Ordinary And Partial Differential Equations

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Ordinary and Partial Differential EquationsAn Introduction to Dynamical SystemsJohn W. Cain, Ph.D. and Angela M. Reynolds, Ph.D.

Mathematics Textbook Series. Editor: Lon Mitchell1. Book of Proof by Richard Hammack2. Linear Algebra by Jim Hefferon3. Abstract Algebra: Theory and Applications by Thomas Judson4. Ordinary and Partial Differential Equations by John W. Cain and Angela M. ReynoldsDepartment of Mathematics & Applied MathematicsVirginia Commonwealth UniversityRichmond, Virginia, 23284Publication of this edition supported by the Center for Teaching Excellence at v c uOrdinary and Partial Differential Equations: An Introduction to Dynamical SystemsEdition 1.0 2010 by John W. Cain and Angela ReynoldsThis work is licensed under the Creative Commons Attribution-NonCommercial-No DerivativeWorks 3.0 License and is published with the express permission of the authors.Typeset in 10pt Palladio L with Pazo Math fonts using PDFLATEX

AcknowledgementsJohn W. Cain expresses profound gratitude to his advisor, Dr. David G. Schaeffer, James B. Duke Professor of Mathematics at Duke University. The firstfive chapters are based in part upon Professor Schaeffer’s introductory graduate course on ordinary differential equations. The material has been adaptedto accommodate upper-level undergraduate students, essentially by omittingtechnical proofs of the major theorems and including additional examples. Othermajor influences on this book include the excellent texts of Perko [8], Strauss [10],and Strogatz [11]. In particular, the material presented in the last five chapters(including the ordering of the topics) is based heavily on Strauss’ book. On theother hand, our exposition, examples, and exercises are more “user-friendly”,making our text more accessible to readers with less background in mathematics.Dr. Reynolds dedicates her portion of this textbook to her mother, father andsisters, she thanks them for all their support and love.Finally, Dr. Cain dedicates his portion of this textbook to his parents Jeanetteand Harry, who he loves more than words can express.iii

ductionInitial and Boundary Value Problems . . . . . . . . . . . . . . . . .4Linear, Constant-Coefficient Systems82.1 Homogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.1Diagonalizable Matrices.2.1.2Algebraic and Geometric Multiplicities of Eigenvalues. . . . 21. . . . . . . . . . . . . . . . . . . . 122.1.3Complex Eigenvalues. . . . . . . . . . . . . . . . . . . . . . . 292.1.4Repeated Eigenvalues and Non-Diagonalizable Matrices. . 372.2Phase Portraits and Planar Systems . . . . . . . . . . . . . . . . . . 452.3Stable, Unstable, and Center Subspaces . . . . . . . . . . . . . . . . 572.4Trace and Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . 652.5Inhomogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . 6778Nonlinear Systems: Local Theory3.1Linear Approximations of Functions of Several Variables . . . . . . 813.2Fundamental Existence and Uniqueness Theorem . . . . . . . . . . 843.3Global Existence, Dependence on Initial Conditions . . . . . . . . . 863.4Equilibria and Linearization . . . . . . . . . . . . . . . . . . . . . . . 943.5The Hartman-Grobman Theorem . . . . . . . . . . . . . . . . . . . . 983.63.7The Stable Manifold Theorem . . . . . . . . . . . . . . . . . . . . . . 100Non-Hyperbolic Equilibria and Lyapunov Functions . . . . . . . . 105Periodic, Heteroclinic, and Homoclinic Orbits4.1122Periodic Orbits and the Poincaré-Bendixon Theorem . . . . . . . . 122iv

