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The Hong Kong University of Science and Technology. Department of Mathematics, Clear Water Bay Kowloon. c 2009 2016 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 http creativecommons org licenses by 3 0 hk or send. a letter to Creative Commons 171 Second Street Suite 300 San Francisco California 94105. 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 2001. 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 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 An associated free. review book with links to YouTube videos is also available from the ebook publisher. bookboon com I welcome any comments suggestions or corrections sent by email. to jeffrey chasnov ust hk Links to my website these lecture notes my YouTube. page and the free ebook from bookboon com are given below. http www math ust hk machas, https www youtube com user jchasnov. Lecture notes, http www math ust hk machas differential equations pdf.

http bookboon com en differential equations with youtube examples ebook. 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 27, 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 Real distinct roots 34. 3 4 2 Complex conjugate distinct roots 36, 3 4 3 Repeated roots 37. 3 5 Inhomogeneous odes 39, 3 6 First order linear inhomogeneous odes revisited 42. 3 7 Resonance 43, 3 8 Damped resonance 46, 4 The Laplace transform 49.

4 1 Definition and properties 49, 4 2 Solution of initial value problems 53. 4 3 Heaviside and Dirac delta functions 55, 4 3 1 Heaviside function 56. 4 3 2 Dirac delta function 58, 4 4 Discontinuous or impulsive terms 59. 5 Series solutions 63, 5 1 Ordinary points 63, 5 2 Regular singular points Cauchy Euler equations 66. 5 2 1 Real distinct roots 68, 5 2 2 Complex conjugate roots 69.

5 2 3 Repeated roots 69, 6 Systems of equations 71. 6 1 Matrices determinants and the eigenvalue problem 71. 6 2 Coupled first order equations 74, 6 2 1 Two distinct real eigenvalues 74. 6 2 2 Complex conjugate eigenvalues 78, 6 2 3 Repeated eigenvalues with one eigenvector 79. 6 3 Normal modes 82, 7 Nonlinear differential equations 85. 7 1 Fixed points and stability 85, 7 1 1 One dimension 85.

7 1 2 Two dimensions 86, 7 2 One dimensional bifurcations 89. 7 2 1 Saddle node bifurcation 89, 7 2 2 Transcritical bifurcation 90. 7 2 3 Supercritical pitchfork bifurcation 91, 7 2 4 Subcritical pitchfork bifurcation 92. 7 2 5 Application a mathematical model of a fishery 94. 7 3 Two dimensional bifurcations 95, 7 3 1 Supercritical Hopf bifurcation 96. 7 3 2 Subcritical Hopf bifurcation 97, 8 Partial differential equations 99.

8 1 Derivation of the diffusion equation 99, 8 2 Derivation of the wave equation 100. 8 3 Fourier series 101, 8 4 Fourier cosine and sine series 103. 8 5 Solution of the diffusion equation 106, 8 5 1 Homogeneous boundary conditions 106. 8 5 2 Inhomogeneous boundary conditions 110, vi CONTENTS. 8 5 3 Pipe with closed ends 111, 8 6 Solution of the wave equation 113.

8 6 1 Plucked string 113, 8 6 2 Hammered string 115. 8 6 3 General initial conditions 115, 8 7 The Laplace equation 116. 8 7 1 Dirichlet problem for a rectangle 116, 8 7 2 Dirichlet problem for a circle 118. 8 8 The Schr dinger equation 121, 8 8 1 Heuristic derivation of the Schr dinger equation 121. 8 8 2 The time independent Schr dinger equation 123. 8 8 3 Particle in a one dimensional box 123, 8 8 4 The simple harmonic oscillator 124.

8 8 5 Particle in a three dimensional box 127, 8 8 6 The hydrogen atom 128. CONTENTS vii, viii CONTENTS, A short mathematical review. A basic understanding of calculus is required to undertake a study of differential. equations This zero chapter presents a short review. 0 1 The trigonometric functions, The Pythagorean trigonometric identity is. sin2 x cos2 x 1, and the addition theorems are, sin x y sin x cos y cos x sin y. cos x y cos x cos y sin x sin y, Also the values of sin x in the first quadrant can be remembered by the rule of.

quarters with 0 0 30 6 45 4 60 3 90 2, sin 0 sin 30 sin 45. sin 60 sin 90, The following symmetry properties are also useful. sin 2 x cos x cos 2 x sin x, sin x sin x cos x cos x. 0 2 The exponential function and the natural logarithm. The transcendental number e approximately 2 71828 is defined as. The exponential function exp x e x and natural logarithm ln x are inverse func. tions satisfying, eln x x ln e x x, The usual rules of exponents apply. e x ey e x y e x ey e x y e x p e px, The corresponding rules for the logarithmic function are.

ln xy ln x ln y ln x y ln x ln y ln x p p ln x, 0 3 DEFINITION OF THE DERIVATIVE. 0 3 Definition of the derivative, The derivative of the function y f x denoted as f x or dy dx is defined as. the slope of the tangent line to the curve y f x at the point x y This slope is. obtained by a limit and is defined as, 0 4 Differentiating a combination of functions. 0 4 1 The sum or difference rule, The derivative of the sum of f x and g x is. Similarly the derivative of the difference is, 0 4 2 The product rule.

