Differential Equations with Linear Algebra

572 Pages·2009·3.42 MB·English

by  Matthew R. Boelkins

Differential Equations with Linear Algebra
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Differential Equations
with Linear Algebra
MatthewR.Boelkins,J.L.Goldberg,andMerleC.Potter
3
2009
3
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LibraryofCongressCataloging-in-PublicationData
Boelkins,MatthewR.
Differentialequationswithlinearalgebra/MatthewR.Boelkins,J.L.Goldberg,MerleC.Potter.
p.cm.
Includesindex.
ISBN978-0-19-538586-1(cloth)
1. Differentialequations,Linear. 2. Algebras,Linear.I.Goldberg,JackL.(JackLeonard),1932–
II.Potter,MerleC.III.Title.
QA372.B6572009
(cid:2)
515.354–dc22 2008050361
9 8 7 6 5 4 3 2 1
PrintedintheUnitedStatesofAmerica
onacid-freepaper
Contents
Introduction xi
1 Essentialsoflinearalgebra 3
1.1 Motivatingproblems 3
1.2 Systemsoflinearequations 8
1.2.1 RowreductionusingMaple 15
1.3 Linearcombinations 21
1.3.1 Markovchains:anapplicationofmatrix-vector
multiplication 26
1.3.2 MatrixproductsusingMaple 29
1.4 Thespanofasetofvectors 33
1.5 Systemsoflinearequationsrevisited 39
1.6 Linearindependence 49
1.7 Matrixalgebra 58
1.7.1 MatrixalgebrausingMaple 62
1.8 Theinverseofamatrix 66
1.8.1 Computergraphics 70
1.8.2 MatrixinversesusingMaple 73
1.9 Thedeterminantofamatrix 78
1.9.1 DeterminantsusingMaple 82
1.10 Theeigenvalueproblem 84
1.10.1 Markovchains,eigenvectors,andGoogle 93
1.10.2 UsingMapletofindeigenvaluesandeigenvectors 94
vi Contents
1.11 Generalizedvectors 99
1.12 Basesanddimensioninvectorspaces 108
1.13 Forfurtherstudy 115
1.13.1 Computergraphics:geometryandlinearalgebraat
work 115
1.13.2 Béziercurves 119
1.13.3 Discretedynamicalsystems 123
2 First-orderdifferentialequations 127
2.1 Motivatingproblems 127
2.2 Definitions,notation,andterminology 129
2.2.1 PlottingslopefieldsusingMaple 135
2.3 Linearfirst-orderdifferentialequations 139
2.4 Applicationsoflinearfirst-orderdifferentialequations 147
2.4.1 Mixingproblems 147
2.4.2 Exponentialgrowthanddecay 148
2.4.3 Newton’slawofCooling 150
2.5 Nonlinearfirst-orderdifferentialequations 154
2.5.1 Separableequations 154
2.5.2 Exactequations 157
2.6 Euler’smethod 162
2.6.1 ImplementingEuler’smethodinExcel 167
2.7 Applicationsofnonlinearfirst-orderdifferential
equations 172
2.7.1 Thelogisticequation 172
2.7.2 Torricelli’slaw 176
2.8 Forfurtherstudy 181
2.8.1 Convertingcertainsecond-orderdesto
first-orderDEs 181
2.8.2 Howraindropsfall 182
2.8.3 Riccati’sequation 183
2.8.4 Bernoulli’sequation 184
3 Linearsystemsofdifferentialequations 187
3.1 Motivatingproblems 187
3.2 Theeigenvalueproblemrevisited 191
3.3 Homogeneouslinearfirst-ordersystems 202
3.4 Systemswithallreallinearlyindependenteigenvectors 211
3.4.1 PlottingdirectionfieldsforsystemsusingMaple 219
3.5 Whenamatrixlackstworeallinearlyindependent
eigenvectors 223
3.6 Nonhomogeneoussystems:undetermined
coefficients 236
3.7 Nonhomogeneoussystems:variationofparameters 245
3.7.1 ApplyingvariationofparametersusingMaple 250
Contents vii