co n t e n t s4.25689Heteroclinic and Homoclinic Orbits . . . . . . . . . . . . . . . . . . 130Bifurcations1405.1Three Basic Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . 1405.2Dependence of Solutions on Parameters . . . . . . . . . . . . . . . . 1485.3Andronov-Hopf Bifurcations . . . . . . . . . . . . . . . . . . . . . . 151Introduction to Delay Differential Equations6.17v166Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1686.2Solving Constant-Coefficient Delay Differential Equations . . . . . 1696.3Characteristic Equations . . . . . . . . . . . . . . . . . . . . . . . . . 1716.4The Hutchinson-Wright Equation . . . . . . . . . . . . . . . . . . . . 172Introduction to Difference Equations1807.1Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1807.2Linear, Constant-Coefficient Difference Equations . . . . . . . . . . 1817.3First-Order Nonlinear Equations and Stability . . . . . . . . . . . . 1917.4Systems of Nonlinear Equations and Stability . . . . . . . . . . . . 1957.5Period-Doubling Bifurcations . . . . . . . . . . . . . . . . . . . . . . 2007.6Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2047.7How to Control Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . 208Introduction to Partial Differential Equations2188.1Basic Classification of Partial Differential Equations . . . . . . . . . 2218.2Solutions of Partial Differential Equations . . . . . . . . . . . . . . . 2278.3Initial Conditions and Boundary Conditions . . . . . . . . . . . . . 2288.4Visualizing Solutions of Partial Differential Equations . . . . . . . . 233Linear, First-Order Partial Differential Equations2369.1Derivation and Solution of the Transport Equation . . . . . . . . . 2399.2Method of Characteristics: More Examples . . . . . . . . . . . . . . 24110 The Heat and Wave Equations on an Unbounded Domain25010.1 Derivation of the Heat and Wave Equations . . . . . . . . . . . . . . 25010.2 Cauchy Problem for the Wave Equation . . . . . . . . . . . . . . . . 25510.3 Cauchy Problem for the Heat Equation . . . . . . . . . . . . . . . . 26510.4 Well-Posedness and the Heat Equation . . . . . . . . . . . . . . . . 27610.5 Inhomogeneous Equations and Duhamel’s Principle . . . . . . . . 284

vi11 Initial-Boundary Value Problems29711.1 Heat and Wave Equations on a Half-Line . . . . . . . . . . . . . . . 29711.2 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 30611.2.1 Wave Equation, Dirichlet Problem. . . . . . . . . . . . . . . . 30711.2.2 Heat Equation, Dirichlet Problem. . . . . . . . . . . . . . . . 31311.2.3 Wave Equation, Neumann Problem. . . . . . . . . . . . . . . 31811.2.4 Heat Equation, Neumann Problem. . . . . . . . . . . . . . . 32411.2.5 Mixed Boundary Conditions: An Example. . . . . . . . . . . 32412 Introduction to Fourier Series33012.1 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33212.1.1 Fourier sine series. . . . . . . . . . . . . . . . . . . . . . . . . 33212.1.2 Fourier cosine series. . . . . . . . . . . . . . . . . . . . . . . . 33712.1.3 Fourier series. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34212.2 Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . 34412.2.1 Norms, distances, inner products, and convergence. . . . . 34712.2.2 Convergence theorems. . . . . . . . . . . . . . . . . . . . . . 35913 The Laplace and Poisson Equations36713.1 Dirchlet and Neumann Problems . . . . . . . . . . . . . . . . . . . . 37013.2 Well-posedness and the Maximum Principle . . . . . . . . . . . . . 37213.3 Translation and Rotation Invariance . . . . . . . . . . . . . . . . . . 37513.4 Laplace’s Equation on Bounded Domains . . . . . . . . . . . . . . . 38313.4.1 Dirichlet problem on a rectangle. . . . . . . . . . . . . . . . . 38313.4.2 Dirichlet problem on a disc. . . . . . . . . . . . . . . . . . . . 390Guide to Commonly Used Notation404References406Index407

