Instant Access to Differential equations and boundary value problems: computing and modeling (edward

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PREFACE PREFACE

Thisisatextbookforthestandardintroductorydifferentialequationscourse takenbyscienceandengineeringstudents.Itsupdatedcontentreflectsthe wideavailabilityoftechnicalcomputingenvironmentslike Maple, Mathematica, andMATLABthatnowareusedextensivelybypracticingengineersandscientists. Thetraditionalmanualandsymbolicmethodsareaugmentedwithcoveragealso ofqualitativeandcomputer-basedmethodsthatemploynumericalcomputationand graphicalvisualizationtodevelopgreaterconceptualunderstanding.Abonusof thismorecomprehensiveapproachisaccessibilitytoawiderrangeofmorerealistic applicationsofdifferentialequations.

PrincipalFeaturesofThisRevision

This5theditionisacomprehensiveandwide-rangingrevision.

Inadditiontofine-tuningtheexposition(bothtextandgraphics)innumerous sectionsthroughoutthebook,newapplicationshavebeeninserted(includingbiological),andwehaveexploitedthroughoutthenewinteractivecomputertechnology thatisnowavailabletostudentsondevicesrangingfromdesktopandlaptopcomputerstosmartphonesandgraphingcalculators.Italsoutilizescomputeralgebra systemssuchas Mathematica,Maple,andMATLABaswellasonlinewebsites suchasWolframjAlpha.

However,withasingleexceptionofanewsectioninsertedinChapter5(noted below),theclasstestedtableofcontentsofthebookremainsunchanged.Therefore, instructors’notesandsyllabiwillnotrequirerevisiontocontinueteachingwiththis newedition.

Aconspicuousfeatureofthiseditionistheinsertionofabout80newcomputergeneratedfigures,manyofthemillustratinghowinteractivecomputerapplications withsliderbarsortouchpadcontrolscanbeusedtochangeinitialvaluesorparametersinadifferentialequation,allowingtheusertoimmediatelyseeinrealtimethe resultingchangesinthestructureofitssolutions.

Someillustrationsofthevarioustypesofrevisionandupdatingexhibitedin thisedition:

NewInteractiveTechnologyandGraphics Newfiguresinsertedthroughoutillustratethefacilityofferedbymoderncomputingtechnologyplatforms fortheusertointeractivelyvaryinitialconditionsandotherparametersin realtime.Thus,usingamouseortouchpad,theinitialpointforaninitial valueproblemcanbedraggedtoanewlocation,andthecorrespondingsolutioncurveisautomaticallyredrawnanddraggedalongwithitsinitialpoint. Forinstance,seetheSections1.3(page28)applicationmoduleand3.1(page 148).Usingsliderbarsinaninteractivegraphic,thecoefficientsorotherparametersinalinearsystemcanbevaried,andthecorrespondingchangesinits directionfieldandphaseplaneportraitareautomaticallyshown;forinstance,

seetheapplicationmoduleforSection5.3(page319).Thenumberofterms usedfromaninfiniteseriessolutionofadifferentialequationcanbevaried, andtheresultinggraphicalchangeinthecorrespondingapproximatesolution isshownimmediately;seetheSection8.2applicationmodule(page516).

NewExposition Inanumberofsections,newtextandgraphicshavebeen insertedtoenhancestudentunderstandingofthesubjectmatter.Forinstance, seethetreatmentsofseparableequationsinSection1.4(page30),linearequationsinSection1.5(page45),isolatedcriticalpointsinSections6.1(page 372)and6.2(page383),andthenewexampleinSection9.6(page618) showingavibratingstringwithamomentary“flatspot.”ExamplesandaccompanyinggraphicshavebeenupdatedinSections2.4–2.6,4.2,and4.3to illustratenewgraphingcalculators.

NewContent ThesingleentirelynewsectionforthiseditionisSection 5.3,whichisdevotedtotheconstructionofa“gallery”ofphaseplaneportraitsillustratingallthepossiblegeometricbehaviorsofsolutionsofthe2dimensionallinearsystem x0 D Ax.Inmotivationandpreparationforthe detailedstudyofeigenvalue-eigenvectormethodsinsubsequentsectionsof Chapter5(whichthenfollowinthesameorderasinthepreviousedition),Section5.3showshowtheparticulararrangementsofeigenvaluesand eigenvectorsofthecoefficientmatrix A correspondtoidentifiablepatterns— “fingerprints,”sotospeak—inthephaseplaneportraitofthesystem x0 D Ax. Theresultinggalleryisshowninthetwopagesofphaseplaneportraitsthat compriseFigure5.3.16(pages315-316)attheendofthesection.Thenew5.3 applicationmodule(ondynamicphaseplaneportraits,page319)showshow studentscanuseinteractivecomputersystemsto“bringtolife”thisgallery,by allowinginitialconditions,eigenvalues,andeveneigenvectorstovaryinreal time.ThisdynamicapproachisthenillustratedwithseveralnewgraphicsinsertedintheremainderofChapter5.Finally,foranewbiologicalapplication, seetheapplicationmoduleforSection6.4,whichnowincludesasubstantialinvestigation(page423)ofthenonlinearFitzHugh-Nagumoequationsin neuroscience,whichwereintroducedtomodelthebehaviorofneuronsinthe nervoussystem.

ComputingFeatures

Thefollowingfeatureshighlightthecomputingtechnologythatdistinguishesmuch ofourexposition.

Over750 computer-generatedfigures showstudentsvividpicturesofdirection fields,solutioncurves,andphaseplaneportraitsthatbringsymbolicsolutions ofdifferentialequationstolife.

About45 applicationmodules followkeysectionsthroughoutthetext.Most oftheseapplicationsoutline“technologyneutral”investigationsillustrating theuseoftechnicalcomputingsystemsandseektoactivelyengagestudents intheapplicationofnewtechnology.

Afresh numericalemphasis thatisaffordedbytheearlyintroductionofnumericalsolutiontechniquesinChapter2(onmathematicalmodelsandnumericalmethods).HereandinChapter4,wherenumericaltechniquesfor systemsaretreated,aconcreteandtangibleflavorisachievedbytheinclusionofnumericalalgorithmspresentedinparallelfashionforsystemsranging fromgraphingcalculatorstoMATLAB.