3.8 Applicationsoflinearsystems 253
3.8.1 Mixingproblems 253
3.8.2 Spring-masssystems 255
3.8.3 RLCcircuits 258
3.9 Forfurtherstudy 268
3.9.1 Diagonalizablematricesandcoupledsystems 268
3.9.2 Matrixexponential 270
4 Higherorderdifferentialequations 273
4.1 Motivatingequations 273
4.2 Homogeneousequations:distinctrealroots 274
4.3 Homogeneousequations:repeatedandcomplexroots 281
4.3.1 Repeatedroots 281
4.3.2 Complexroots 283
4.4 Nonhomogeneousequations 288
4.4.1 Undeterminedcoefficients 289
4.4.2 Variationofparameters 295
4.5 Forcedmotion:beatsandresonance 300
4.6 Higherorderlineardifferentialequations 309
4.6.1 SolvingcharacteristicequationsusingMaple 316
4.7 Forfurtherstudy 319
4.7.1 Dampedmotion 319
4.7.2 Forcedoscillationswithdamping 321
4.7.3 TheCauchy–Eulerequation 323
4.7.4 Companionsystemsandcompanionmatrices 325
5 Laplacetransforms 329
5.1 Motivatingproblems 329
5.2 Laplacetransforms:gettingstarted 331
5.3 GeneralpropertiesoftheLaplacetransform 337
5.4 Piecewisecontinuousfunctions 347
5.4.1 TheHeavisidefunction 347
5.4.2 TheDiracdeltafunction 353
5.4.3 TheHeavisideandDiracfunctionsinMaple 357
5.5 SolvingIVPswiththeLaplacetransform 359
5.6 MoreontheinverseLaplacetransform 371
5.6.1 Laplacetransformsandinversetransforms
usingMaple 375
5.7 Forfurtherstudy 378
5.7.1 Laplacetransformsofinfiniteseries 378
5.7.2 Laplacetransformsofperiodicforcingfunctions 380
5.7.3 Laplacetransformsofsystems 384
6 Nonlinearsystemsofdifferentialequations 387
6.1 Motivatingproblems 387
viii Contents
6.2 Graphicalbehaviorofsolutionsfor2×2nonlinear
systems 391
6.2.1 Plottingdirectionfieldsofnonlinearsystems
usingMaple 397
6.3 Linearapproximationsofnonlinearsystems 400
6.4 Euler’smethodfornonlinearsystems 409
6.4.1 ImplementingEuler’smethodforsystemsinExcel 413
6.5 Forfurtherstudy 417
6.5.1 Thedampedpendulum 417
6.5.2 Competitivespecies 418
7 Numericalmethodsfordifferentialequations 421
7.1 Motivatingproblems 421
7.2 BeyondEuler’smethod 423
7.2.1 Heun’smethod 424
7.2.2 ModifiedEuler’smethod 427
7.3 Higherordermethods 430
7.3.1 Taylormethods 431
7.3.2 Runge–Kuttamethods 434
7.4 Methodsforsystemsandhigherorderequations 439
7.4.1 Euler’smethodforsystems 440
7.4.2 Heun’smethodforsystems 442
7.4.3 Runge–Kuttamethodforsystems 443
7.4.4 MethodsforhigherorderIVPs 445
7.5 Forfurtherstudy 449
7.5.1 Predator–Preyequations 449
7.5.2 Competitivespecies 450
7.5.3 Thedampedpendulum 450
8 Seriessolutionsfordifferentialequations 453
8.1 Motivatingproblems 453
8.2 AreviewofTaylorandpowerseries 455
8.3 Powerseriessolutionsoflinearequations 463
8.4 Legendre’sequation 471
8.5 Threeimportantexamples 477
8.5.1 TheHermiteequation 477
8.5.2 TheLaguerreequation 480
8.5.3 TheBesselequation 482
8.6 ThemethodofFrobenius 485
8.7 Forfurtherstudy 491
8.7.1 Taylorseriesforfirst-orderdifferentialequations 491
8.7.2 TheGammafunction 491
Contents ix
Appendix A Reviewofintegrationtechniques 493
Appendix B Complexnumbers 503
Appendix C Rootsofpolynomials 509
AppendixD Lineartransformations 513
Appendix E Solutionstoselectedexercises 523
Index 549