CHAPTER 1IntroductionThe mathematical sub-discipline of differential equations and dynamical systemsis foundational in the study of applied mathematics. Differential equationsarise in a variety of contexts, some purely theoretical and some of practicalinterest. As you read this textbook, you will find that the qualitative andquantitative study of differential equations incorporates an elegant blend of linearalgebra and advanced calculus. For this reason, it is expected that the reader hasalready completed courses in (i) linear algebra; (ii) multivariable calculus; and(iii) introductory differential equations. Familiarity with the following topics isespecially desirable: From basic differential equations: separable differential equations and separation of variables; and solving linear, constant-coefficient differential equationsusing characteristic equations. From linear algebra: solving systems of m algebraic equations with n unknowns; matrix inversion; linear independence; and eigenvalues/eigenvectors. From multivariable calculus: parametrized curves; partial derivatives andgradients; and approximating a surface using a tangent plane.Some of these topics will be reviewed as we encounter them later—in thischapter, we will recall a few basic notions from an introductory course indifferential equations. Readers are encouraged to supplement this book with theexcellent textbooks of Hubbard and West [5], Meiss [7], Perko [8], Strauss [10],and Strogatz [11].Question: Why study differential equations?1

2Answer: When scientists attempt to mathematically model various naturalphenomena, they often invoke physical “laws” or biological “principles” whichgovern the rates of change of certain quantities of interest. Hence, the equationsin mathematical models tend to include derivatives. For example, supposethat a hot cup of coffee is placed in a room of constant ambient temperature α.Newton’s Law of Cooling states that the rate of change of the coffee temperatureT (t) is proportional to the difference between the coffee’s temperature and theroom temperature. Mathematically, this can be expressed asdTdt k ( T α ),where k is a proportionality constant.Solution techniques for differential equations (de s) depend in part upon howmany independent variables and dependent variables the system has.Example 1.0.1. One independent variable and one independent variable. Inwriting the equationd2 y cos( xy) 3,dx2it is understood that y is the dependent variable and x is the independentvariable.When a differential equation involves a single independent variable, we referto the equation as an ordinary differential equation (ode).Example 1.0.2. If there are several dependent variables and a single independentvariable, we might have equations such asdy x2 y xy2 z,dxdz z y cos x.dxThis is a system of two ode s, and it is understood that x is the independentvariable.Example 1.0.3. One dependent variable, several independent variables. Considerthe de u 2 u 2 u 2 2. t x yThis equation involves three independent variables (x, y, and t) and one dependent variable, u. This is an example of a partial differential equation (pde). If thereare several independent variables and several dependent variables, one may havesystems of pde s.

in t ro du c t i o n3Although these concepts are probably familiar to the reader, we give a moreexact definition for what we mean by ode. Suppose that x and y are independentand dependent variables, respectively, and let y(k) ( x ) denote the kth derivativeof y with respect to x. (If k 3, we will use primes.)Definition 1.0.4. Any equation of the form F ( x, y, y0 , y00 , . . . , y(n) ) 0 is calledan ordinary differential equation. If y(n) is the highest derivative appearing in theequation, we say that the ode is of order n.Example 1.0.5. d3 ydx3 2 (cos x )dyd2 y y 2dxdxcan be written as (y000 )2 yy00 (cos x )y0 0, so using the notation in the aboveDefinition, we would have F ( x, y, y0 , y00 , y000 ) (y000 )2 yy00 (cos x )y0 . This is athird-order ode.Definition 1.0.6. A solution of the ode F ( x, y, y0 , y00 , . . . , y(n) ) 0 on an intervalI is any function y( x ) which is n-times differentiable and satisfies the equationon I.Example 1.0.7. For any choice of constant A, the functiony( x ) Aex1 Aexis a solution of the first-order ode y0 y y2 for all real x. To see why, we usethe quotient rule to calculatey0 Aex (1 Aex ) ( Aex )2Aex .(1 Aex )2(1 Aex )2By comparison, we calculate thaty y2 Aex( Aex )2Aex .xx2(1 Ae ) (1 Ae )(1 Aex )2Therefore, y0 y y2 , as claimed.The definition of a solution of an ode is easily extended to systems of ode s(see below). In what follows, we will focus solely on systems of first-order ode s.This may seem overly restrictive, until we make the following observation.

