Differential Equations, Third Edition

CONTENTS

1

FIRST-ORDER

DIFFERENTIAL

EQUATIONS 2

1.1 Modeling via Differential Equations 1.2

Analytic

Technique: Separation of Variables

1.3 Qualitative Technique: Slope Fields

36

1.4 Numerical Technique: Euler's Method

53

1.5 Existence and Uniqueness of Solutions

65

1.6

Equilibria

and the Phase Line

112

1.8 Linear Equations 1.9

Integrating

Factors for Linear Equations

Review Exercises for Chapter 1 Labs for Chapter 1

FIRST-ORDER

138

SYSTEMS

169

Methods for Special Systems

2.4 Euler's Method for Systems 2.5 The Lorenz Equations

Labs for Chapter 2

224

198

213

Review Exercises for Chapter 2

xv

151

152

2.2 The Geometry of Systems Analytic

126

144

2.1 Modeling via Systems

2.3

76

96

1.7 Bifurcations

2

20

220

187

1

xvi

CONTENTS

3

LINEAR

233

SYSTEMS

3.1 Properties of Linear Systems and the Linearity

Principle

258

3.2 Straight-Line Solutions

3.3 Phase Planes for Linear Systems with Real Eigenvalues 3.4 Complex Eigenvalues

290

3.5 Special Cases: Repeated and Zero Eigenvalues 3.6 Second-Order Linear Equations 3.7 The Trace-Determinant Plane

Review Exercises for Chapter 3

341

375

FORCING

RESONANCE

4.1 Forced Harmonic Oscillators 4.2 Sinusoidal Forcing

354

370

Labs for Chapter 3 AND

381

382

397

4.3 Undamped Forcing and Resonance 4.4

4.5 The Tacoma Narrows Bridge Review Exercises for Chapter 4 Labs for Chapter 4 5

NONLINEAR 5.1

Equilibrium

5.2 Qualitative 5.3

Hamiltonian

Nonlinear

433 443

SYSTEMS

451 452

Point Analysis 471

Analysis

Systems

484 502

Systems in Three Dimensions

5.6 Periodic Forcing of Nonlinear Review Exercises for Chapter 5 Labs for Chapter 5

421

446

5.4 Dissipative Systems 5.5

409

and Phase of the Steady State

Amplitude

309

324

3.8 Linear Systems in Three Dimensions

4

552

234

524

Systems and Chaos 549

532

274

CONTENTS

559

TRANSFORMS

LAPLACE

6

6.1 Laplace Transforms

560

6.2 Discontinuous Functions

572

6.3 Second-Order Equations

581

6.4 Delta Functions 6.5 Convolutions

and

595

Impulse Forcing

603

6.6 The Qualitative Theory of Laplace Transforms Table of Laplace Transforms

620

Review Exercises for Chapter 6 Labs for Chapter 6

7

621

624

METHODS

NUMERICAL

627

7.1 Numerical Error in Euler's Method 7.2 Improving Euler's Method 7.3 The Runge-Kutta

Method

7.4 The Effects of Finite

8

DISCRETE

628

641 649 660

Arithmetic

Review Exercises for Chapter 7 Labs for Chapter 7

664

665

DYNAMICAL

SYSTEMS

8.1 The Discrete Logistic Equation

670

8.2 Fixed Points and Periodic Points 8.3 Bifurcations 8.4 Chaos 8.5 Chaos in

683

692 701

the

Lorenz System

Review Exercises for Chapter 8 Labs for Chapter 8

612

717

709 715

669

xvii

xviii

CONTENTS

723

APPENDICES A Changing Variables

724

B The Ultimate Guess

736

C Complex Numbers and Euler's Formula

HINTS

INDEX

AND

ANSWERS

819

749

744

INDEX

Air

resistance, 4

Lorenz, Edward N., 215 McKenna, Joseph, 440 Smale, Stephen. 63 Xia , Zhihong , 495 Bob, See Boat. Boston, 95 Braun M., 152, 433 Bridges Millennium , 417 Tacoma Narrows , 433 , 448

Alligood , Kathleen, 347

methods, 13, 62 systems, 164, 187 Angular frequency, 299 Annihilation parameter, 671 ApoUo 13, 666 Asymptotic stability , 86 Analytic

