Differential Equations by R. Decker and B. Albright Contents 1 Introduction to Differential Equations 3 1.1 A Brief Overview of Differential Equations . 3 1.2 Explicit, Numerical, and Graphical Solutions . 20 1.3 Mathematical Modeling with Differential Equations: General Principles . 35 1.4 Deriving Models of Electrical Circuits . 48 1.5 Using Lagrange’s Equations to Model Mechanical Systems . 53 2 First-order Differential Equations 61 2.1 Linear first-order differential equations . 61 2.2 Existence, uniqueness, and portraits for first-order equations . 74 2.3 Separable Differential Equations . 83 2.4 Numerical methods for first-order equations . 96 2.5 Autonomous first-order equations and bifurcations . 105 3 Second-order Differential Equations 121 3.1 Linear Second-order Equations . 124 3.2 Harmonic Oscillators . 137 3.3 Nonhomogeneous Equations and The Method of Undetermined Coefficients 152 3.4 Driven Mass-Spring Systems, Beats, and Resonance . 162 3.5 Numerical Methods for Second-Order DE’s and Systems . 174 i ii CONTENTS 3.6 Qualitative Methods and the Phase Plane . 184 4 Linear Algebra Interlude 207 4.1 Matrices, Vectors, Scalars . 208 4.2 Eigenvalues and Eigenvectors . 214 5 Systems of First-order Differential Equations 219 5.1 Explicit Solutions of Constant-coefficient Linear Systems . 219 5.2 Stability of Autonomous Linear Systems . 224 5.3 Fixed Points and Stability of Nonlinear Autonomous Systems . 232 5.4 Bifurcations in Systems . 235 6 Laplace Transforms 243 6.1 Simple Laplace Transforms . 243 6.2 Deriving More Laplace Transforms . 256 6.3 The Unit Step and Delta functions . 266 6.4 Convolution and Circuits . 283 Preface test The classical approach to introductory differential equations textbooks is to present tech- niques for analytically solving different categories of differential equations and then ana- lyzing the solutions algebraically. In this book we take a more modern approach, utilizing software to graphically and numerically solve differential equations. The focus of this text is on the setting up, or modeling, of the equations and the analysis of their solutions. This text is intended for a one semester introduction course to differential equations for math, science, and engineering majors. The prerequisite is two semesters of calculus. This book is intended to be used with available software such as Maple, Mathematica, Mat- lab, Maxima, Wolfram Alpha, TI CAS enabled calculators, and websites. Interactive java graphing applets for first-order differential equations andfirst-order systems of two equations are available at uhaweb.hartford.edu/rdecker/DeckerDEbook/DeckerDEbook.html (no www at the beginning). Other applets specifically related to examples in the text are located there also. 1 2 CONTENTS Chapter 1 Introduction to Differential Equations In this chapter we introduce the main concepts behind differential equations, why they are important, how they can be derived (created), and how information can be extracted from them in the form of various types of solutions (exact, graphical and numerical). The rest of the text will develop these ideas further by categorizing differential equations and introducing techniques specific to those categories. 1.1 A Brief Overview of Differential Equations The physical laws of the universe are written in the language of differential equations. The classical mechanics of Newton, Lagrange and Hamilton, the fluid mechanics of Bernoulli and Euler, and Maxwell’s theory of electricity and magnetism are all expressed via differential equations - and form much of the theoretical basis of the engineering disciplines. In the area of modern physics, Einstein’s theory of general relativity and the quantum mechanics of Schrodinger and Dirac are based on differential equations. Differential equations have invaded many other branches of science, including (but not limited to) chemistry, biology, economics and finance, and meteorology. It is no exaggeration to claim that themodern world as we know it could not have come into being without the development of this branch of mathematics. 