Engineering Mathematics II : For WBUT

393 Pages·2011·2.472 MB·English

by  Babu Ram

Engineering Mathematics-II
W.B. University of Technology, Kolkata
BABU RAM
Formerly Dean, Faculty of Physical Sciences,
Maharshi Dayanand University, Rohtak
Copyright 2011 Dorling Kindersley (India) Pvt. Ltd
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In memory of
my parents
Smt. Manohari Devi and Sri Makhan Lal
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Contents
Preface vii 1.20 Applications of Linear Differential
Symbols and Basic Formulae viii Equations 1.67
Roadmap to the Syllabus xii
1.21 Mass-Spring System 1.69
1.22 Simple Pendulum 1.70
1 Ordinary Differential Equations 1.1 1.23 Solution in Series 1.71
1.24 Bessel’s Equation and Bessel’s
1.1 Defi nitions and Examples 1.1
Function 1.81
1.2 Formulation of Differential Equation 1.1
1.25 Fourier–Bessel Expansion of
1.3 Solution of Differential Equation 1.3
a Continuous Function 1.88
1.4 Differential Equations of First Order 1.4
1.26 Legendre’s Equation and Legendre’s
1.5 Separable Equations 1.5 Polynomial 1.89
1.6 Homogeneous Equations 1.8 1.27 Fourier–Legendre Expansion of a
1.7 Equations Reducible to Function 1.95
Homogeneous Form 1.11 1.28 Miscellaneous Examples 1.96
1.8 Linear Differential Equations 1.12 Exercises 1.114
1.9 Equations Reducible to
Linear Differential Equations 1.14 2 Graphs 2.1
1.10 Exact Differential Equation 1.15
2.1 Defi nitions and Basic Concepts 2.1
1.11 The Solution of Exact Differential
2.2 Special Graphs 2.3
Equation 1.16
2.3 Subgraphs 2.6
1.12 Equations Reducible to Exact
Equation 1.18 2.4 Isomorphisms of Graphs 2.9
1.13 Applications of First Order and 2.5 Walks, Paths and Circuits 2.11
First Degree Equations 1.25 2.6 Eulerian Paths and Circuits 2.15
1.14 Linear Differential Equations 1.37 2.7 Hamiltonian Circuits 2.21
1.15 Solution of Homogeneous 2.8 Matrix Representation of Graphs 2.29
Linear Differential Equation with
2.9 Planar Graphs 2.30
Constant Coeffi cients 1.39
2.10 Colouring of Graph 2.37
1.16 Complete Solution of Linear Differential
2.11 Directed Graphs 2.41
Equation with Constant Coeffi cients 1.42
2.12 Trees 2.44
1.17 Method of Variation of Parameters to
Find Particular Integral 1.51 2.13 Isomorphism of Trees 2.49
1.18 Differential Equations with 2.14 Representation of Algebraic Expressions
Variable Coeffi cients 1.54 by Binary Trees 2.52
1.19 Simultaneous Linear Differential 2.15 Spanning Tree of a Graph 2.55
Equations with Constant Coeffi cients 1.65 2.16 Shortest Path Problem 2.57
vi (cid:132) Contents
2.17 Minimal Spanning Tree 2.63 5.4 Miscellaneous Examples 5.23
2.18 Cut Sets 2.68 Exercises 5.26
2.19 Tree Searching 2.72
6 Inverse Laplace Transform 6.1
2.20 Transport Networks 2.75
Exercises 2.82 6.1 Defi nition and Examples of Inverse
Laplace Transform 6.1
6.2 Properties of Inverse
3 Improper Integrals 3.1
Laplace Transform 6.2
3.1. Improper Integral 3.1 6.3 Partial Fractions Method to Find Inverse
3.2. Convergence of Improper Integral with Laplace Transform 6.10
Unbounded Integrand 3.1 6.4 Heaviside’s Expansion Theorem 6.13