ModelingFeatures

Mathematicalmodelingisagoalandconstantmotivationforthestudyofdifferentialequations.Tosampletherangeofapplicationsinthistext,takealookatthe followingquestions:

Whatexplainsthecommonlyobservedtimelagbetweenindoorandoutdoor dailytemperatureoscillations?(Section1.5)

Whatmakesthedifferencebetweendoomsdayandextinctioninalligatorpopulations?(Section2.1)

Howdoaunicycleandatwoaxlecarreactdifferentlytoroadbumps?(Sections3.7and5.4)

Howcanyoupredictthetimeofnextperihelionpassageofanewlyobserved comet?(Section4.3)

Whymightanearthquakedemolishonebuildingandleavestandingtheone nextdoor?(Section5.4)

Whatdetermineswhethertwospecieswillliveharmoniouslytogether,or whethercompetitionwillresultintheextinctionofoneofthemandthesurvivaloftheother?(Section6.3)

Whyandwhendoesnon-linearityleadtochaosinbiologicalandmechanical systems?(Section6.5)

Ifamassonaspringisperiodicallystruckwithahammer,howdoesthe behaviorofthemassdependonthefrequencyofthehammerblows?(Section 7.6)

Whyareflagpoleshollowinsteadofsolid?(Section8.6)

Whatexplainsthedifferenceinthesoundsofaguitar,axylophone,anddrum? (Sections9.6,10.2,and10.4) OrganizationandContent

Wehavereshapedtheusualapproachandsequenceoftopicstoaccommodatenew technologyandnewperspectives.Forinstance:

Afteraprecisoffirst-orderequationsinChapter1(thoughwiththecoverageofcertaintraditionalsymbolicmethodsstreamlinedabit),Chapter2offersanearlyintroductiontomathematicalmodeling,stabilityandqualitative propertiesofdifferentialequations,andnumericalmethods—acombination oftopicsthatfrequentlyaredispersedlaterinanintroductorycourse.Chapter 3includesthestandardmethodsofsolutionoflineardifferentialequationsof higherorder,particularlythosewithconstantcoefficients,andprovidesanespeciallywiderangeofapplicationsinvolvingsimplemechanicalsystemsand electricalcircuits;thechapterendswithanelementarytreatmentofendpoint problemsandeigenvalues.

Chapters4and5provideaflexibletreatmentoflinearsystems.Motivated bycurrenttrendsinscienceandengineeringeducationandpractice,Chapter4offersanearly,intuitiveintroductiontofirst-ordersystems,models,and numericalapproximationtechniques.Chapter5beginswithaself-contained

treatmentofthelinearalgebrathatisneeded,andthenpresentstheeigenvalue approachtolinearsystems.Itincludesawiderangeofapplications(ranging fromrailwaycarstoearthquakes)ofallthevariouscasesoftheeigenvalue method.Section5.5includesafairlyextensivetreatmentofmatrixexponentials,whichareexploitedinSection5.6onnonhomogeneouslinearsystems. Chapter6onnonlinearsystemsandphenomenarangesfromphaseplaneanalysistoecologicalandmechanicalsystemstoaconcludingsectiononchaos andbifurcationindynamicalsystems.Section6.5presentsanelementaryintroductiontosuchcontemporarytopicsasperiod-doublinginbiologicaland mechanicalsystems,thepitchforkdiagram,andtheLorenzstrangeattractor (allillustratedwithvividcomputergraphics).

Laplacetransformmethods(Chapter7)andpowerseriesmethods(Chapter8) followthematerialonlinearandnonlinearsystems,butcanbecoveredatany earlierpoint(afterChapter3)theinstructordesires.

Chapters9and10treattheapplicationsofFourierseries,separationofvariables,andSturm-Liouvilletheorytopartialdifferentialequationsandboundaryvalueproblems.AftertheintroductionofFourierseries,thethreeclassicalequations—thewaveandheatequationsandLaplace’sequation—are discussedinthelastthreesectionsofChapter9.Theeigenvaluemethodsof Chapter10aredevelopedsufficientlytoincludesomerathersignificantand realisticapplications.

Thisbookincludesenoughmaterialappropriatelyarrangedfordifferentcourses varyinginlengthfromonequartertotwosemesters.ThebrieferversionDifferentialEquations:ComputingandModeling(0-321-81625-0)endswithChapter7on Laplacetransformmethods(andthusomitsthematerialonpowerseriesmethods, Fourierseries,separationofvariablesandpartialdifferentialequations).

Theanswersectionhasbeenexpandedconsiderablytoincreaseitsvalueasalearningaid.Itnowincludestheanswerstomostodd-numberedproblemsplusagood manyeven-numberedones.The Instructor’sSolutionsManual (0-321-797019)availableat www.pearsonhighered.com/irc providesworked-outsolutions formostoftheproblemsinthebook,andthe StudentSolutionsManual (0-32179700-0)containssolutionsformostoftheodd-numberedproblems.Thesemanualshavebeenreworkedextensivelyforthiseditionwithimprovedexplanationsand moredetailsinsertedinthesolutionsofmanyproblems.

Theapproximately45applicationmodulesinthetextcontainadditionalproblemandprojectmaterialdesignedlargelytoengagestudentsintheexploration andapplicationofcomputationaltechnology.Theseinvestigationsareexpanded considerablyinthe ApplicationsManual (0-321-79704-3)thataccompaniesthe textandsupplementsitwithadditionalandsometimesmorechallenginginvestigations.Eachsectioninthismanualhasparallelsubsections Using Maple, Using Mathematica,and UsingMATLAB thatdetailtheapplicablemethodsandtechniquesofeachsystem,andwillaffordstudentusersanopportunitytocomparethe meritsandstylesofdifferentcomputationalsystems.Thesematerials—aswellas thetextofthe ApplicationsManual itself—arefreelyavailableatthewebsite www.pearsonhighered.com/mathstatsresources

Acknowledgments

Inpreparingthisrevision,weprofitedgreatlyfromtheadviceandassistanceofthe followingverycapableandperceptivereviewers:

