Omicron Pi for all

P Σ T Y Φ X Ψ Ω

ρ σ τ υ ϕ χ ψ ω

Rho Sigma Tau Upsilon Phi Chi Psi Omega

UNIT 1 Vector Calculus 1.1. SCALAR AND VECTOR FUNCTIONS Scalar function f(x, y, z) is a function defined at each point in a certain domain D in space. Its value is real and depends only on the point P(x, y, z) in space but not on any particular coordinate system being used. →

→

If to each value of a scalar variable t, there corresponds a value of a vector r , then r is →

→

→

→

called a vector function of the scalar variable t and we write r = r (t) or r = f (t) →

For example, the position vector r of a particle moving along a curved path is a vector function of time t, a scalar. Since every vector can be uniquely expressed as a linear combination of three fixed non→

coplanar vectors, therefore we may write f (t) = f1(t) i + f2(t) j + f3(t) k where i , j , k denote unit vectors along the axis of x, y, z respectively and f1(t), f2(t), f3(t) are called the components →

of the vector f (t) along the coordinate axes. 1.2. DERIVATIVE OF A VECTOR FUNCTION WITH RESPECT TO A SCALAR →

→

Let r = f (t) be a vector function of the scalar variable t. Let δt be a small increment in t and →

→

δ r , the corresponding increment in r . Then →

→

→

→

→

→

r + δ r = f (t + δt) so that δ r = f (t + δt) – f (t) and →

If Lt

δt → 0

→

→

→

→

δr f (t + δt) − f (t) = δt δt

→

→

δr f (t + δt) − f (t) dr = Lt exists, then the value of this limit is denoted by δt δt → 0 δt dt →

and is called the derivative of r with respect to t. →

→

dr d2 r Since is itself a vector function of t, its derivative is denoted by and is called dt dt 2 →

the second derivative of r with respect to t. Similarly, we can define higher order derivatives →

of r .

1