x Contents
51.6 Worked problems on differential equations
of the form a
d 2 y
dx 2 + b
dy
dx
+ cy = f (x)
where f (x) is a sum or a product
490
52 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
511
53 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
525
Revision Test 15
528
54 Presentation of statistical data
529
54.1 Some statistical terminology
529
54.2 Presentation of ungrouped data
530
54.3 Presentation of grouped data
534
55 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
546
56 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
551
Revision Test 16
554
57 The binomial and Poisson distributions
556
57.1 The binomial distribution
556
57.2 The Poisson distribution
559
58 The normal distribution
562
58.1 Introduction to the normal distribution
562
58.2 Testing for a normal distribution
566
59 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
576
Revision Test 17
581
61 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
583
62 Properties of Laplace transforms
587
62.1 The Laplace transform of e at f (t)
587
62.2 Laplace transforms of the form e at f (t)
587
62.3 The Laplace transforms of derivatives
589
62.4 The initial and final value theorems
591
63 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
598
64 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
51.6 Worked problems on differential equations
of the form a
d 2 y
dx 2 + b
dy
dx
+ cy = f (x)
where f (x) is a sum or a product
490
52 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
511
53 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
525
Revision Test 15
528
54 Presentation of statistical data
529
54.1 Some statistical terminology
529
54.2 Presentation of ungrouped data
530
54.3 Presentation of grouped data
534
55 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
546
56 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
551
Revision Test 16
554
57 The binomial and Poisson distributions
556
57.1 The binomial distribution
556
57.2 The Poisson distribution
559
58 The normal distribution
562
58.1 Introduction to the normal distribution
562
58.2 Testing for a normal distribution
566
59 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
576
Revision Test 17
581
61 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
583
62 Properties of Laplace transforms
587
62.1 The Laplace transform of e at f (t)
587
62.2 Laplace transforms of the form e at f (t)
587
62.3 The Laplace transforms of derivatives
589
62.4 The initial and final value theorems
591
63 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
598
64 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