x Contents
51.6 Worked problems on differential equations
of the forma
d^2 y
dx^2+b
dy
dx
+cy=f(x)
wheref(x)is a sum or a product 49052 Power series methods of solving ordinary
differential equations 493
52.1 Introduction 493
52.2 Higher order differential coefficients as
series 493
52.3 Leibniz’s theorem 495
52.4 Power series solution by the
Leibniz–Maclaurin method 497
52.5 Power series solution by the Frobenius
method 500
52.6 Bessel’s equation and Bessel’s functions 506
52.7 Legendre’s equation and Legendre
polynomials 51153 An introduction to partial differential equations 515
53.1 Introduction 515
53.2 Partial integration 515
53.3 Solution of partial differential equations
by direct partial integration 516
53.4 Some important engineering partial
differential equations 518
53.5 Separating the variables 518
53.6 The wave equation 519
53.7 The heat conduction equation 523
53.8 Laplace’s equation 525Revision Test 15 52854 Presentation of statistical data 529
54.1 Some statistical terminology 529
54.2 Presentation of ungrouped data 530
54.3 Presentation of grouped data 53455 Measures of central tendency and dispersion 541
55.1 Measures of central tendency 541
55.2 Mean, median and mode for discrete data 541
55.3 Mean, median and mode for grouped data 542
55.4 Standard deviation 544
55.5 Quartiles, deciles and percentiles 54656 Probability 548
56.1 Introduction to probability 548
56.2 Laws of probability 549
56.3 Worked problems on probability 549
56.4 Further worked problems on probability 551Revision Test 16 55457 The binomial and Poisson distributions 556
57.1 The binomial distribution 556
57.2 The Poisson distribution 55958 The normal distribution 562
58.1 Introduction to the normal distribution 562
58.2 Testing for a normal distribution 56659 Linear correlation 570
59.1 Introduction to linear correlation 570
59.2 The product-moment formula for
determining the linear correlation
coefficient 570
59.3 The significance of a coefficient of
correlation 571
59.4 Worked problems on linear correlation 571
60 Linear regression 575
60.1 Introduction to linear regression 575
60.2 The least-squares regression lines 575
60.3 Worked problems on linear regression 576Revision Test 17 58161 Introduction to Laplace transforms 582
61.1 Introduction 582
61.2 Definition of a Laplace transform 582
61.3 Linearity property of the Laplace
transform 582
61.4 Laplace transforms of elementary
functions 582
61.5 Worked problems on standard Laplace
transforms 58362 Properties of Laplace transforms 587
62.1 The Laplace transform of eatf(t) 587
62.2 Laplace transforms of the form eatf(t) 587
62.3 The Laplace transforms of derivatives 589
62.4 The initial and final value theorems 59163 Inverse Laplace transforms 593
63.1 Definition of the inverse Laplace transform 593
63.2 Inverse Laplace transforms of simple
functions 593
63.3 Inverse Laplace transforms using partial
fractions 596
63.4 Poles and zeros 59864 The solution of differential equations using
Laplace transforms 600
64.1 Introduction 600
64.2 Procedure to solve differential equations
by using Laplace transforms 600
64.3 Worked problems on solving differential
equations using Laplace transforms 600