AnthonyAidoo, EasternConnecticutStateUniversity

BrentSolie, KnoxCollege

ElizabethBradley, UniversityofLouisville

GregoryDavis, UniversityofWisconsin-GreenBay

ZoranGrujic, UniversityofVirginia

RichardJardine, KeeneStateCollege

YangKuang, ArizonaStateUniversity

DeningLi, WestVirginiaUniversity

FranciscoSayas-Gonzalez, UniversityofDelaware LutherWhite, UniversityofOklahoma

Hong-MingYin, WashingtonStateUniversity

MortezaShafii-Mousavi, IndianaUniversity-SouthBend

Itisapleasureto(onceagain)creditDennisKletzingandhisextraordinaryTEXpertise fortheattractivepresentationofthetextandtheartinthisbook.Wearegrateful tooureditor,WilliamHoffman,forhissupportandinspirationofthisrevision;to SalenaCashaforhercoordinationoftheeditorialprocessandBethHoustonforher supervisionoftheproductionofthisbook;andtoJoeVetereforhisassistancewith technicalaspectsofthedevelopmentofitsextensivesupplementaryresources.Finally,wededicatethiseditiontoourcolleagueDavidE.Penneywhopassedaway onJune3,2014.

HenryEdwards h.edwards@mindspring.com DavidCalvis dcalvis@bw.edu

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1 1

First-Order DifferentialEquations

1.1 DifferentialEquationsandMathematicalModels

Thelawsoftheuniversearewritteninthelanguageofmathematics.Algebra issufficienttosolvemanystaticproblems,butthemostinterestingnatural phenomenainvolvechangeandaredescribedbyequationsthatrelatechanging quantities.

Becausethederivative dx=dt D f 0 .t/ ofthefunction f istherateatwhich thequantity x D f.t/ ischangingwithrespecttotheindependentvariable t ,it isnaturalthatequationsinvolvingderivativesarefrequentlyusedtodescribethe changinguniverse.Anequationrelatinganunknownfunctionandoneormoreof itsderivativesiscalleda differentialequation

Example1 Thedifferentialequation

dx dt D x 2 C t 2 involvesboththeunknownfunction x.t/ anditsfirstderivative x 0 .t/ D dx=dt .Thedifferential equation

d 2 y dx 2 C 3 dy dx C 7y D 0 involvestheunknownfunction y oftheindependentvariable x andthefirsttwoderivatives y 0 and y 00 of y

Thestudyofdifferentialequationshasthreeprincipalgoals:

1. Todiscoverthedifferentialequationthatdescribesaspecifiedphysical situation.

2. Tofind—eitherexactlyorapproximately—theappropriatesolutionofthat equation.

3. Tointerpretthesolutionthatisfound.

Inalgebra,wetypicallyseektheunknown numbers thatsatisfyanequation suchas x 3 C 7x 2 11x C 41 D 0.Bycontrast,insolvingadifferentialequation,we

arechallengedtofindtheunknown functions y D y.x/ forwhichanidentitysuch as y 0 .x/ D 2xy.x/—thatis,thedifferentialequation

dy dx D 2xy

—holdsonsomeintervalofrealnumbers.Ordinarily,wewillwanttofind all solutionsofthedifferentialequation,ifpossible.

Example2 If C isaconstantand

(1) then

Thuseveryfunction y.x/ oftheforminEq.(1) satisfies—andthusisasolutionof—the differentialequation

dy dx D 2xy (2) forall x .Inparticular,Eq.(1)definesan infinite familyofdifferentsolutionsofthisdifferentialequation,oneforeachchoiceofthearbitraryconstant C .Bythemethodofseparationof variables(Section1.4)itcanbeshownthateverysolutionofthedifferentialequationin(2) isoftheforminEq.(1).

DifferentialEquationsandMathematicalModels

Thefollowingthreeexamplesillustratetheprocessoftranslatingscientificlawsand principlesintodifferentialequations.Ineachoftheseexamplestheindependent variableistime t ,butwewillseenumerousexamplesinwhichsomequantityother thantimeistheindependentvariable.

Example3

FIGURE1.1.1. Newton’slawof cooling,Eq.(3),describesthecooling ofahotrockinwater.

Newton’slawofcoolingmaybestatedinthisway:The timerateofchange (therateof changewithrespecttotime t )ofthetemperature T.t/ ofabodyisproportionaltothedifferencebetween T andthetemperature A ofthesurroundingmedium(Fig.1.1.1).Thatis,

dt D k.T A/; (3)

where k isapositiveconstant.Observethatif T>A,then dT=dt<0,sothetemperatureis adecreasingfunctionof t andthebodyiscooling.Butif T<A,then dT=dt>0,sothat T isincreasing.

Thusthephysicallawistranslatedintoadifferentialequation.Ifwearegiventhe valuesof k and A,weshouldbeabletofindanexplicitformulafor T.t/,andthen—withthe aidofthisformula—wecanpredictthefuturetemperatureofthebody. Temperature T Temperature A

Example4

Torricelli’slawimpliesthatthe timerateofchange ofthevolume V ofwaterinadraining tank(Fig.1.1.2)isproportionaltothesquarerootofthedepth y ofwaterinthetank: dV dt D k py;

(4)

where k isaconstant.Ifthetankisacylinderwithverticalsidesandcross-sectionalarea A, then V D Ay ,so dV=dt D A .dy=dt/.InthiscaseEq.(4)takestheform dy dt D hpy;

where h D k=A isaconstant.

(5)

Example5 The timerateofchange ofapopulation P.t/ withconstantbirthanddeathratesis,inmany simplecases,proportionaltothesizeofthepopulation.Thatis,

where k istheconstantofproportionality.

LetusdiscussExample5further.Notefirstthateachfunctionoftheform

isasolutionofthedifferentialequation

FIGURE1.1.2. Newton’slawof cooling,Eq.(3),describesthecooling ofahotrockinwater.

Example6

in(6).Weverifythisassertionasfollows:

= –12

FIGURE1.1.3. Graphsof

P.t/ D Ce kt with k D ln 2

forallrealnumbers t .Becausesubstitutionofeachfunctionoftheformgivenin (7)intoEq.(6)producesanidentity,allsuchfunctionsaresolutionsofEq.(6).

