Lectures on Differential Equations1 Craig A. Tracy2 Department of Mathematics University of California Davis, CA 95616 March 2017 1c Craig A. Tracy, 2000, 2017 Davis, CA 95616 2email: [email protected] 2 Contents 1 Introduction 1 1.1Whatisadifferentialequation?............................... 2 1.2 Differential equation for the pendulum . 4 1.3Introductiontocomputersoftware............................. 7 1.4Exercises........................................... 11 2 First Order Equations & Conservative Systems 13 2.1Linearfirstorderequations................................. 13 2.2Conservativesystems..................................... 18 2.3Levelcurvesoftheenergy.................................. 30 2.4Exercises........................................... 32 3 Second Order Linear Equations 41 3.1Theoryofsecondorderequations.............................. 42 3.2Reductionoforder...................................... 46 3.3Constantcoefficients..................................... 46 3.4 Forced oscillations of the mass-spring system . 51 3.5Exercises........................................... 55 4 Difference Equations 57 4.1Introduction.......................................... 58 4.2Constantcoefficientdifferenceequations.......................... 58 4.3Inhomogeneousdifferenceequations............................ 60 4.4Exercises........................................... 61 i ii CONTENTS 5 Matrix Differential Equations 65 5.1Thematrixexponential................................... 66 5.2ApplicationofmatrixexponentialtoDEs......................... 68 5.3 Relation to earlier methods of solving constant coefficient DEs . 71 5.4ProblemfromMarkovprocesses............................... 72 5.5ApplicationofmatrixDEtoradioactivedecays...................... 75 5.6Inhomogenousmatrixequations............................... 76 5.7Exercises........................................... 80 6 Weighted String 85 6.1Derivationofdifferentialequations............................. 86 6.2Reductiontoaneigenvalueproblem............................ 88 6.3Computationoftheeigenvalues............................... 89 6.4Theeigenvectors....................................... 90 6.5Determinationofconstants................................. 93 6.6Continuumlimit:Thewaveequation............................ 95 6.7Inhomogeneousproblem................................... 99 6.8Vibratingmembrane..................................... 100 6.9Exercises........................................... 105 7 Quantum Harmonic Oscillator 115 7.1 Schr¨odingerequation..................................... 116 7.2 Harmonic oscillator . 116 7.3 Some properties of the harmonic oscillator . 125 7.4TheHeisenbergUncertaintyPrinciple........................... 128 7.5Comparisonofthreeproblems................................ 130 7.6Exercises........................................... 131 8HeatEquation 133 8.1Introduction.......................................... 134 8.2Fouriertransform....................................... 134 8.3SolvingtheheatequationbytheFouriertransform.................... 135 CONTENTS iii 8.4Heatequationonthehalf-line................................ 141 8.5Heatequationonthecircle................................. 142 8.6Exercises........................................... 145 9 Laplace Transform 147 9.1Matrixversion........................................ 148 −1 9.2 Structure of (sIn − A) ................................... 151 9.3Exercises........................................... 153 iv CONTENTS Preface Figure 1: Sir Isaac Newton, December 25, 1642–March 20, 1727 (Julian Calendar). These notes are for a one-quarter course in differential equations. The approach is to tie the study of differential equations to specific applications in physics with an emphasis on oscillatory systems. The following two quotes by V. I. Arnold express the philosophy of these notes. Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap. In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy’s warning that ugly mathematics has no permanent place under the Sun). Since scholastic mathematics that is cut off from physics is fit neither for teaching nor for application in any other science, the result was the universal hate towards mathematicians— both on the part of the poor schoolchildren (some of whom in the meantime became ministers) and of the users. V. I. Arnold, On Teaching Mathematics CONTENTS v Newton’s fundamental discovery, the one which he considered necessary to keep secret and published only in the form of an anagram, consists of the following: Data aequatione quotcunque fluentes quantitae involvente fluxions invenire et vice versa. In contemporary mathematical language, this means: “It is useful to solve differential equations”. V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations. I thank Eunghyun (Hyun) Lee for his help with these notes during the 2008–09 academic year. Also thanks to Andrew Waldron for his comments on the notes. Craig Tracy, Sonoma, California vi CONTENTS Notation Symbol Definition of Symbol R field of real numbers Rn the n-dimensional vector space with each component a real number C field of complex numbers x˙ the derivative dx/dt, t is interpreted as time x¨ the second derivative d2x/dt2, t is interpreted as time := equals by definition Ψ=Ψ(x, t) wave function in quantum mechanics ODE ordinary differential equation PDE partial differential equation KE kinetic energy PE potential energy det determinant δij the Kronecker delta, equal to 1 if i = j and 0 otherwise L the Laplace transform operator n k The binomial coefficient n choose k. Maple is a registered trademark of Maplesoft. Mathematica is a registered trademark of Wolfram Research. MatLab is a registered trademark of the MathWorks, Inc. Chapter 1 Introduction Figure 1.1: Galileo Galilei, 1564–1642. From The Galileo Project: “Galileo’s discovery was that the period of swing of a pendulum is independent of its amplitude–the arc of the swing–the isochronism of the pendulum. Now this discovery had important implications for the measurement of time intervals. In 1602 he explained the isochronism of long pendulums in a letter to a friend, and a year later another friend, Santorio Santorio, a physician in Venice, began using a short pendulum, which he called “pulsilogium,” to measure the pulse of his patients. The study of the pendulum, the first harmonic oscillator, date from this period.” See the You Tube video http://youtu.be/MpzaCCbX-z4. 1 2 CHAPTER 1. INTRODUCTION 1.1 What is a differential equation? From Birkhoff and Rota [3] A differential equation is an equation between specified derivative on an unknown function, its values, and known quantities and functions. Many physical laws are most simply and naturally formulated as differential equations (or DEs, as we will write for short). For this reason, DEs have been studied by the greatest mathematicians and mathematical physicists since the time of Newton. Ordinary differential equations are DEs whose unknowns are functions of a single variable; they arise most commonly in the study of dynamical systems and electrical networks. They are much easier to treat that partial differential equations, whose unknown functions depend on two or more independent variables. Ordinary DEs are classified according to their order. The order of a DE is defined as the largest positive integer, n, for which an nth derivative occurs in the equation. Thus, an equation of the form φ(x, y, y)=0 is said to be of the first order. From Wikipedia A differential equation is a mathematical equation that relates some function of one or more variables with its derivatives. Differential equations arise whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions the set of functions that satisfy the equation. Only the simplest differential equations admit solutions given by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy. Many fundamental laws of physics and chemistry can be formulated as differential equa- tions. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed to- gether with the sciences where the equations had originated and where the results found application. However, diverse problems,