3 4 CHAPTER 1 Introduction to Differential Equations Laplace’s dream The beginning student may be surprised to find that differential equations can be usedto predict the future - and they have a much better track record than any psychic. To quote the great mathematician Pierre-Simon Laplace1 We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes. —Pierre Simon Laplace, A Philosophical Essay on Probabilities2 The engineer and the crumpled paper The process described by Laplace goes something like this. Imagine that an engineer has accidently knocked a crumpled piece of paper out of her third story window, 8 meters above the ground. Being a good environmentalist, she wonders how much time she has before the paper hits the ground. She immediately recalls Newton’s second law F = ma. The law says that the sum of all of the forces acting on a body are equal to the mass of the body multiplied by its acceleration. There are two forces acting on the paper; the force of gravity (pulling it downward) and the force of air resistance (which acts in the opposite direction of the motion). Working furiously, she assigns the variable x to the position of the paper (distance above the ground in meters), lets t (in seconds) represent the time elapsed since she dropped the paper, and recalls from calculus that acceleration is the second derivative of position. Newton’s second law becomes F = mx00. She also knows that the force of gravity is given by mg where m is the mass of the body and g is the acceleration due to gravity (in meters per second2). 1Laplace, Pierre Simon, A Philosophical Essay on Probabilities, translated into English from the original French 6th ed. by Truscott, F.W. and Emory, F.L., Dover Publications (New York, 1951) p.4 2Due to the development of quantum theory in the 1920’s and 1930’s,this statement must be modified a bit - the differential equations of quantum mechanics make predictions about the probabilities of certain events occurring, at least on a microscopic scale. The spirit of the statement still holds, as extremely accurate predictions of such probabilities can be made. 1.1 A Brief Overview of Differential Equations 5 She also assumes that the force due to air resistance is proportional to the velocity (this is a common assumption). This means that the force due to air resistance is equal to cx0 where c > 0 is a constant. If she takes the upward direction to be positive, Newton’s second law yields the equation −mg − cx0 = mx00 (the negative sign on the cx0 term reflects the idea that the force of friction is opposite to the direction of motion, so that if the paper is falling, x0 is negative and hence −cx0 points in the upward direction). Rearranging this equation and dividing by m our engineer obtains the equation c x00 = − x0 − g: (1.1) m This equation, called a differential equation, describes a relationship between the paper’s velocity x0(t) as a function of time, its acceleration x00(t) and the constants c, m, and g. The engineer also knows two other pieces of information. The fact that the window is 8 meters above the ground means that x(0) = 8 (taking t = 0 to be the time the paper starts its fall). Also, the downward velocity of the paper is initially zero, since the paper is knocked off a stationary surface. Thus x0(0) = 0. The equations x(0) = 8 and x0(0) = 0 are called initial conditions, and a DE along with one or more initial conditions is called an initial value problem. The engineer needs to estimate the values of the constants c, m, and g in order to get a good prediction of when the paper hits the ground. The last is easy, as it is well know meters that the acceleration of gravity is 9:8 sec2 . Also, she knows the mass of one piece of paper is about 4:5 grams, or 0:0045 kilograms. The value of c, the proportionality constant for air resistance is harder, but fortunately she is an airplane designer, and has measured this constant for many different objects, including crumpled paper. The value isabout newtons c = 0:01 meter/sec . Differential equation (1.1) with the values of the constants substituted in becomes x00 = −2x0 − 9:8: (1.2) Now comes the key step. The engineer wants to find a function that solves the initial value problem. Specifically, she wants a function x(t) that solves the differential equation (this means the second derivative of x must equal −2 times the first derivative minus 9.8) and satisfies the conditions x(0) = 8 and x0(0) = 0 (how this is done in general is the subject of much of the rest of this text). This function is called a solution to the initial value problem. 6 CHAPTER 1 Introduction to Differential Equations Using her knowledge of differential equations, she obtains the following solution: x(t) = −2:45 exp(−2t) − 4:9t + 10:45 This function predicts the height above the ground of the crumpled paper for any value of t (in seconds). One can easily verify that this function solves the differential equation and satisfies the initial conditions (this will be done later).