Thus,evenifthevalueoftheconstant k isknown,thedifferentialequation dP=dt D kP has infinitelymany differentsolutionsoftheform P.t/ D Ce kt ,onefor eachchoiceofthe“arbitrary”constant C .Thisistypicalofdifferentialequations. Itisalsofortunate,becauseitmayallowustouseadditionalinformationtoselect fromamongallthesesolutionsaparticularonethatfitsthesituationunderstudy.

Supposethat P.t/ D Ce kt isthepopulationofacolonyofbacteriaattime t ,thatthepopulationattime t D 0 (hours,h)was1000,andthatthepopulationdoubledafter 1 h.This additionalinformationabout P.t/ yieldsthefollowingequations:

1000 D P.0/ D Ce 0 D C;

2000 D P.1/ D Ce k :

Itfollowsthat C D 1000 andthat e k D 2,so k D ln 2 0:693147.Withthisvalueof k the differentialequationin(6)is

dP

dt D .ln 2/P .0:693147/P:

Substitutionof k D ln 2 and C D 1000 inEq.(7)yieldstheparticularsolution

P.t/ D 1000e .ln 2/t D 1000.e ln 2 /t D 1000 2t (because e ln 2 D 2) thatsatisfiesthegivenconditions.Wecanusethisparticularsolutiontopredictfuturepopulationsofthebacteriacolony.Forinstance,thepredictednumberofbacteriainthepopulation afteroneandahalfhours(when t D 1:5)is

P.1:5/ D 1000 23=2 2828:

Thecondition P.0/ D 1000 inExample6iscalledan initialcondition because wefrequentlywritedifferentialequationsforwhich t D 0 isthe“startingtime.” Figure1.1.3showsseveraldifferentgraphsoftheform P.t/ D Ce kt with k D ln 2 Thegraphsofalltheinfinitelymanysolutionsof dP=dt D kP infactfilltheentire two-dimensionalplane,andnotwointersect.Moreover,theselectionofanyone point P0 onthe P -axisamountstoadeterminationof P.0/.Becauseexactlyone solutionpassesthrougheachsuchpoint,weseeinthiscasethataninitialcondition P.0/ D P0 determinesauniquesolutionagreeingwiththegivendata.

MathematicalModels

OurbriefdiscussionofpopulationgrowthinExamples5and6illustratesthecrucial processof mathematicalmodeling (Fig.1.1.4),whichinvolvesthefollowing:

1. Theformulationofareal-worldprobleminmathematicalterms;thatis,the constructionofamathematicalmodel.

2. Theanalysisorsolutionoftheresultingmathematicalproblem.

3. Theinterpretationofthemathematicalresultsinthecontextoftheoriginal real-worldsituation—forexample,answeringthequestionoriginallyposed.

FIGURE1.1.4. Theprocessofmathematicalmodeling.

Inthepopulationexample,thereal-worldproblemisthatofdeterminingthe populationatsomefuturetime.A mathematicalmodel consistsofalistofvariables(P and t )thatdescribethegivensituation,togetherwithoneormoreequations relatingthesevariables(dP=dt D kP , P.0/ D P0 )thatareknownorareassumedto hold.Themathematicalanalysisconsistsofsolvingtheseequations(here,for P as afunctionof t ).Finally,weapplythesemathematicalresultstoattempttoanswer theoriginalreal-worldquestion.

Asanexampleofthisprocess,thinkoffirstformulatingthemathematical modelconsistingoftheequations dP=dt D kP , P.0/ D 1000,describingthebacteriapopulationofExample6.Thenourmathematicalanalysisthereconsistedof solvingforthesolutionfunction P.t/ D 1000e .ln 2/t D 1000 2t asourmathematicalresult.Foraninterpretationintermsofourreal-worldsituation—theactual bacteriapopulation—wesubstituted t D 1:5 toobtainthepredictedpopulationof P.1:5/ 2828 bacteriaafter1.5hours.If,forinstance,thebacteriapopulationis growingunderidealconditionsofunlimitedspaceandfoodsupply,ourprediction maybequiteaccurate,inwhichcaseweconcludethatthemathematicalmodelis adequateforstudyingthisparticularpopulation.

Ontheotherhand,itmayturnoutthatnosolutionoftheselecteddifferential equationaccuratelyfitstheactualpopulationwe’restudying.Forinstance,for no choiceoftheconstants C and k doesthesolution P.t/ D Ce kt inEq.(7)accurately describetheactualgrowthofthehumanpopulationoftheworldoverthepastfew centuries.Wemustconcludethatthedifferentialequation dP=dt D kP isinadequate formodelingtheworldpopulation—whichinrecentdecadeshas“leveledoff”as comparedwiththesteeplyclimbinggraphsintheupperhalf(P>0)ofFig.1.1.3. Withsufficientinsight,wemightformulateanewmathematicalmodelincluding aperhapsmorecomplicateddifferentialequation,onethattakesintoaccountsuch factorsasalimitedfoodsupplyandtheeffectofincreasedpopulationonbirthand deathrates.Withtheformulationofthisnewmathematicalmodel,wemayattempt totraverseonceagainthediagramofFig.1.1.4inacounterclockwisemanner.If wecansolvethenewdifferentialequation,wegetnewsolutionfunctionstocom-

Real-world situation
Mathematical model
Mathematical results
Mathematical analysis Formulation Interpretation

parewiththereal-worldpopulation.Indeed,asuccessfulpopulationanalysismay requirerefiningthemathematicalmodelstillfurtherasitisrepeatedlymeasured againstreal-worldexperience.