Attractor

Lorenz, 710 strange, 710 Australia , 15 Autonomous equations, 24 , 40 , 76 , 96 Autonomous systems, 181

Carbon dating , 17 Carrying capacity, 9- 10 CD stores model, 235 Cell phones, 142 Center, 299, 301 Changing variables. 455 , 724 w-substitution , 724 Chaos, 218, 532, 543, 700, 701 sensitive dependence on initial conditions, 703, 706 Characteristic polynomial , 265 three-dimensional matrix , 358, 361 Checking solutions, 21, 188 Chemical reactions, 169, 483 Chopping function , 704 Circuit theory , See RC and RLC circuits. Cobweb, 683 Coefficient matrix , 238 three-dimensional , 354 Coefficient of damping , 193 Competing species system, 179, 225, 453, 472 Complex numbers , 291, 744

Baby bottle, 379 Beating, 411 Bernoulli equation, 729 Beth, 16 Bifurcation , 96- 107 creating equilibrium points , 98, 101 definition , 97 discrete dynamical systems, 692 linear systems, 348, 375 period-doubling , 694 pitchfork , 693 tangent , 692 value , 100, 103 Bifurcation diagram , 100 Biography

Alligood, Kathleen, 347 Curry , James H., 202 Kopell, Nancy , 85 819

820

Index

Compound interest, 28. 36, 125 Computers, 628, 642, 653 Conservation of energy, 492 Conserved quantity , 486, 488 Constant -coefficient equations, 325 Convolution , 605, 625 Cool picture, 163 Cooling model, 35, 141 Cooperative species system, 225, 483 Critical loci, 347 Critically damped harmonic oscillator, 331 Curry , James H., 202 Cycles, 674, 679, 683- 690 classification, 689 D'Ancona , U., 13 Damping , 193 adjustable, See MR fluids . Decoupled systems, 189 Delayed logistic equation , 718 Delta function , See Dirac delta function . Dempster, J. P., 18, 152 Dense subset, 705 Dependent variables, 3- 4 , 20 systems, 173 Determinant , 242, 342 three-dimensional matrix , 357 Wronskian , 258 Dick, See Glen. Difference equation, See Discrete dynamical system. Dirac delta function , 597 Direction field, 173 Discontinuous forcing, 572 Discrete dynamical system, 634, 670 bifurcation , 692- 696 logistic difference equation , 696 period-doubling , 694 pitchfork , 693 tangent , 692 cycles, 674, 679, 683- 690 attracting , 689 classification, 689 neutral , 689 repelling, 689 eventually fixed points , 673

eventually periodic points . 674 fixed points , 673 , 683- 690 attracting , 685

classification, 686 finding , 677 neutral , 685, 689 repelling, 685 graphical iteration , 683 histograms , 676 Newton's method , 717 orbits, 673 period-doubling route to chaos, 698 periodic orbits, 674 renormalization , 699 seed, 673 time series, 676 web diagram , 683 Discrete logistic equation, See Logistic difference equation. Dissipative systems, 502- 518 Dottie number , 678 Duffing equation

periodically forced, 532 Dynamical system discrete time , 670 Eigenvalue , 263, 264

complex. 290 real distinct , 268 repeated, 309, 310 three-dimensional matrix , 357 zero, 319 Eigenvector, 263, 266 three-dimensional matrix , 357 Electric circuit theory , See RC and RLC circuits. Energy , 489 , 492 dissipation, 505 Equilibrium points , 5, 70, 78 bifurcation , 102- 104. 348 center, 299, 301 classification, 86, 284, 462 creation, 101 Hamiltonian systems, 496-497 linear systems, 240- 242 linearization , 87, 729

node, 86 saddle, 274, 362, 463 sink , 86, 279, 362, 463 source, 86, 283, 362, 463 spiral saddle, 360 spiral sink , 299, 463 spiral source, 299, 463 stable, 86, 285 systems, 156, 177 unstable , 285 Equilibrium solutions, 5 , 10, 70 systems, 152, 177 Eric, See Initial value . Euler's formula , 293, 746 Euler's method , 53- 60, 641 errors, 62, 72 , 628 first step, 630 second step, 633 total , 635 formula , 56 improved , 641- 647 order, 637 step size, 54, 629 systems, 198- 204 Eventually fixed points , 673 Eventually periodic points, 674 Existence and Uniqueness Theorem applications, 70- 73 first -order equations, 68 systems, 205 Existence Theorem, 66 Explicit solutions, 13, 23 Exponential growth . See Population models., 562, 670 of solutions, 66, 82 Extendability Extended Linearity Principle. 116. 384 External forcing, 383 Extinction , 104 Extra firm mattress , 168 Finite arithmetic , 660- 663 First -order differential equation, 5 , 20