3.3. Convergence of Improper Integral with 6.5 Series Method to Determine Inverse
Infi nite Limits 3.5 Laplace Transform 6.14
Exercises 3.7 6.6 Convolution Theorem 6.15
6.7 Complex Inversion Formula 6.20
4 Beta and Gamma Functions 4.1 6.8 Miscellaneous Examples 6.25
4.1 Beta Function 4.1 Exercises 6.29
4.2 Properties of Beta Function 4.1
7
Applications of Laplace Transform 7.1
4.3 Gamma Function 4.5
7.1 Ordinary Differential Equations 7.1
4.4 Properties of Gamma Function 4.5
7.2 Simultaneous Differential Equations 7.13
4.5 Relation Between Beta and Gamma
Functions 4.5 7.3 Difference Equations 7.16
4.6 Dirichlet’s and Liouville’s Theorems 4.11 7.4 Integral Equations 7.21
4.7 Miscellaneous Examples 4.13 7.5 Integro-Differential Equations 7.24
7.6 Solution of Partial Differential
Exercises 4.14
Equation 7.25
7.7 Evaluation of Integrals 7.29
5
Laplace Transform 5.1
7.8 Miscellaneous Examples 7.31
5.1 Defi nition and Examples of Laplace
Exercises 7.36
Transform 5.1
5.2 Properties of Laplace Transforms 5.8 Solved Question Papers Q.1
5.3 Limiting Theorems 5.22 Index I.1
Preface
All branches of Engineering, Technology and Science require mathematics as a tool for the description
of their contents. Therefore, a thorough knowledge of various topics in mathematics is essential to pursue
courses in Engineering, Technology and Science. The aim of this book is to provide students with sound
mathematics skills and their applications. Although the book is designed primarily for use by engineering
students, it is also suitable for students pursuing bachelor degrees with mathematics as one of the sub-
jects and also for those who prepare for various competitive examinations. The material has been arranged
to ensure the suitability of the book for class use and for individual self study. Accordingly, the contents
of the book have been divided into seven chapters covering the complete syllabus prescribed for B.Tech.
Semester-II of West Bengal University of Technology. A number of examples, fi gures, tables and exercises
have been provided to enable students to develop problem-solving skills. The language used is simple and
lucid. Suggestions and feedback on this book are welcome.
Acknowledgements
I am extremely grateful to the reviewers for their valuable comments. My family members provided moral
support during the preparation of this book. My son, Aman Kumar, software engineer, Adobe India Ltd,
offered wise comments on some of the contents of the book. I am thankful to Sushma S. Pradeep for
excellently typing the manuscript.
Special thanks are due to Thomas Mathew Rajesh, Anita Yadav, Ravi Vishwakarma, M. E. Sethurajan and
Jennifer Sargunar at Pearson Education for their constructive support.
BABU RAM
Symbols and Basic Formulae
1 Greek Letters (f) b2 − 4ac < 0 ⇒ the roots are complex
(g) if b2 − 4ac is a perfect square, the roots
a alpha f phi
are rational
b beta Φ capital phi
g gamma y psi 3
Properties of Logarithm
Γ capital gamma Ψ capital psi
(i) log 1 = 0, log 0 = −∞ for a > 1, log a = 1
d delta x xi a a a
log 2 = 0.6931, log 10 = 2.3026,
Δ capital delta h eta loge e = 0.4343 e
e epsilon z zeta (ii) log1 0p + log q = log pq