ButinExample6wesimplyignoredanycomplicatingfactorsthatmightaffectourbacteriapopulation.Thismadethemathematicalanalysisquitesimple, perhapsunrealisticallyso.Asatisfactorymathematicalmodelissubjecttotwocontradictoryrequirements:Itmustbesufficientlydetailedtorepresentthereal-world situationwithrelativeaccuracy,yetitmustbesufficientlysimpletomakethemathematicalanalysispractical.Ifthemodelissodetailedthatitfullyrepresentsthe physicalsituation,thenthemathematicalanalysismaybetoodifficulttocarryout. Ifthemodelistoosimple,theresultsmaybesoinaccurateastobeuseless.Thus thereisaninevitabletradeoffbetweenwhatisphysicallyrealisticandwhatismathematicallypossible.Theconstructionofamodelthatadequatelybridgesthisgap betweenrealismandfeasibilityisthereforethemostcrucialanddelicatestepin theprocess.Waysmustbefoundtosimplifythemodelmathematicallywithout sacrificingessentialfeaturesofthereal-worldsituation.

Mathematicalmodelsarediscussedthroughoutthisbook.Theremainderof thisintroductorysectionisdevotedtosimpleexamplesandtostandardterminology usedindiscussingdifferentialequationsandtheirsolutions.

ExamplesandTerminology

Example7 If C isaconstantand y.x/ D 1=.C x/,then dy dx D 1 .C x/

if x 6D C .Thus

definesasolutionofthedifferentialequation

onanyintervalofrealnumbersnotcontainingthepoint x D C .Actually,Eq.(8)definesa one-parameterfamily ofsolutionsof dy=dx D y 2 ,oneforeachvalueofthearbitraryconstant or“parameter” C .With C D 1 wegettheparticularsolution y.x/ D 1

thatsatisfiestheinitialcondition y.0/ D 1.AsindicatedinFig.1.1.5,thissolutioniscontinuousontheinterval . 1;1/ buthasaverticalasymptoteat x D 1

Example8 Verifythatthefunction y.x/ D 2x 1=2 x 1=2 ln x satisfiesthedifferentialequation

forall x>0

Solution Firstwecomputethederivatives

ThensubstitutionintoEq.(10)yields

if x ispositive,sothedifferentialequationissatisfiedforall x>0

1/(1 – x)

5 (0, 1)

Thesolutionof y 0 D y 2 definedby y.x/ D 1=.1 x/. Continued

Example7

Thefactthatwecanwriteadifferentialequationisnotenoughtoguarantee thatithasasolution.Forexample,itisclearthatthedifferentialequation

has no (real-valued)solution,becausethesumofnonnegativenumberscannotbe negative.Foravariationonthistheme,notethattheequation

obviouslyhasonlythe(real-valued)solution y.x/ 0.Inourpreviousexamples anydifferentialequationhavingatleastonesolutionindeedhadinfinitelymany. The order ofadifferentialequationistheorderofthehighestderivativethat appearsinit.ThedifferentialequationofExample8isofsecondorder,thosein Examples2through7arefirst-orderequations,and

.4/ C x 2 y .3/ C x 5 y D sin x

isafourth-orderequation.Themostgeneralformofan nth-order differential equationwithindependentvariable x andunknownfunctionordependentvariable y D y.x/ is

F x;y;y 0 ;y 00 ;:::;y .n/ D 0; (13)

where F isaspecificreal-valuedfunctionof n C 2 variables.

Ouruseoftheword solution hasbeenuntilnowsomewhatinformal.Tobe precise,wesaythatthecontinuousfunction u D u.x/ isa solution ofthedifferential equationin(13) ontheinterval I providedthatthederivatives u0 , u00 , ::: , u.n/ exist on I and

F x;u;u 0 ;u 00 ;:::;u.n/ D 0

forall x in I .Forthesakeofbrevity,wemaysaythat u D u.x/ satisfies the differentialequationin(13)on I

Remark Recallfromelementarycalculusthatadifferentiablefunctiononanopeninterval isnecessarilycontinuousthere.Thisiswhyonlyacontinuousfunctioncanqualifyasa (differentiable)solutionofadifferentialequationonaninterval.

Figure1.1.5showsthetwo“connected”branchesofthegraph y D 1=.1 x/.Theleft-hand branchisthegraphofa(continuous)solutionofthedifferentialequation y 0 D y 2 thatis definedontheinterval . 1;1/.Theright-handbranchisthegraphofa different solutionof thedifferentialequationthatisdefined(andcontinuous)onthedifferentinterval .1; 1/.So thesingleformula y.x/ D 1=.1 x/ actuallydefinestwodifferentsolutions(withdifferent domainsofdefinition)ofthesamedifferentialequation y 0 D y 2

Example9 If A and B areconstantsand

(14) thentwosuccessivedifferentiationsyield

forall x .Consequently,Eq.(14)defineswhatitisnaturaltocalla two-parameterfamily of solutionsofthesecond-orderdifferentialequation y 00 C 9y D 0 (15) onthewholerealnumberline.Figure1.1.6showsthegraphsofseveralsuchsolutions.

FIGURE1.1.5.

FIGURE1.1.6. Thethreesolutions

Althoughthedifferentialequationsin(11)and(12)areexceptionstothegeneralrule,wewillseethatan nth-orderdifferentialequationordinarilyhasan nparameterfamilyofsolutions—oneinvolving n differentarbitraryconstantsorpa-

rameters.

InbothEqs.(11)and(12),theappearanceof y 0 asanimplicitlydefinedfunctioncausescomplications.Forthisreason,wewillordinarilyassumethatanydifferentialequationunderstudycanbesolvedexplicitlyforthehighestderivativethat appears;thatis,thattheequationcanbewrittenintheso-called normalform

where G isareal-valuedfunctionof n C 1 variables.Inaddition,wewillalways seekonlyreal-valuedsolutionsunlesswewarnthereaderotherwise.

Allthedifferentialequationswehavementionedsofarare ordinary differentialequations,meaningthattheunknownfunction(dependentvariable)depends ononlya single independentvariable.Ifthedependentvariableisafunctionof twoormoreindependentvariables,thenpartialderivativesarelikelytobeinvolved; iftheyare,theequationiscalleda partial differentialequation.Forexample,the temperature u D u.x;t/ ofalongthinuniformrodatthepoint x attime t satisfies (underappropriatesimpleconditions)thepartialdifferentialequation

@t D k @2 u @x 2 ;

where k isaconstant(calledthe thermaldiffusivity oftherod).InChapters1 through8wewillbeconcernedonlywith ordinary differentialequationsandwill refertothemsimplyasdifferentialequations.

