HIGHER
ENGINEERING
MATHEMATICS
TH
5 EDITION
JOHN BIRD
SAMPLE OF WORKED SOLUTIONS
TO EXERCISES
, INTRODUCTION
In ‘Higher Engineering Mathematics 5th Edition’ are some 1750 further problems arranged
regularly throughout the text within 250 Exercises. A sample of solutions for over 1000 of these
further problems has been prepared in this document. The reader should be able to cope with
the remainder by referring to similar worked problems contained in the text.
CONTENTS
Page
Chapter 1 Algebra 1
Chapter 2 Inequalities 13
Chapter 3 Partial fractions 19
Chapter 4 Logarithms and exponential functions 25
Chapter 5 Hyperbolic functions 41
Chapter 6 Arithmetic and geometric progressions 48
Chapter 7 The binomial series 55
Chapter 8 Maclaurin’s series 65
Chapter 9 Solving equations by iterative methods 71
Chapter 10 Computer numbering systems 85
Chapter 11 Boolean algebra and logic circuits 94
Chapter 12 Introduction to trigonometry 110
Chapter 13 Cartesian and polar co-ordinates 131
Chapter 14 The circle and its properties 135
Chapter 15 Trigonometric waveforms 144
Chapter 16 Trigonometric identities and equations 155
Chapter 17 The relationship between trigonometric and hyperbolic functions 163
Chapter 18 Compound angles 168
Chapter 19 Functions and their curves 181
Chapter 20 Irregular areas, volumes and mean values of waveforms 197
ii
,Chapter 21 Vectors, phasors and the combination of waveforms 202
Chapter 22 Scalar and vector products 212
Chapter 23 Complex numbers 219
Chapter 24 De Moivre’s theorem 232
Chapter 25 The theory of matrices and determinants 238
Chapter 26 The solution of simultaneous equations by matrices and determinants 246
Chapter 27 Methods of differentiation 257
Chapter 28 Some applications of differentiation 266
Chapter 29 Differentiation of parametric equations 281
Chapter 30 Differentiation of implicit functions 287
Chapter 31 Logarithmic differentiation 291
Chapter 32 Differentiation of hyperbolic functions 295
Chapter 33 Differentiation of inverse trigonometric and hyperbolic functions 297
Chapter 34 Partial differentiation 306
Chapter 35 Total differential, rates of change and small changes 312
Chapter 36 Maxima, minima and saddle points for functions of two variables 319
Chapter 37 Standard integration 327
Chapter 38 Some applications of integration 332
Chapter 39 Integration using algebraic substitutions 350
Chapter 40 Integration using trigonometric and hyperbolic substitutions 356
Chapter 41 Integration using partial fractions 365
Chapter 42 The t = tan /2 substitution 372
Chapter 43 Integration by parts 376
Chapter 44 Reduction formulae 384
Chapter 45 Numerical integration 390
Chapter 46 Solution of first order differential equations by separation of variables 398
Chapter 47 Homogeneous first order differential equations 410
Chapter 48 Linear first order differential equations 417
Chapter 49 Numerical methods for first order differential equations 424
dy
Chapter 50 Second order differential equations of the form dy
2
435
b
cy 0
a 2 dx
dx dy
Chapter 51 Second order differential equations of the form d2y 441
b
cy f(x)
a dx
dx2
Chapter 52 Power series methods of solving ordinary differential equations 458
iii
, Chapter 53 An introduction to partial differential equations 474
Chapter 54 Presentation of statistical data 489
Chapter 55 Measures of central tendency and dispersion 497
Chapter 56 Probability 504
Chapter 57 The binomial and Poisson distributions 508
Chapter 58 The normal distribution 513
Chapter 59 Linear correlation 523
Chapter 60 Linear regression 527
Chapter 61 Sampling and estimation theories 533
Chapter 62 Significance testing 543
Chapter 63 Chi-square and distribution-free tests 553
Chapter 64 Introduction to Laplace transforms 566
Chapter 65 Properties of Laplace transforms 569
Chapter 66 Inverse Laplace transforms 575
Chapter 67 The solution of differential equations using Laplace transforms 582
Chapter 68 The solution of simultaneous differential equations using Laplace transforms
590
Chapter 69 Fourier series for periodic functions of period 2 595
Chapter 70 Fourier series for a non-periodic functions over period 2 601
Chapter 71 Even and odd functions and half-range Fourier series 608
Chapter 72 Fourier series over any range 616
Chapter 73 A numerical method of harmonic analysis 623
Chapter 74 The complex or exponential form of a Fourier series 627
iv
ENGINEERING
MATHEMATICS
TH
5 EDITION
JOHN BIRD
SAMPLE OF WORKED SOLUTIONS
TO EXERCISES
, INTRODUCTION
In ‘Higher Engineering Mathematics 5th Edition’ are some 1750 further problems arranged
regularly throughout the text within 250 Exercises. A sample of solutions for over 1000 of these
further problems has been prepared in this document. The reader should be able to cope with
the remainder by referring to similar worked problems contained in the text.