autonomous , 24 Bernoulli equation, 729 bifurcation changing

, 96 variables, 724

Euler's method , 53 existence, 68 linear, 112 one-parameter family , 97 power series, 736 Riccati equation, 731 separable, 20 slope fields, 36 uniqueness, 68 First -order numerical method , 637 Fixed points , 673, 683- 690 attracting , 685 classification, 686 finding , 677 neutral , 685 , 689 Poincare return map , 536 repelling, 685 Floating cube, 441 Food chain model, 525 Forced equation, 383 Forced harmonic oscillator, 382 beating, 411 discontinuous forcing, 586 glass harmonica , 430 impulse forcing, 595, 598 phase angle, 421 phase plane, 401 resonance, 415 sinusoidal forcing, 397, 413 , 421, 589 undamped , 409 Forced response, 387 Fourier transform , 560 Frequency angular , 299 natural , 299 WEEI , 299 General solution , 7, 23 systems, 190 Gib, 187, 212 Gift , 296, 325 Glass armonica, See glass harmonica. Glass harmonica , 430 Glen, See Dick. Glider, 556

822

Index

Gradient systems, 513 definition , 516 lobster motion , 513 Lyapunov functions , 513 Graphical iteration , 683 Gravitational force , 489 Growth -rate coefficient, 4 Guess-and-test method , 118, 194, 388 ultimate guess , 736

Half-life, 17 Hamilton , William R., 488 Hamiltonian function , 488 construction , 493 Hamiltonian systems, 484-497 conserved quantity , 486 definition , 488 energy, 489 equilibrium points , 496 Hamiltonian function , 488 ideal pendulum , 490 phase portraits , 488 undamped harmonic oscillator, 488 Hammer strike , 421 , 595 Harmonic oscillator, 159, 160, 193, 234 , 250, 269, 315, 377, 379 as Hamiltonian system, 488 classification, 330 clock, 340 coefficient of damping , 193 critically damped, 331, 334 335 507 damped, forced, See Forced harmonic oscillator. hard spring , 552 modified damping , 227 natural frequency, 207, 330 overdamped, 331. 333- 334 resonance, 415 sinusoidal forcing, 397, 421 soft spring , 552 spring constant , 160 trace-determinant plane , 346 undamped , 301, 330 underdamped , 331- 332 Harry , 187, 212

, 146 Heaviside function , 573 derivative of , 602 Heaviside, Oliver, 573 Hedgpath , J.W ., 15 Hermite equation , 739 Hermite polynomials , 739 Hey, Jude , 420 Histograms , 676 Homogeneous equations, 114, 325, 383, 734 Hooke's law of springs , 160 Harvesting

Ideal pendulum , 490 Identity matrix , 264 Implicit solutions, 28 Improved Euler's method , 641- 647 cost, 646 definition , 644 errors, 644 order, 646 Impulse forcing, 595 Independent variable, 3 4, 20 Initial condition, 5 , 22, 154 discrete dynamical system, See seed. Initial point , 155 Initial -value problem, 6 , 22- 23 system, 162 Integral equation , 642 Integral transform , 560 Integrating factor, 128 Inverse Laplace transform , 566 Iteration , 670, 673 graphical, 683 notation , 679 \342\200\224

Jacobian matrix , 460 Kopell, Nancy , 85 Laplace transform , 560 convolution , 605, 625 definition , 561 Dirac delta function , 597 discontinuous functions , 572 functions , 562 exponential

