Differential Equations - Department of Mathematics, HKUST

1 Introduction to DifferentialEquationsLecture notes for MATH 2351/2352 Jeffrey R. Chasnov 10 8 6 4 202 2 1012y0 Airy s functions 10 8 6 4 202 2 1012xy1 TheHongKongUniversity ofScience andTechnologyiiThe Hong Kong University of Science and TechnologyDepartment of MathematicsClear Water Bay, KowloonHong KongCopyrightc 2009 2016 by Jeffrey Robert ChasnovThis work is licensed under the Creative Commons Attribution Hong Kong License. Toview a copy of this license, visit or senda letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, follows are my lecture notes for a first course in Differential Equations ,taught at the Hong Kong University of Science and Technology.

2 Included in thesenotes are links to short tutorial videos posted on of the material of Chapters 2-6 and 8 has been adapted from the widelyused textbook elementary Differential Equations and boundary value problems by Boyce & DiPrima (John Wiley & Sons, Inc., Seventh Edition,c 2001). Many ofthe examples presented in these notes may be found in this book. The material ofChapter 7 is adapted from the textbook Nonlinear dynamics and chaos by StevenH. 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.

3 An associated freereview book with links to YouTube videos is also available from the ebook I welcome any comments, suggestions or corrections sent by emailto Links to my website, these lecture notes, my YouTubepage, and the free ebook from are given : ~machasYouTube: notes: ~ : A short mathematical trigonometric functions .. exponential function and the natural logarithm .. of the derivative .. a combination of functions .. sum or difference rule .. product rule .. quotient rule .. chain rule .. elementary functions .. power rule .. functions.

4 And natural logarithm functions .. of the integral .. fundamental theorem of calculus .. and indefinite integrals .. integrals of elementary functions .. Substitution .. Integration by parts .. Taylor series .. Functions of several variables .. Complex numbers ..81 Introduction to simplest type of Differential equation .. 132 First-order Euler method .. Equations .. Equations .. interest .. reactions .. velocity .. velocity .. circuit .. logistic equation .. 293 Second-order odes, constant Euler method.

5 Principle of superposition .. Wronskian .. odes .. real roots .. complex-conjugate roots .. roots .. odes .. linear first-order odes revisited .. circuit .. on a spring .. resonance .. 504 The Laplace and properties .. of initial value problems .. and Dirac delta functions .. function .. delta function .. or impulsive terms .. 635 Series points .. singular points: Cauchy-Euler Equations .. real roots .. complex-conjugate roots .. roots .. 736 Systems of , determinants and the eigenvalue problem.

6 First-order Equations .. real eigenvalues .. complex-conjugate eigenvalues .. eigenvalues with one eigenvector .. modes .. 867 Nonlinear Differential points and stability .. dimension .. dimensions .. bifurcations .. bifurcation .. bifurcation .. pitchfork bifurcation .. pitchfork bifurcation .. : a mathematical model of a fishery .. bifurcations .. Hopf bifurcation .. Hopf bifurcation .. 101viiiCONTENTSCONTENTS8 Partial Differential of the diffusion equation .. of the wave equation.

7 Series .. sine and cosine series .. of the diffusion equation .. boundary conditions .. boundary conditions .. with closed ends .. of the wave equation .. string .. string .. initial conditions .. Laplace equation .. problem for a rectangle .. problem for a circle .. Schr dinger equation .. derivation of the Schr dinger equation .. time-independent Schr dinger equation .. in a one-dimensional box .. simple harmonic oscillator .. in a three-dimensional box .. hydrogen atom .. 132 CONTENTSixCONTENTSxCONTENTSC hapter 0A short mathematical reviewA basic understanding of calculus is required to undertake a study of differentialequations.

8 This zero chapter presents a short The trigonometric functionsThe Pythagorean trigonometric identity issin2x+cos2x=1,and the addition theorems aresin(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 sinxin the first quadrant can be remembered by the rule ofquarters, with 0 =0, 30 = /6, 45 = /4, 60 = /3, 90 = /2:sin 0 = 04,sin 30 = 14,sin 45 = 24,sin 60 = 34,sin 90 = following symmetry properties are also useful:sin( /2 x) =cosx,cos( /2 x) =sinx;andsin( x) = sin(x),cos( x) =cos(x). The exponential function and the natural logarithmThe transcendental numbere, approximately , is defined ase=limn (1+1n) exponential function exp(x) =exand natural logarithm lnxare inverse func-tions satisfyingelnx=x,lnex= usual rules of exponents apply:exey=ex+y,ex/ey=ex y,(ex)p= corresponding rules for the logarithmic function areln(xy) =lnx+lny,ln(x/y) =lnx lny,lnxp= DEFINITION OF THE Definition of the derivativeThe derivative of the functiony=f(x), denoted asf (x)ordy/dx, is defined asthe slope of the tangent line to the curvey=f(x)at the point(x,y).

9 This slope isobtained by a limit, and is defined asf (x) =limh 0f(x+h) f(x)h.(1) Differentiating a combination of The sum or difference ruleThe derivative of the sum off(x)andg(x)is(f+g) =f +g .Similarly, the derivative of the difference is(f g) =f g . The product ruleThe derivative of the product off(x)andg(x)is(f g) =f g+f g ,and should be memorized as the derivative of the first times the second plus thefirst times the derivative of the second. The quotient ruleThe derivative of the quotient off(x)andg(x)is(fg) =f g f g g2,and should be memorized as the derivative of the top times the bottom minus thetop times the derivative of the bottom over the bottom squared.

10 The chain ruleThe derivative of the composition off(x)andg(x)is(f(g(x))) =f (g(x)) g (x),and should be memorized as the derivative of the outside times the derivative ofthe inside. 2 CHAPTER 0. A SHORT MATHEMATICAL DIFFERENTIATING elementary Differentiating elementary The power ruleThe derivative of a power ofxis given byddxxp=pxp Trigonometric functionsThe derivatives of sinxand cosxare(sinx) =cosx,(cosx) = thus say that the derivative of sine is cosine, and the derivative of cosine isminus sine. Notice that the second derivatives satisfy(sinx) = sinx,(cosx) = Exponential and natural logarithm functionsThe derivative ofexand lnxare(ex) =ex,(lnx) = Definition of the integralThe definite integral of a functionf(x)>0 fromx=atob(b>a) is definedas the area bounded by the vertical linesx=a,x=b, the x-axis and the curvey=f(x).