Inthischapterweconcentrateon first-order differentialequationsoftheform

Example10

Wealsowillsamplethewiderangeofapplicationsofsuchequations.Atypical mathematicalmodelofanappliedsituationwillbean initialvalueproblem,consistingofadifferentialequationoftheformin(17)togetherwithan initialcondition y.x0 / D y0 .Notethatwecall y.x0 / D y0 aninitialconditionwhetherornot x0 D 0.To solve theinitialvalueproblem

dy dx D f.x;y/;y.x0 / D y0 (18)

meanstofindadifferentiablefunction y D y.x/ thatsatisfiesbothconditionsin Eq.(18)onsomeintervalcontaining x0 .

Giventhesolution y.x/ D 1=.C x/ ofthedifferentialequation dy=dx D y 2 discussedin Example7,solvetheinitialvalueproblem

dy dx D y 2 ;y.1/ D 2:

Solution Weneedonlyfindavalueof C sothatthesolution y.x/ D 1=.C x/ satisfiestheinitial condition y.1/ D 2.Substitutionofthevalues x D 1 and y D 2 inthegivensolutionyields

2 D y.1/ D 1 C 1 ;

0 5 x y –5 –5

y = 2/(3 – 2x) x = 3/2

so 2C 2 D 1,andhence C D 3 2 .Withthisvalueof C weobtainthedesiredsolution (1, 2) (2, –2) 05

FIGURE1.1.7. Thesolutionsof

y 0 D y 2 definedby y.x/ D 2=.3 2x/

Figure1.1.7showsthetwobranchesofthegraph y D 2=.3 2x/.Theleft-handbranchis thegraphon . 1; 3 2 / ofthesolutionofthegiveninitialvalueproblem y 0 D y 2 , y.1/ D 2. Theright-handbranchpassesthroughthepoint .2; 2/ andisthereforethegraphon . 3 2 ; 1/ ofthesolutionofthedifferentinitialvalueproblem y 0 D y 2 , y.2/ D 2

Thecentralquestionofgreatestimmediateinteresttousisthis:Ifwearegiven adifferentialequationknowntohaveasolutionsatisfyingagiveninitialcondition, howdoweactually find or compute thatsolution?And,oncefound,whatcanwedo withit?Wewillseethatarelativelyfewsimpletechniques—separationofvariables (Section1.4),solutionoflinearequations(Section1.5),elementarysubstitution methods(Section1.6)—areenoughtoenableustosolveavarietyoffirst-order equationshavingimpressiveapplications.

1.1 Problems

InProblems1through12,verifybysubstitutionthateach givenfunctionisasolutionofthegivendifferentialequation. Throughouttheseproblems,primesdenotederivativeswithrespectto x

1. y 0 D 3x 2 ; y D x 3 C 7

2. y 0 C 2y D 0; y D 3e 2x

3. y 00 C 4y D 0; y1 D cos 2x

5.

9. y 0 C 2xy 2 D 0; y D 1 1 C x 2

10. x 2 y 00 C xy 0 y D ln x ; y1 D x ln x , y2 D 1 x ln x 11. x 2 y 00 C 5xy 0 C 4y D 0; y1 D 1 x 2 , y2 D ln x x 2

12. x 2 y 00 xy 0 C 2y D 0; y1 D x cos.ln x/, y2 D x sin.ln x/

InProblems13through16,substitute y D e rx intothegiven differentialequationtodetermineallvaluesoftheconstant r forwhich y D e rx isasolutionoftheequation.

13. 3y 0 D 2y 14. 4y 00 D y 15. y 00 C y 0 2y D 0 16. 3y 00 C 3y 0 4y D 0

InProblems17through26,firstverifythat y.x/ satisfiesthe givendifferentialequation.Thendetermineavalueoftheconstant C sothat y.x/ satisfiesthegiveninitialcondition.Usea computerorgraphingcalculator(ifdesired)tosketchseveral typicalsolutionsofthegivendifferentialequation,andhighlighttheonethatsatisfiesthegiveninitialcondition.

17. y 0 C y D 0; y.x/ D Ce x , y.0/ D 2

18. y 0 D 2y ; y.x/ D Ce 2x , y.0/ D 3

19. y 0 D y C 1; y.x/ D Ce x 1, y.0/ D 5

20. y 0 D x y ; y.x/ D Ce x C x 1, y.0/ D 10

21. y 0 C 3x 2 y D 0; y.x/ D Ce x 3 , y.0/ D 7

22. e y y 0 D 1; y.x/ D ln.x C C/, y.0/ D 0

23. x dy dx C 3y D 2x 5 ; y.x/ D 1 4 x 5 C Cx 3 , y.2/ D 1

24. xy 0 3y D x 3 ; y.x/ D x 3 .C C ln x/, y.1/ D 17

25. y 0 D 3x 2 .y 2 C 1/; y.x/ D tan.x 3 C C/, y.0/ D 1

26. y 0 C y tan x D cos x ; y.x/ D .x C C/ cos x , y. / D 0

InProblems27through31,afunction y D g.x/ isdescribed bysomegeometricpropertyofitsgraph.Writeadifferential equationoftheform dy=dx D f.x;y/ havingthefunction g as itssolution(orasoneofitssolutions).

27. Theslopeofthegraphof g atthepoint .x;y/ isthesum of x and y

28. Thelinetangenttothegraphof g atthepoint .x;y/ intersectsthe x -axisatthepoint .x=2;0/

29. Everystraightlinenormaltothegraphof g passesthrough thepoint .0;1/.Canyou guess whatthegraphofsucha function g mightlooklike?

30. Thegraphof g isnormaltoeverycurveoftheform y D x 2 C k (k isaconstant)wheretheymeet.

31. Thelinetangenttothegraphof g at .x;y/ passesthrough thepoint . y;x/.

InProblems32through36,write—inthemannerofEqs.(3) through(6)ofthissection—adifferentialequationthatisa mathematicalmodelofthesituationdescribed.