Index first

-order equations, 565

Heaviside function , 573 inverse, 566 linearity , 564 periodic functions , 580 poles, 614, 624 product rule , 603 qualitative theory , 612 ramp function , 579 rules, 620 sawtooth function , 581 second-order equations, 581 square wave , 581 table, 620 trigonometric functions , 582 Lazer, A . C , 433 Learning model, 15, 144 Legendre equation, 743 Legendre polynomials , 743 Levy, Matthys , 208, 433 Linear approximation , See Linearization. Linear combinations , 243 Linear equations first -order, 112 integrating factor, 128 Linearity Principle, 114, 243 Extended, 116, 384 nonhomogeneous, 114, 383 second-order, 325 Linear independence, 249 in three dimensions, 355 Linear systems angular frequency, 299 bifurcation , 348, 375 center, 299 characteristic polynomial , 265 classification, 284 three-dimensional, 361 coefficient matrix , 238 complex eigenvalues, 290 constant coefficients, 237 critical loci , 347 determinant , 242 dimension, 240

eigenvalues, 263 eigenvectors, 263

-

equilibrium points , 240 242 general solution , 250, 268, 292, 313

harmonic oscillator, See Harmonic oscillator.

Principle , 243 Extended, 384 linearly independent solutions, 250 matrix notation , 237 natural frequency, 299, 330 natural period , 299 Principle of Superposition, See Linearity Principle. real distinct eigenvalues, 268 repeated eigenvalues, 309, 318 saddle, 274, 362 sink , 279, 362 source, 283, 362 spiral sink , 299 spiral source, 299 straight -line solutions, 267 three dimensions, 354- 365 trace-determinant plane, 342 Wronskian , 258 zero eigenvalue, 319 Linearity Principle , 114, 243 Extended, 116, 384 three -dimensional systems, 355 Linearization, 87, 453, 458, 727 failure , 464- 465 Linearly independent solutions, 250 Logistic difference equation, 671 annihilation parameter, 671 bifurcation , 696 chaotic example, 702 delayed logistic, 718 fixed points , 687 without chaos , 701 Logistic equation, 9- 11, 148, 727 harvesting , 104 modified version, 89 predator-prey system, 157 qualitative analysis, 10 Long-term behavior, 10, 13 Linearity

823

824

Index

Lorenz equations, 214 , 368, 528, 709- 714 attractor , 710 Poincare return map , 712 template , 711 Lorenz, Edward N ., 214, 215 Lori, See Gib and Harry . Lost in space, 666 Lucky guess method , 329 Lyapunov function , 507 damped harmonic oscillator, 507 nonlinear pendulum , 505 Lyapunov , Aleksandr M., 507 Magic fingers , 340 Magnetorheological fluid , See MR fluid . Maple, 112 Mass-spring , See Harmonic oscillator. Mathematica , 112 Matrices characteristic polynomial , 265 degenerate, 242 determinant , 342 diagonal, 358 eigenvalue, 263 eigenvector, 263 identity , 264 Jacobian, 460 singular , 242 symmetric , 273 trace, 342 upper triangular , 273 Matrix notation , 237 McKenna, Joseph , 433, 440 Memorization model, 144 Metaphor of the parking lot, 175 Metaphor of the rope, 77 Millennium Bridge, 417 Miracle, 729, 732 Missing solutions, 27 Mixing problems, 31- 32, 43 , 131 Modeling, 2- 13 Models baby bottle, 379 bridges Millennium , 417

Tacoma Narrows , 433 , 448 carbon dating , 17 CD stores, 235 cellphones , 142 chemical reactions, 169, 483 cholesterol, 34 circuit. See RC and RLC circuits, clock, 340 competing species, 179, 225, 453 , 472 compound interest, 28 , 36, 125 cooperative species, 225 , 483 delayed logistic, 718 electric circuits, See RC and RLC circuits, exponential growth , 670 extinction , 104 extra firm mattress, 168 food chain, 525 forced harmonic oscillator, 382 glass harmonica, 430 glider, 556 hammer strike , 421 , 595 harmonic oscillator, See Harmonic oscillator, harvesting , 104, 146 housing market , 254 ideal pendulum , 490 learning , 15, 144 lobster motion , 513 logistic population model, See Logistic equation and Logistic difference equation , measuring mass in space, 377 memorization , 144 Millennium Bridge, 417 mixing problems , 31- 32, 43, 131 modified logistic, 89 mold growth , 145 MR fluid , 231, 426 Newton's law of cooling, 35, 141 oil production , 149 opera singer, 382, 420, 430 pendulum , 489 , 502 population , See Population models, predator-prey, See Predator-prey system, radioactive decay, 16 RC circuit , 44 , 61, 568, 577 RLC circuit , 375, 391, 447