CONTENTS
Page
Chapter 1 Algebra 1
Chapter 2 Inequalities 13
Chapter 3 Partial fractions 19
Chapter 4 Logarithms and exponential functions 25
Chapter 5 Hyperbolic functions 41
Chapter 6 Arithmetic and geometric progressions 48
Chapter 7 The binomial series 55
Chapter 8 Maclaurin’s series 65
Chapter 9 Solving equations by iterative methods 71
Chapter 10 Computer numbering systems 85
Chapter 11 Boolean algebra and logic circuits 94
Chapter 12 Introduction to trigonometry 110
Chapter 13 Cartesian and polar co-ordinates 131
Chapter 14 The circle and its properties 135
Chapter 15 Trigonometric waveforms 144
Chapter 16 Trigonometric identities and equations 155
Chapter 17 The relationship between trigonometric and hyperbolic functions 163
Chapter 18 Compound angles 168
Chapter 19 Functions and their curves 181
Chapter 20 Irregular areas, volumes and mean values of waveforms 197
ii
,Chapter 21 Vectors, phasors and the combination of waveforms 202
Chapter 22 Scalar and vector products 212
Chapter 23 Complex numbers 219
Chapter 24 De Moivre’s theorem 232
Chapter 25 The theory of matrices and determinants 238
Chapter 26 The solution of simultaneous equations by matrices and determinants 246
Chapter 27 Methods of differentiation 257
Chapter 28 Some applications of differentiation 266
Chapter 29 Differentiation of parametric equations 281
Chapter 30 Differentiation of implicit functions 287
Chapter 31 Logarithmic differentiation 291
Chapter 32 Differentiation of hyperbolic functions 295
Chapter 33 Differentiation of inverse trigonometric and hyperbolic functions 297
Chapter 34 Partial differentiation 306
Chapter 35 Total differential, rates of change and small changes 312
Chapter 36 Maxima, minima and saddle points for functions of two variables 319
Chapter 37 Standard integration 327
Chapter 38 Some applications of integration 332
Chapter 39 Integration using algebraic substitutions 350
Chapter 40 Integration using trigonometric and hyperbolic substitutions 356
Chapter 41 Integration using partial fractions 365
Chapter 42 The t = tan /2 substitution 372
Chapter 43 Integration by parts 376
Chapter 44 Reduction formulae 384
Chapter 45 Numerical integration 390
Chapter 46 Solution of first order differential equations by separation of variables 398
Chapter 47 Homogeneous first order differential equations 410
Chapter 48 Linear first order differential equations 417
Chapter 49 Numerical methods for first order differential equations 424
dy
Chapter 50 Second order differential equations of the form dy
2
435
b
cy 0
a 2 dx
dx dy
Chapter 51 Second order differential equations of the form d2y 441
b
cy f(x)
a dx
dx2
Chapter 52 Power series methods of solving ordinary differential equations 458
iii
, Chapter 53 An introduction to partial differential equations 474
Chapter 54 Presentation of statistical data 489
Chapter 55 Measures of central tendency and dispersion 497
Chapter 56 Probability 504
Chapter 57 The binomial and Poisson distributions 508
Chapter 58 The normal distribution 513
Chapter 59 Linear correlation 523
Chapter 60 Linear regression 527
Chapter 61 Sampling and estimation theories 533
Chapter 62 Significance testing 543
Chapter 63 Chi-square and distribution-free tests 553
Chapter 64 Introduction to Laplace transforms 566
Chapter 65 Properties of Laplace transforms 569
Chapter 66 Inverse Laplace transforms 575
Chapter 67 The solution of differential equations using Laplace transforms 582
Chapter 68 The solution of simultaneous differential equations using Laplace transforms
590
Chapter 69 Fourier series for periodic functions of period 2 595
Chapter 70 Fourier series for a non-periodic functions over period 2 601
Chapter 71 Even and odd functions and half-range Fourier series 608
Chapter 72 Fourier series over any range 616
Chapter 73 A numerical method of harmonic analysis 623
Chapter 74 The complex or exponential form of a Fourier series 627
iv