4i n i t i a l a n d b o u n da r y va l u e p ro b l e m sObservation. Any nth-order ode can be written as a system of n first-orderode s. The process of doing so is straightforward, as illustrated in the followingexample:Example 1.0.8. Consider the second-order ode y00 (cos x )y0 y2 ex . To avoidusing second derivatives, we introduce a new dependent variable z y0 so thatz0 y00 . Our ode can be re-written as z0 (cos x )z y2 ex . Thus, we haveobtained a system of two first-order ode s:dy z,dxdz (cos x )z y2 ex .dxA solution of the above system of ode s on an open interval I is any vectorof differentiable functions [y( x ), z( x )] which simultaneously satisfy both ode swhen x I.Example 1.0.9. Consider the systemdz y.dtdy z,dtWe claim that for any choices of constants C1 and C2 ,"y(t)z(t)#" C1 cos t C2 sin t# C1 sin t C2 cos tis a solution of the system. To verify this, assume that y and z have this form.Differentiation reveals that y0 C1 sin t C2 cos t and z0 C1 cos t C2 sin t.Thus, y0 z and z0 y, as required.1.1. Initial and Boundary Value ProblemsIn the previous example, the solution of the system of ode s contains arbitraryconstants C1 and C2 . Therefore, the system has infinitely many solutions. Inpractice, one often has additional information about the underlying system,allowing us to select a particular solution of practical interest. For example,suppose that a cup of coffee is cooling off and obeys Newton’s Law of Cooling.In order to predict the coffee’s temperature at future times, we would need tospecify the temperature of the coffee at some reference time (usually consideredto be the “initial” time). By specifying auxiliary conditions that solutions of an

in t ro du c t i o n5ode must satisfy, we may be able to single out a particular solution. There aretwo usual ways of specifying auxiliary conditions.Initial conditions. Suppose F ( x, y, y0 , y00 , . . . , y(n) ) 0 is an nth order odewhich has a solution on an open interval I containing x x0 . Recall fromyour course on basic differential equations that, under reasonable assumptions,we would expect the general solution of this ode to contain n arbitrary constants.One way to eliminate these constants and single out one particular solution is tospecify n initial conditions. To do so, we may specify values fory( x0 ), y0 ( x0 ), y00 ( x0 ), . . . y(n 1) ( x0 ).We regard x0 as representing some “initial time”. An ode together with its initialconditions (ic s) forms an initial value problem (ivp). Usually, initial conditionswill be specified at x0 0.Example 1.1.1. Consider the second-order ode y00 ( x ) y( x ) 0. You can checkthat the general solution is y( x ) C1 cos x C2 sin( x ), where C1 and C2 arearbitrary constants. To single out a particular solution, we would need to specifytwo initial conditions. For example, if we require that y(0) 1 and y0 (0) 0, wefind that C1 1 and C2 0. Hence, we obtain a particular solution y( x ) cos x.If we have a system of n first-order ode s, we will specify one initial conditionfor each independent variable. If the dependent variables arey1 ( x ), y2 ( x ), . . . y n ( x ),we typically specify the values ofy1 (0), y2 (0), . . . , y n (0).Boundary conditions. Instead of specifying requirements that y and its derivatives must satisfy at one particular value of the independent variable x, we couldinstead impose requirements on y and its derivatives at different x values. Theresult is called a boundary value problem (bvp).Example 1.1.2. Consider the boundary value problem y00 y 0 with boundaryconditions y(0) 1 and y(π/2) 0. The general solution of the ode isy( x ) C1 cos x C2 sin x. Using the first boundary condition, we find that