32. Thetimerateofchangeofapopulation P isproportional tothesquarerootof P

33. Thetimerateofchangeofthevelocity v ofacoasting motorboatisproportionaltothesquareof v .

34. Theacceleration dv=dt ofaLamborghiniisproportional tothedifferencebetween 250 km/handthevelocityofthe car.

35. Inacityhavingafixedpopulationof P persons,thetime rateofchangeofthenumber N ofthosepersonswhohave heardacertainrumorisproportionaltothenumberof thosewhohavenotyetheardtherumor.

36. Inacitywithafixedpopulationof P persons,thetimerate ofchangeofthenumber N ofthosepersonsinfectedwith acertaincontagiousdiseaseisproportionaltotheproduct ofthenumberwhohavethediseaseandthenumberwho donot.

InProblems37through42,determinebyinspectionatleast onesolutionofthegivendifferentialequation.Thatis,use yourknowledgeofderivativestomakeanintelligentguess. Thentestyourhypothesis.

37. y 00 D 0

Problems43through46concernthedifferentialequation

dt D kx 2 ; where k isaconstant.

43. (a) If k isaconstant,showthatageneral(one-parameter) solutionofthedifferentialequationisgivenby x.t/ D 1=.C kt/,where C isanarbitraryconstant.

(b) Determinebyinspectionasolutionoftheinitialvalue problem x 0 D kx 2 , x.0/ D 0

44. (a) Assumethat k ispositive,andthensketchgraphsof solutionsof x 0 D kx 2 withseveraltypicalpositive valuesof x.0/.

(b) Howwouldthesesolutionsdifferiftheconstant k werenegative?

45. Supposeapopulation P ofrodentssatisfiesthedifferentialequation dP=dt D kP 2 .Initially,thereare P.0/ D 2

rodents,andtheirnumberisincreasingattherateof dP=dt D 1 rodentpermonthwhenthereare P D 10 rodents.BasedontheresultofProblem43,howlongwillit takeforthispopulationtogrowtoahundredrodents?To athousand?What’shappeninghere?

46. Supposethevelocity v ofamotorboatcoastinginwater satisfiesthedifferentialequation dv=dt D kv 2 .Theinitialspeedofthemotorboatis v.0/ D 10 meterspersecond(m/s),and v isdecreasingattherateof1m/s2 when v D 5 m/s.BasedontheresultofProblem43,longdoes ittakeforthevelocityoftheboattodecreaseto1m/s?To 1 10 m/s?Whendoestheboatcometoastop?

47. InExample7wesawthat y.x/ D 1=.C x/ definesa one-parameterfamilyofsolutionsofthedifferentialequation dy=dx D y 2 (a) Determineavalueof C sothat y.10/ D 10 (b) Isthereavalueof C suchthat y.0/ D 0? Canyouneverthelessfindbyinspectionasolutionof dy=dx D y 2 suchthat y.0/ D 0? (c) Figure1.1.8shows typicalgraphsofsolutionsoftheform y.x/ D 1=.C x/ Doesitappearthatthesesolutioncurvesfilltheentire xyplane?Canyouconcludethat,givenanypoint .a;b/ in theplane,thedifferentialequation dy=dx D y 2 hasexactlyonesolution y.x/ satisfyingthecondition y.a/ D b ?

48.(a) Showthat y.x/ D Cx 4 definesaone-parameterfamilyofdifferentiablesolutionsofthedifferentialequation xy 0 D 4y (Fig.1.1.9). (b) Showthat y.x/ D ( x 4 if x<0, x 4 if x = 0

definesadifferentiablesolutionof xy 0 D 4y forall x ,butis notoftheform y.x/ D Cx 4 (c) Givenanytworealnumbers a and b ,explainwhy—incontrasttothesituationin part(c)ofProblem47—thereexistinfinitelymanydifferentiablesolutionsof xy 0 D 4y thatallsatisfythecondition y.a/ D b

FIGURE1.1.8. Graphsofsolutionsofthe equation dy=dx D y 2

FIGURE1.1.9. Thegraph y D Cx 4 for variousvaluesof C

1.2 IntegralsasGeneralandParticularSolutions

Thefirst-orderequation dy=dx D f.x;y/ takesanespeciallysimpleformifthe right-hand-sidefunction f doesnotactuallyinvolvethedependentvariable y ,so dy dx D

InthisspecialcaseweneedonlyintegratebothsidesofEq.(1)toobtain

Thisisa generalsolution ofEq.(1),meaningthatitinvolvesanarbitraryconstant C ,andforeverychoiceof C itisasolutionofthedifferentialequationin(1).If G.x/ isaparticularantiderivativeof f —thatis,if G 0 .x/ f.x/—then

FIGURE1.2.1. Graphsof y D 1 4 x 2 C C forvariousvaluesof C . C2 onthesameinterval I are“parallel”inthesenseillustratedbyFigs.1.2.1and 1.2.2.Thereweseethattheconstant C isgeometricallytheverticaldistancebetweenthetwocurves y.x/ D G.x/ and y.x/ D G.x/ C C .

Thegraphsofanytwosuchsolutions

Tosatisfyaninitialcondition y.x0 / D y0 ,weneedonlysubstitute x D x0 and y D y0 intoEq.(3)toobtain y0 D G.x0 / C C ,sothat C D y0 G.x0 /.Withthis choiceof C ,weobtainthe particularsolution ofEq.(1)satisfyingtheinitialvalue problem

Wewillseethatthisisthetypicalpatternforsolutionsoffirst-orderdifferential equations.Ordinarily,wewillfirstfinda generalsolution involvinganarbitrary constant C .Wecanthenattempttoobtain,byappropriatechoiceof C ,a particular solution satisfyingagiveninitialcondition y.x0 / D y0 .

FIGURE1.2.2. Graphsof y D sin x C C forvariousvaluesof C .

Remark Asthetermisusedinthepreviousparagraph,a generalsolution ofafirst-order differentialequationissimplyaone-parameterfamilyofsolutions.Anaturalquestionis whetheragivengeneralsolutioncontains every particularsolutionofthedifferentialequation.Whenthisisknowntobetrue,wecallit the generalsolutionofthedifferentialequation. Forexample,becauseanytwoantiderivativesofthesamefunction f.x/ candifferonlybya constant,itfollowsthateverysolutionofEq.(1)isoftheformin(2).ThusEq.(2)servesto define the generalsolutionof(1).

