Differential Equations

[Pages:149]Differential Equations

Jeffrey R. Chasnov

Adapted for

:

Differential Equations for Engineers

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The Hong Kong University of Science and Technology Department of Mathematics Clear Water Bay, Kowloon Hong Kong

Copyright c 2009?2019 by Jeffrey Robert Chasnov This work is licensed under the Creative Commons Attribution 3.0 Hong Kong License. To view a copy of this license, visit or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.

Preface

What follows are my lecture notes for a first course in differential equations, taught at the Hong Kong University of Science and Technology. Included in these notes are links to short tutorial videos posted on YouTube.

Much of the material of Chapters 2-6 and 8 has been adapted from the widely used textbook "Elementary differential equations and boundary value problems" by Boyce & DiPrima (John Wiley & Sons, Inc., Seventh Edition, c 2001). Many of the examples presented in these notes may be found in this book. The material of Chapter 7 is adapted from the textbook "Nonlinear dynamics and chaos" by Steven H. Strogatz (Perseus Publishing, c 1994).

All web surfers are welcome to download these notes, watch the YouTube videos, and to use the notes and videos freely for teaching and learning.

I also have some online courses on Coursera. A lot of time and effort has gone into their production, and the video lectures have better video quality than the ones prepared for these notes. You can click on the links below to explore these courses.

If you want to learn differential equations, have a look at

Differential Equations for Engineers

If your interests are matrices and elementary linear algebra, try

Matrix Algebra for Engineers

If you want to learn vector calculus (also known as multivariable calculus, or calculus three), you can sign up for

Vector Calculus for Engineers

And if your interest is numerical methods, have a go at

Numerical Methods for Engineers

Jeffrey R. Chasnov Hong Kong

February 2021

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Contents

0 A short mathematical review

1

0.1 The trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . 1

0.2 The exponential function and the natural logarithm . . . . . . . . . . . 1

0.3 Definition of the derivative . . . . . . . . . . . . . . . . . . . . . . . . . 2

0.4 Differentiating a combination of functions . . . . . . . . . . . . . . . . 2

0.4.1 The sum or difference rule . . . . . . . . . . . . . . . . . . . . . 2

0.4.2 The product rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

0.4.3 The quotient rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

0.4.4 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

0.5 Differentiating elementary functions . . . . . . . . . . . . . . . . . . . . 3

0.5.1 The power rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

0.5.2 Trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . 3

0.5.3 Exponential and natural logarithm functions . . . . . . . . . . . 3

0.6 Definition of the integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

0.7 The fundamental theorem of calculus . . . . . . . . . . . . . . . . . . . 4

0.8 Definite and indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . 5

0.9 Indefinite integrals of elementary functions . . . . . . . . . . . . . . . . 5

0.10 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

0.11 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

0.12 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

0.13 Functions of several variables . . . . . . . . . . . . . . . . . . . . . . . . 7

0.14 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1 Introduction to odes

13

1.1 The simplest type of differential equation . . . . . . . . . . . . . . . . . 13

2 First-order odes

15

2.1 The Euler method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Separable equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Compound interest . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.2 Chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.3 Terminal velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.4 Escape velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.5 RC circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.6 The logistic equation . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Second-order odes, constant coefficients

31

3.1 The Euler method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 The principle of superposition . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Homogeneous odes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.1 Distinct real roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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3.4.2 Distinct complex-conjugate roots . . . . . . . . . . . . . . . . . . 36 3.4.3 Repeated roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Inhomogeneous odes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.6 Inhomogeneous linear first-order odes revisited . . . . . . . . . . . . . 42 3.7 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.8.1 RLC circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.8.2 Mass on a spring . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.8.3 Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.9 Damped resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 The Laplace transform

53

4.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Solution of initial value problems . . . . . . . . . . . . . . . . . . . . . . 57

4.3 Heaviside and Dirac delta functions . . . . . . . . . . . . . . . . . . . . 59

4.3.1 Heaviside function . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.2 Dirac delta function . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Discontinuous or impulsive terms . . . . . . . . . . . . . . . . . . . . . 63

5 Series solutions

67

5.1 Ordinary points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Regular singular points: Cauchy-Euler equations . . . . . . . . . . . . 70

5.2.1 Distinct real roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.2 Distinct complex-conjugate roots . . . . . . . . . . . . . . . . . . 73

5.2.3 Repeated roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 Systems of equations

75

6.1 Matrices, determinants and the eigenvalue problem . . . . . . . . . . . 75

6.2 Coupled first-order equations . . . . . . . . . . . . . . . . . . . . . . . . 78

6.2.1 Distinct real eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 78

6.2.2 Distinct complex-conjugate eigenvalues . . . . . . . . . . . . . . 82

6.2.3 Repeated eigenvalues with one eigenvector . . . . . . . . . . . 83

6.3 Normal modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7 Nonlinear differential equations

89

7.1 Fixed points and stability . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.1.1 One dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.1.2 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.2 One-dimensional bifurcations . . . . . . . . . . . . . . . . . . . . . . . . 93

7.2.1 Saddle-node bifurcation . . . . . . . . . . . . . . . . . . . . . . . 93

7.2.2 Transcritical bifurcation . . . . . . . . . . . . . . . . . . . . . . . 94

7.2.3 Supercritical pitchfork bifurcation . . . . . . . . . . . . . . . . . 95

7.2.4 Subcritical pitchfork bifurcation . . . . . . . . . . . . . . . . . . 96

7.2.5 Application: a mathematical model of a fishery . . . . . . . . . 98

7.3 Two-dimensional bifurcations . . . . . . . . . . . . . . . . . . . . . . . . 99

7.3.1 Supercritical Hopf bifurcation . . . . . . . . . . . . . . . . . . . 100

7.3.2 Subcritical Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . 101

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8 Partial differential equations

103

8.1 Derivation of the diffusion equation . . . . . . . . . . . . . . . . . . . . 103

8.2 Derivation of the wave equation . . . . . . . . . . . . . . . . . . . . . . 104

8.3 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

8.4 Fourier sine and cosine series . . . . . . . . . . . . . . . . . . . . . . . . 107

8.5 Example solutions of the diffusion equation . . . . . . . . . . . . . . . 110

8.5.1 Homogeneous boundary conditions . . . . . . . . . . . . . . . . 110

8.5.2 Inhomogeneous boundary conditions . . . . . . . . . . . . . . . 114

8.5.3 Pipe with closed ends . . . . . . . . . . . . . . . . . . . . . . . . 115

8.6 Example solutions of the wave equation . . . . . . . . . . . . . . . . . . 117

8.6.1 Plucked string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.6.2 Hammered string . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8.6.3 General initial conditions . . . . . . . . . . . . . . . . . . . . . . 119

8.7 The Laplace equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.7.1 Dirichlet problem for a rectangle . . . . . . . . . . . . . . . . . . 120

8.7.2 Dirichlet problem for a circle . . . . . . . . . . . . . . . . . . . . 122

8.8 The Schr?dinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . 125

8.8.1 Heuristic derivation of the Schr?dinger equation . . . . . . . . 125

8.8.2 The time-independent Schr?dinger equation . . . . . . . . . . . 127

8.8.3 Particle in a one-dimensional box . . . . . . . . . . . . . . . . . 127

8.8.4 The simple harmonic oscillator . . . . . . . . . . . . . . . . . . . 128

8.8.5 Particle in a three-dimensional box . . . . . . . . . . . . . . . . 131

8.8.6 The hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . 133

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