Index band, 229, 448 , 28 shock absorbers, 339 active, 231 state populations , 148 suspension bridge, 417 , 433 , 448 suspension system , 339 swaying skyscraper, 206 209 Tacoma Narrows Bridge, 433 , 448 trolley cars, 95 truck seats, 231 tuned -mass damper, 509 Volterra -Lotka, 482 washing machines, 426 weather, 214 Moldy bread, 145 rubber

saving

Mozart

The Marriage of Figaro , 487 MR fluid , 231, 426 frequency, 207, 299 , 330 period, 299 response, 387 Newton's law of cooling, 35, 141 Newton's method , 717 Newton's second law of motion , 159, 382, 490 Node first -order equation , 86 Nonhomogeneous equations. 114. 383 Nonlinear equation , 9 Nullcline , 472 , 476 - 479 Numerical integration , 642 Riemann sums , 642 Simpson's rule, 650 trapezoid rule, 642, 650 Numerical methods , 13, 62 errors, 62, 72 , 628, 644, 653 Euler's method , 53- 60, 198, 628 finite arithmetic , 660- 663 improved Euler's method , 644 in practice, 662 order, 637, 641, 646 round -off error, See Finite arithmetic . Runge - Kutta , 649- 658 Mathematica program , 655 Natural Natural Natural

C program , 656 order, 653 TI calculator program . 654 systems, 164, 198, 657 Oil Production, 149 Old joke , 2 Opera singer, 382, 420, 430 Orbit, 673 Order differential equation first -order, 5, 20

second-order, 161

numerical method , 637, 641 Euler's method , 637 Improved Euler's method , 646

-

, 653 equation, 5 harmonic oscillator, 331 Overdamped Oxford, 18 Runge Kutta Ordinary differential

P-Delta effect, 207 Parameterized family of differentia] Parameters, 3 - 4, 20 bifurcation value , 100 family of equations. 97 Parking lot, 175 Particular

equations, 96

solution

-value problem, 7 nonhomogeneous equations, 384 Pendulum , 489 damped, 457, 502 higher order approximation of , 553 ideal, 490 periodically forced, 540 Poincare return map , 540 Period, 155 forcing, 397 natural , 299 Period-doubling bifurcation , 694 Period-doubling route to chaos, 698 Periodic solution , 154 Phase angle, 421 Phase line, 77 equilibrium points, 78 , 84 initial

825

Index

826

metaphor of the rope, 77 sketching solutions, 81

Phase plane, 156 metaphor of the parking lot, 175 Phase portrait , 156 Phase shift , 401, 425 Phase space three-dimensional , 356 Phasor, 408 Pitchfork bifurcation , 693 Poincare return map , 533 fixed point , 536 Lorenz equations, 712 Poincare, Henri, 214, 534, 543 Poles of Laplace transform , 614, 624 Population models carrying capacity, 9 10 competing species, 179, 225, 453 , 472 cooperative species, 225, 483 delayed logistic , 718 exponential growth , 4, 148, 670 growth -rate coefficient, 4 harvesting , 104, 146 logistic equation , See Logistic equation and Logistic difference equation . modified logistic, 89 predator-prey, See Predator-prey system, modified, 157 state populations, 148 three species, 525 U.S. population , 7 Volterra -Lotka, 482 Power series methods, 736 Predator-prey system, 11- 13, 152- 158 modified, 157 pesticides, 167 Runge Kutta method , 658 three species, 525 methods, 6, 13, 62 systems, 164, 181