ι iota c chi a a a
p
q theta π pi (iii) loga p + loga q = loga q
l lambda s sigma (iv) log pq=qlog p
a a
m mu Σ capital sigma log n
(v) log n=log b⋅log n= b
n nu t tau a a b logba
w omega r rho 4
Angles Relations
Ω capital omega k kapha
180°
(i) 1radian=
2 π
Algebraic Formulae
(ii) 1°=0.0174radian
(i) Arithmetic progression a, a + d, a + 2d,
nth term T = a + (n − 1) d
n 5 Algebraic Signs of Trigonometrical Ratios
n
Sum of n terms = [2a+(n−1)d]
2 (a) First quadrant: All trigonometric ratios are
(ii) Geometrical progression: a, ar, ar2, positive
nth term Tn = ar n−1 (b) Second quadrant: sin q and cosec q are positive,
Sum of n terms = a(1−rn) all others negative
1−r
(c) T hird quadrant: tan q and cot q are positive, all
(iii) Arithmetic mean of two numbers a and b is
others negative
1
(a+b) (d) Fouth quadrant: cos q and sec q are positive, all
2
(iv) Geometric mean of two numbers a and b others negative
is ab
2ab 6 Commonly Used Values of Trigonometrical Ratios
(v) Harmonic mean of two numbers a and b is
a+b
π π π
(vi) I f ax2 + bx + c = 0 is quadratic, then sin2 =1,cos2 =0,tan2 =∞
(a) its roots are given by −b± 2ba2−4ac cosecπ2 =1,secπ2 =∞,cosπ2 =0
(b) the sum of the roots is equal to −ba sinπ=1,cosπ= 3,tanπ= 1
6 2 6 2 6 3
c
(c) product of the roots is equal to π π 2 π
a cosec =2,sec = ,cot = 3
(d) b2 − 4ac = 0 ⇒ the roots are equal 6 6 3 6
(e) b2 − 4ac > 0 ⇒ the roots are real sinπ= 3,cosπ=1,tanπ= 3
and distinct 3 2 3 2 3
Symbols and Basic Formulae (cid:132) ix
cosecπ= 2 ,secπ=2,cotπ= 1 (l) tan3A=3tanA−tan3A
3 3 3 3 3 1−3tan2A
π 1 π 1 π A+B A−B
sin = ,cos = ,tan =1 (m) sinA+sinB=2sin cos
4 2 4 2 4 2 2
π π π A+B A−B
cosec = 2,sec = 2,cot =1 (n) sinA−sinB=2cos sin
4 4 4 2 2
A+B A−B
(o) cosA+cosB=2cos cos
7
Trigonometric Ratios of Allied Angles 2 2
A+B B−A
(a) sin(−θ)=−sinθ,cos(−θ)=cosθ (p) cosA−cosB=2sin sin
tan(−θ)=−tanθ 2 2
1
cosec(−θ)=−cosecθ,sec(−θ)=secθ (q) sinAcosB= [sin(A+B)+sin(A−B)]
2
cot(−θ)=−cotθ
1
(r) cosAsinB= [sin(A+B)−sin(A−B)]
(b) Anytrigonometric ratio of 2
(n.90±θ)= (s) cosAcosB=1[cos(A+B)+cos(A−B)]
⎧±sametrigonometric ratio of θ 2
⎪⎨when n is even (t) sinAsinB=1[cos(A−B)−cos(A+B)]
⎪⎩±co-ratio of θ whennisodd 2
For example: sin(4620)=sin[90°(52)−60°] 9 Expressions for sin A; cos A and tan A
2 2 2
3
=sin(−60°)=−sin60°=− 2 . (a) sinA=± 1−cosA
Similarly,cosec(270°−θ)=cosec(90°(3)−θ) 2 2
=−secθ. 1+cosA
(b) cosA=±
2 2
8
Transformations of Products and Sums
1−cosA
(c) tanA=±
(a) sin(A+B)=sinAcosB+cosAsinB 2 1+cosA
(b) sin(A−B)=sinAcosB−cosAsinB (d) sinA+cosA=± 1+sinA
2 2
(c) cos(A+B)=cosAcosB−sinAsinB (e) sinA−cosA=± 1−sinA
(d) cos(A−B)=cosAcosB+sinAsinB 2 2
(e) tan(A+B)= tanA+tanB 10 Relations Between Sides and Angles of a Triangle
1−tanAtanB
a b c
tanA−tanB (a) = = (sine formulae)
(f) tan(A−B)= sinA sinB sinC
1+tanAtanB
b2+c2−a2⎫
2tanA (b) cosA= ⎪
(g) sin2A=2sinAcosA=1+tan2A 2bc ⎪
c2+a2−b2⎪
(h) cos2A=cos2A−sin2A=1−2sin2A cosB= ⎬cosine formulae
2ca ⎪
=2cos2A−1=11+−ttaann22AA cosC= a2+2ba2b−c2⎪⎪⎭
sin2A 2tanA
(i) tan2A= cos2A=1−tan2A (c) a=bcosC+ccosB⎫⎪
(j) sin3A=3sinA−4sin3A b=ccosA+acosC⎬Projection formulae.
⎪
(k) cos3A=4cos3A−3cosA c=acosB+bcosA⎭