Since the time of Newton differential equations have become an
essential mathematical tool for the solution of many physical problems.
Accordingly the reader is introduced to this subject by considering
problems derived from several scientific disciplines.
It is assumed that the reader has sufficient knowledge of calculus to
evaluate the various integrals which arise. Care has been taken to avoid
the use of the relation e^{ia} = cos a + i sin a in Chapter 5 in order that
readers unfamiliar with this relation can follow all the developments of
that chapter. However, such readers are advised to omit Sections 62
and 64 of Chapter 7.
The contents of this book are adequate for the requirements of under-
graduate scientists and engineers and for a first-year course for under-
graduate mathematicians. It is hoped that it may also be useful to school
scholarship candidates. Hence the text is presented without emphasis
on rigour.
Author(s): Barry Spain
Publisher: Van Nostrand Reinhold
Year: 1969
Language: English
Pages: 152
Preface v
CHAPTER 1 INTRODUCTION
1 Radioactivity 1
2 Cooling 2
3 Epidemic 3
4 Simple Pendulum 3
5 Resisted Motion of a Particle 5
6 Electric Circuit 6
7 Differential Equations 7
8 Primitives 8
9 General and Particular Solutions 9
10 Integral Curves 10
11 Singular Solutions 11
CHAPTER 2 FIRST-ORDER DIFFERENTIAL EQUATIONS
12 Separable Equations 15
13 Linear Equations 16
14 Bernoulli Equations 18
15 Homogeneous Equations 19
16 Exact Equations 21
17 Integrating Factors 23
18 Riccati Equations 25
19 Clairaut Equations 26
20 Equations Linear in x and y 27
CHAPTER 3 ApPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS
21 Tangents and Normals to Curves 28
22 Orthogonal Trajectories 29
23 Salt Solution 31
24 Chemical Compound-Law of Mass Action 32
25 Flow from an Orifice 34
26 Motion against Friction 35
27 Viscoelasticity 38
28 Simple Electric Circuits 39
CHAPTER 4 LINEAR EQUATIONS OF THE SECOND ORDER
29 Introduction 40
30 Complementary Functions 41
31 General Solution when tIle Completnentary Function is known 43
32 Superposition Principle 44
33 General Solution when one Particular Solution of the Reduced Equation is known 45
34 Variation of Parameters 47
CHAPTER 5 SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS
35 Auxiliary Equation 49
36 Auxiliary Equation witll Real Distinct Roots (b 2 > ac) 49
37 Auxiliary Equation with Coincident Roots (b 2 == ac) 50
38 Auxiliary Equation with Complex Roots (b 2 < ac) 51
39 Complementary Functions 52
40 Particular Integral whenf(x) == e kro 53
41 Particular Integral when I( x) == A cas Lt.X + f1 sin
42 Particular Integral whenf(x) = L arx r , (an 7"':0) 56
43 Particular Integral when I(x) == (A cos
44 Particular Integral whenf(x) = ek 59 r=O
45 Particular Integral when f(x) = (A. cos ax + It sin (Xx) L arx r 60
46 Euler Linear Differential Equations of the Second Order 62
CHAPTER 6 ApPLICATIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS
47 Electric Circuit 64
48 Free Oscillation with no Damping 64
49 Damped Free Oscillations 65
50 Charging of a Capacitor 66
51 Alternating Electromagnetic Force 67
52 Resonance 68
53 Freely-Hanging Uniform Chain 69
54 Curvature 71
CHAPTER 7 D-OPERATOR METHODS
55 D-Operator 73
56 Shift Theorem 74
57 Homogeneous Equations 74
58 Non-Homogeneous Equations 77
59 Inverse Operators 77
60 Shift Theorem for Inverse Operators 78
61 l/F(D) Operating on an Exponential 79
62 l/F(D) Operating on cos kx and sin kx 80
63 l/F(D) Operating on a Polynomial 81
64 1/ F(D) Operating on Products of Polynomials, Exponentials and Circular Functions 82
CHAPTER 8 LAPLACE TRANSFORMS
65 Introduction 84-
66 Laplace Transforms 85
67 Laplace Transforms of Elen1entary :Functions 87
68 Shift Theorem 88
69 Heaviside Unit Function 89
70 Periodic Functions 90
71 Laplace Transforms of Derivatives 92
72 Inverse Laplace Transforms 93
CHAPTER 9 ApPLICATION OF LAPLACE 'l-'RANSFORMS TO THE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
73 Second-Order Equations 95
74 Higher-Order Differential Equations 98
75 Systems of Linear Differential Equations 100
CHAPTER 10 STABILITY
76 Poincare Phase Plane 103
77 Stability at a Singular Point 103
78 Stability of Linear System of First-Order Equations 104
79 General Consideration of Stability 108
CHAPTER 11 SOLUTION IN SERIES
80 Power Series 110
81 Power Series Solutions 111
82 Series Solutions near an Ordinary Point 112
83 Solutions near a Regular Singular Point 114
84 Solutions near a Regular Singularity when the Roots of the Indicial Equation are Equal 118
85 Solutions near a Regular Singularity when the Roots of the Indicial Equation Differ by an Integer 120
CHAPTER 12 TWO-POINT BOUNDARY PROBLElVIS
86 Introduction 123
87 Sturm-Liouville Boundary Problems 124
88 Orthogonal Property of Sturm-Liouville Systems 128
89 Non-Homogeneous Systems 129
Table of Laplace Transforms 132
Solutions 133
Index 141