Qualitative

Radioactive decay, 16 Ramp function , 579 RC circuit, 44, 61, 568, 577

constant input voltage, 45 on-off input voltage, 46 periodic input voltage, 47 zero input voltage, 45 Red Sox, 299 , 1919- 2003 Renormalization , 699 Resonance, 415 Millennium Bridge, 417 Restoring force , 159 Return map , See Poincare return map . Riccati equation , 731 Riemann sums, 642 RLC circuit , 375, 391, 447 Robo-lobster, 513 Rock and Roll , See Paul . Rope, 77 Round -off error, See Finite arithmetic . Rubber band, 229, 448 Runge Kutta method , 649 658 Mathematica program , 655 C program , 656 definition , 650 error, 653 finite arithmetic , 660 order, 653 step size, 653 systems, 657 TI calculator program , 654 Saddle, 274 , 277, 362 connection, 487 nonlinear , 463 separatrix, 287, 463 Saddle node bifurcation , See Tangent . Salvadori, Mario, 208, 433 Saving model, 28 Sawtooth function , 581 Second-order differential equation , 161 forced, 383 Hermite equation , 739 homogeneous, 383 Legendre equation , 743 linear, 325 nonhomogeneous, 383 power series, 739

Index Second-order numerical methods, 646 Seed, 673 Sensitive dependence on initial conditions, 703, 706 Separable equations, 23- 28 missing solutions, 27 Separation of variables, 20- 32 Separatrix, 287, 463 stable, 463 unstable , 463 Shock absorbers, 339 active, 231, 426 truck seats, 231 washing machines, 426 Simple harmonic motion , 331 Simpson's rule, 650 Singular matrix , 242 Sink first -order equation

, 86 linear systems, 279, 362 nonlinear , 463 spiral, 299, 463 Sinusoidal forcing, 397, 413, 421 , 589 Skyscraper P-Delta effect, 207 swaying , 206 tuned -mass damper, 509 Slope fields, 36-43 sketching , 37 special cases autonomous equations, 40 dy/dt = fit) , 39- 40 Smale, Stephen, 63 Sock absorbers, See washing machines. Solution , 6- 7. 20 asymptotic , 82 checking, 21 , 188 domain of definition , 66, 82 existence and uniqueness, 65 general, 7, 23, 190 implicit , 28 initial -value problem , 7, 22- 23 , 162, 188 linearly independent, 355 missing , 27 periodic, 154. 299 separatrix, 287 system , 13, 154

Solution

827

curve, 155

Source first -order equation

, 86 linear systems, 283, 362 nonlinear , 463 spiral, 299, 463 Speed bumps , 421 Spiral saddle, 360 Spiral sink , 299, 463 Spiral source, 299, 463 Spring constant, 160 Springs, 160, See Harmonic oscillator. hard , 552 soft, 552 Square wave , 581 Stable equilibrium point , 86, 285 Stable separatrix, 463 Steady-state response, 387 Steady-state solution , 120 Step function , See Heaviside function . Step size, 54 Straight line solutions, 267 Superposition, Principle of . See Linearity Principle . Suspension bridge , 417 , 433 , 448 Lazer-McKenna light flexible long span, 441 Suspension system model, 339 Systems, 152 autonomous , 181 checking solutions, 188 coupled, 12 decoupled, 189 dimension , 240 direction field, 173 Euler's method , 198 Existence and Uniqueness Theorem, 205 first -order, 12 general solution , 190 linear, 237 Metaphor of the parking lot, 175 nullcline , 472 phase plane , 156 phase portrait , 156 Runge Kutta method , 657 three-dimensional, 354- 365 vector field, 171

828

Index

Tacoma Narrows Bridge, 433- 440, 448 Tangent bifurcation , 692 Taylor polynomial , 459 power series methods, 736 Taylor's Theorem, 457, 631, 641 Template Lorenz, 711 Tent function , 716 Time-series, 676 Trace, 342 Trace-determinant plane, 342 critical loci, 347 Trapezoid rule, 642. 650 Trolley cars, 95 Truck seats, 231 Tuned-mass damper, 509 H- substitution

, 26 , 724

Undamped harmonic oscillator, 330 Underdamped harmonic oscillator, 331 Undetermined coefficients, method of , 118, 388

Unforced equation, 383 Uniqueness Theorem first -order equations applications, 70- 73 systems, 205 Unstable equilibrium point , 285, 299 Unstable separatrix, 463 Van

der Pol equation , 201 , 452

Vector field, 171, 173 Volterra , V , 13 Volterra Washing

-Lotka system. 482 machines, 426

Weather , 214, 539 Web diagram , 683 WEEI , 299 World Series, 2004 Wronskian , 258 Wytham Woods , 18

Xia, Zhihong , 495