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Mathematics made easy
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  • Engineering Mathematics-1

This document has the step by step full workings and example for basic Mathematics for First year student, we know mathematics is hard so we made it easy for even a beginner to understand and prepare well for your examination.

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  • May 14, 2024
  • 502
  • 2022/2023
  • Class notes
  • Mr. nwankwo matthew
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  • mathematics
  • basic mathematics
  • engineering mathematics
  • first year mathematics
  • mathematics made easy
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  • mathematics e
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LECTURE NOTES ON
MATHEMATICAL METHODS

Mihir Sen
Joseph M. Powers

Department of Aerospace and Mechanical Engineering
University of Notre Dame
Notre Dame, Indiana 46556-5637
USA

updated
29 July 2012, 2:31pm

,2




CC BY-NC-ND. 29 July 2012, Sen & Powers.

,Contents

Preface 11

1 Multi-variable calculus 13
1.1 Implicit functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Functional dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 Coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.1 Jacobian matrices and metric tensors . . . . . . . . . . . . . . . . . . 22
1.3.2 Covariance and contravariance . . . . . . . . . . . . . . . . . . . . . . 31
1.3.3 Orthogonal curvilinear coordinates . . . . . . . . . . . . . . . . . . . 41
1.4 Maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.4.1 Derivatives of integral expressions . . . . . . . . . . . . . . . . . . . . 44
1.4.2 Calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.5 Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2 First-order ordinary differential equations 57
2.1 Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.2 Homogeneous equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.3 Exact equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.4 Integrating factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.5 Bernoulli equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.6 Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.7 Reduction of order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.7.1 y absent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.7.2 x absent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.8 Uniqueness and singular solutions . . . . . . . . . . . . . . . . . . . . . . . . 71
2.9 Clairaut equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3 Linear ordinary differential equations 79
3.1 Linearity and linear independence . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2 Complementary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2.1 Equations with constant coefficients . . . . . . . . . . . . . . . . . . . 82

3

, 4 CONTENTS


3.2.1.1 Arbitrary order . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2.1.2 First order . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2.1.3 Second order . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.2.2 Equations with variable coefficients . . . . . . . . . . . . . . . . . . . 85
3.2.2.1 One solution to find another . . . . . . . . . . . . . . . . . . 85
3.2.2.2 Euler equation . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3 Particular solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3.1 Method of undetermined coefficients . . . . . . . . . . . . . . . . . . 88
3.3.2 Variation of parameters . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.3.3 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.3.4 Operator D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4 Series solution methods 103
4.1 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1.1 First-order equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.1.2 Second-order equation . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.1.2.1 Ordinary point . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.1.2.2 Regular singular point . . . . . . . . . . . . . . . . . . . . . 108
4.1.2.3 Irregular singular point . . . . . . . . . . . . . . . . . . . . 114
4.1.3 Higher order equations . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.2 Perturbation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2.1 Algebraic and transcendental equations . . . . . . . . . . . . . . . . . 115
4.2.2 Regular perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.2.3 Strained coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2.4 Multiple scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.2.5 Boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.2.6 WKBJ method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.2.7 Solutions of the type eS(x) . . . . . . . . . . . . . . . . . . . . . . . . 139
4.2.8 Repeated substitution . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5 Orthogonal functions and Fourier series 147
5.1 Sturm-Liouville equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.1.1 Linear oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.1.2 Legendre’s differential equation . . . . . . . . . . . . . . . . . . . . . 153
5.1.3 Chebyshev equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.1.4 Hermite equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.1.4.1 Physicists’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.1.4.2 Probabilists’ . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.1.5 Laguerre equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.1.6 Bessel’s differential equation . . . . . . . . . . . . . . . . . . . . . . . 165

CC BY-NC-ND. 29 July 2012, Sen & Powers.

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