6i n i t i a l a n d b o u n da r y va l u e p ro b l e m sC1 1. Since y0 ( x ) C1 sin x C2 cos x, the second boundary condition tellsus that C1 0. Notice that the two boundary conditions produce conflictingrequirements on C1 . Consequently, the bvp has no solutions.As the previous example suggests, boundary value problems can be a trickymatter. In the ode portion of this text, we consider only initial value problems.Exercises1. Write the equation of the line that passes through the points ( 1, 2, 3) and(4, 0, 1) in R3 , three-dimensional Euclidean space.2. Find the general solution of the differential equationd3 yd2 ydy 2 2 5 0.3dxdxdx3. Find the general solution of the differential equationdyd2 y 6 9y 0.2dxdx4. Solve the ivpy00 3y0 2y 0,y(0) 1,y0 (0) 1.y(0) 0,y(1) e.5. Solve (if possible) the bvpy00 3y0 2y 0,6. Solve the ivpy(4) y00 0,y(0) 1,y0 (0) 0,y00 (0) 1,y000 (0) 0.7. Solve the differential equationdy (y 2)(y 1).dx

in t ro du c t i o n8. Solve the ivp7dy ey sin x,dxy(0) 0.9. Find the equations of the planes tangent to the surfacez f ( x, y) x2 2x y2 2y 2at the points ( x, y, z) (1, 1, 0) and ( x, y, z) (0, 2, 2).10. Find the eigenvalues of the matrix"A 1441#and, for each eigenvalue, find a corresponding eigenvector.11. Find the eigenvalues of the matrix 1 A 0 3 0 0 1 213and, for each eigenvalue, find a corresponding eigenvector.12. Write the following differential equations as systems of first-order ode s:y00 5y0 6y 0 y00 2y0 7 cos(y0 )y(4) y00 8y0 y2 ex .

CHAPTER 2Linear, Constant-Coefficient SystemsThere are few classes of ode s for which exact, analytical solutions can beobtained by hand. However, for many systems which cannot be solvedexplicitly, we may approximate the dynamics by using simpler systems of ode swhich can be solved exactly. This often allows us to extract valuable qualitativeinformation about complicated dynamical systems. We now introduce techniques for systematically solving linear systems of first-order ode s with constantcoefficients.Notation. Because we will be working with vectors of dependent variables, weshould establish (or recall) some commonly used notation. We denote the set ofreal numbers by R. We let Rn denote the set of all vectors with n components,each of which is a real number. Usually, vectors will be denoted by bold letterssuch as x, y, and we will use capital letters such as A to denote n n matricesof real numbers. Generally, we shall not distinguish between row vectors andcolumn vectors, as our intentions will usually be clear from the context. Forexample, if we write the product xA, then x should be treated as a row vector,whereas if we write Ax, then x is understood to be a column vector. If we writex(t), we mean a vector of functions, each of which depends on a variable t. Insuch cases, the vector x(0) would be a constant vector in which each componentfunction has been evaluated at t 0. Moreover, the vector x0 (t) is the vectorconsisting of the derivatives of the functions which form the components of x(t).Systems with constant coefficients. Suppose that y1 , y2 , . . . yn are variableswhich depend on a single variable t. The general form of a linear, constant-8

li n e a r, c o n s ta n t -c o e f f i c i e n t s y s t e m s9coefficient system of first-order ode s is as follows:dy1 a11 y1 (t) a12 y2 (t) · · · a1n yn (t) f 1 (t)dtdy2 a21 y1 (t) a22 y2 (t) · · · a2n yn (t) f 2 (t)dt.(2.1)dyn an1 y1 (t) an2 y2 (t) · · · ann yn (t) f n (t).dtHere, each aij is a constant (1 i, j n), and f i (t) (i 1, 2, . . . n) are functionsof t only.Example 2.0.3. Soon, we will learn how to solve the li

(iii) introductory differential equations. Familiarity with the following topics is especially desirable: From basic differential equations: separable differential equations and separa-tion of variables; and solving linear, constant-coefficient differential equations using characteristic equations.

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