The derivative of the product of f x and g x is, f g f g f g. and should be memorized as the derivative of the first times the second plus the. first times the derivative of the second, 0 4 3 The quotient rule. The derivative of the quotient of f x and g x is, and should be memorized as the derivative of the top times the bottom minus the. top times the derivative of the bottom over the bottom squared. 0 4 4 The chain rule, The derivative of the composition of f x and g x is. f g x f g x g x, and should be memorized as the derivative of the outside times the derivative of.

the inside, 2 CHAPTER 0 A SHORT MATHEMATICAL REVIEW. 0 5 DIFFERENTIATING ELEMENTARY FUNCTIONS, 0 5 Differentiating elementary functions. 0 5 1 The power rule, The derivative of a power of x is given by. 0 5 2 Trigonometric functions, The derivatives of sin x and cos x are. sin x cos x cos x sin x, We thus say that the derivative of sine is cosine and the derivative of cosine is.

minus sine Notice that the second derivatives satisfy. sin x sin x cos x cos x, 0 5 3 Exponential and natural logarithm functions. The derivative of e x and ln x are, e x e x ln x, 0 6 Definition of the integral. The definite integral of a function f x 0 from x a to b b a is defined. as the area bounded by the vertical lines x a x b the x axis and the curve. y f x This area under the curve is obtained by a limit First the area is. approximated by a sum of rectangle areas Second the integral is defined to be the. limit of the rectangle areas as the width of each individual rectangle goes to zero. and the number of rectangles goes to infinity This resulting infinite sum is called a. Riemann Sum and we define, f x dx lim, f a n 1 h h 2. where N b a h is the number of terms in the sum The symbols on the left. hand side of 2 are read as the integral from a to b of f of x dee x The Riemann. Sum definition is extended to all values of a and b and for all values of f x positive. and negative Accordingly, Z a Z b Z b Z b, f x dx f x dx and f x dx f x dx. Z c Z b Z c, f x dx f x dx f x dx, which states when f x 0 and a b c that the total area is equal to the sum of.

CHAPTER 0 A SHORT MATHEMATICAL REVIEW 3, 0 7 THE FUNDAMENTAL THEOREM OF CALCULUS. 0 7 The fundamental theorem of calculus, View tutorial on YouTube. Using the definition of the derivative we differentiate the following integral. a f s ds a f s ds, f s ds lim, dx a h 0 h, This result is called the fundamental theorem of calculus and provides a connection. between differentiation and integration, The fundamental theorem teaches us how to integrate functions Let F x be a. function such that F x f x We say that F x is an antiderivative of f x Then. from the fundamental theorem and the fact that the derivative of a constant equals. F x f s ds c, Now F a c and F b a f s ds F a Therefore the fundamental theorem.

shows us how to integrate a function f x provided we can find its antiderivative. f s ds F b F a 3, Unfortunately finding antiderivatives is much harder than finding derivatives and. indeed most complicated functions cannot be integrated analytically. We can also derive the very important result 3 directly from the definition of. the derivative 1 and the definite integral 2 We will see it is convenient to choose. the same h in both limits With F x f x we have, f s ds F s ds. F a n 1 h h, N F a nh F a n 1 h, F a nh F a n 1 h, The last expression has an interesting structure All the values of F x evaluated. at the points lying between the endpoints a and b cancel each other in consecutive. terms Only the value F a survives when n 1 and the value F b when. n N yielding again 3, 4 CHAPTER 0 A SHORT MATHEMATICAL REVIEW. 0 8 DEFINITE AND INDEFINITE INTEGRALS, 0 8 Definite and indefinite integrals.

The Riemann sum definition of an integral is called a definite integral It is convenient. to also define an indefinite integral by, f x dx F x. where F x is the antiderivative of f x, 0 9 Indefinite integrals of elementary functions. From our known derivatives of elementary functions we can determine some sim. ple indefinite integrals The power rule gives us, x n dx c n 1. When n 1 and x is positive we have, If x is negative using the chain rule we have. Therefore since, we can generalize our indefinite integral to strictly positive or strictly negative x.

Trigonometric functions can also be integrated, cos xdx sin x c sin xdx cos x c. Easily proved identities are an addition rule, f x g x dx f x dx g x dx. and multiplication by a constant, A f x dx A f x dx. This permits integration of functions such as, x2 7x 2 dx 2x c. 5 cos x sin x dx 5 sin x cos x c, CHAPTER 0 A SHORT MATHEMATICAL REVIEW 5.

0 10 SUBSTITUTION, 0 10 Substitution, More complicated functions can be integrated using the chain rule Since. f g x f g x g x, f g x g x dx f g x c, This integration formula is usually implemented by letting y g x Then one. writes dy g x dx to obtain, f g x g x dx f y dy, 0 11 Integration by parts. Another integration technique makes use of the product rule for differentiation. f g f g f g, f g f g f g, Therefore Z Z, f x g x dx f x g x f x g x dx. Commonly the above integral is done by writing, u g x dv f x dx.

du g x dx v f x, Then the formula to be memorized is. udv uv vdu, 0 12 Taylor series, A Taylor series of a function f x about a point x a is a power series repre. sentation of f x developed so that all the derivatives of f x at a match all the. derivatives of the power series Without worrying about convergence here we have. f x f a f a x a x a 2 x a 3, Notice that the first term in the power series matches f a all other terms vanishing. the second term matches f a all other terms vanishing etc Commonly the Taylor. 6 CHAPTER 0 A SHORT MATHEMATICAL REVIEW, 0 13 FUNCTIONS OF SEVERAL VARIABLES. series is developed with a 0 We will also make use of the Taylor series in a. slightly different form with x x e and a x, f x 2 f x 3.

f x e f x f x e e e, Another way to view this series is that of g e f x e expanded about e 0. Taylor series that are commonly used include, 1 x x2 for x 1. ln 1 x x for x 1, 0 13 Functions of several variables. For simplicity we consider a function f f x y of two variabl. Introduction to Differential Equations Lecture notes for MATH 2351 2352 Jeffrey R Chasnov m m k K k x 1 x 2 The Hong Kong University of Science and Technology