Example1 Solvetheinitialvalueproblem

dy dx D 2x C 3;y.1/ D 2:

Solution IntegrationofbothsidesofthedifferentialequationasinEq.(2)immediatelyyieldsthe generalsolution

y.x/ D Z .2x C 3/dx D x 2 C 3x C C:

Figure1.2.3showsthegraph y D x 2 C 3x C C forvariousvaluesof C .Theparticularsolution weseekcorrespondstothecurvethatpassesthroughthepoint .1;2/,therebysatisfyingthe initialcondition

y.1/ D .1/2 C 3 .1/ C C D 2:

Itfollowsthat C D 2,sothedesiredparticularsolutionis y.x/ D x 2 C 3x 2:

FIGURE1.2.3. Solutioncurvesfor thedifferentialequationinExample1.

Second-orderequations. Theobservationthatthespecialfirst-orderequation

dy=dx D f.x/ isreadilysolvable(providedthatanantiderivativeof f canbefound) extendstosecond-orderdifferentialequationsofthespecialform

(4)

inwhichthefunction g ontheright-handsideinvolvesneitherthedependentvariable y noritsderivative dy=dx .Wesimplyintegrateoncetoobtain

where G isanantiderivativeof g and C1 isanarbitraryconstant.Thenanother integrationyields

where C2 isasecondarbitraryconstant.Ineffect,thesecond-orderdifferential equationin(4)isonethatcanbesolvedbysolvingsuccessivelythe first-order equations

D g.x/ and

VelocityandAcceleration

Directintegrationissufficienttoallowustosolveanumberofimportantproblems concerningthemotionofaparticle(or masspoint )intermsoftheforcesacting onit.Themotionofaparticlealongastraightline(the x -axis)isdescribedbyits positionfunction x D f.t/ (5) givingits x -coordinateattime t .The velocity oftheparticleisdefinedtobe v.t/ D f 0 .t/I thatis, v D dx dt : (6) Its acceleration a.t/ is a.t/ D v 0 .t/ D x 00 .t/;inLeibniznotation,

(7)

Equation(6)issometimesappliedeitherintheindefiniteintegralform x.t/ D R v.t/dt orinthedefiniteintegralform x.t/ D x.t0 / C Z t t0 v.s/ds;

whichyoushouldrecognizeasastatementofthefundamentaltheoremofcalculus (preciselybecause dx=dt D v ).

Newton’s secondlawofmotion saysthatifaforce F.t/ actsontheparticle andisdirectedalongitslineofmotion,then

ma.t/ D F.t/I thatis, F D ma; (8)

Example2

where m isthemassoftheparticle.Iftheforce F isknown,thentheequation x 00 .t/ D F.t/=m canbeintegratedtwicetofindthepositionfunction x.t/ interms oftwoconstantsofintegration.Thesetwoarbitraryconstantsarefrequentlydeterminedbythe initialposition x0 D x.0/ andthe initialvelocity v0 D v.0/ ofthe particle.

Constantacceleration. Forinstance,supposethattheforce F ,andthereforethe acceleration a D F=m,are constant.Thenwebeginwiththeequation

dt D a (a isaconstant)

andintegratebothsidestoobtain v.t/ D Z adt D at C C1 :

Weknowthat v D v0 when t D 0,andsubstitutionofthisinformationintothe precedingequationyieldsthefactthat C1 D v

Asecondintegrationgives

t

Thus,withEq.(10)wecanfindthevelocity,andwithEq.(11)theposition,of theparticleatanytime t intermsofits constant acceleration a ,itsinitialvelocity v0 ,anditsinitialposition x0 .

Alunarlanderisfallingfreelytowardthesurfaceofthemoonataspeedof450metersper second(m=s).Itsretrorockets,whenfired,provideaconstantdecelerationof2.5metersper secondpersecond(m=s2 )(thegravitationalaccelerationproducedbythemoonisassumed tobeincludedinthegivendeceleration).Atwhatheightabovethelunarsurfaceshouldthe retrorocketsbeactivatedtoensurea“softtouchdown”(v D 0 atimpact)?

Solution Wedenoteby x.t/ theheightofthelunarlanderabovethesurface,asindicatedinFig.1.2.4. Welet t D 0 denotethetimeatwhichtheretrorocketsshouldbefired.Then v0 D 450

(m=s,negativebecausetheheight x.t/ isdecreasing),and a DC2:5,becauseanupward thrustincreasesthevelocity v (althoughitdecreasesthe speed jv j).ThenEqs.(10)and(11) become

v.t/ D 2:5t 450 (12) and

x.t/ D 1:25t 2 450t C x0 ; (13) where x0 istheheightofthelanderabovethelunarsurfaceatthetime t D 0 whenthe retrorocketsshouldbeactivated.

FIGURE1.2.4. Thelunarlanderof Example2.

FromEq.(12)weseethat v D 0 (softtouchdown)occurswhen t D 450=2:5 D 180 s (thatis, 3 minutes);thensubstitutionof t D 180, x D 0 intoEq.(13)yields

x0 D 0 .1:25/.180/2 C 450.180/ D 40;500

meters—thatis, x0 D 40.5km 25 1 6 miles.Thustheretrorocketsshouldbeactivatedwhen thelunarlanderis40.5kilometersabovethesurfaceofthemoon,anditwilltouchdown softlyonthelunarsurfaceafter3minutesofdeceleratingdescent.

Lunar

PhysicalUnits

Numericalworkrequiresunitsforthemeasurementofphysicalquantitiessuchas distanceandtime.Wesometimesuseadhocunits—suchasdistanceinmilesor kilometersandtimeinhours—inspecialsituations(suchasinaprobleminvolving anautotrip).However,thefoot-pound-second(fps)andmeter-kilogram-second (mks)unitsystemsareusedmoregenerallyinscientificandengineeringproblems. Infact,fpsunitsarecommonlyusedonlyintheUnitedStates(andafewother countries),whilemksunitsconstitutethestandardinternationalsystemofscientific units.

fpsunitsmksunits

Time g pound(lb) slug foot(ft) second(s) 32ft/s2 newton(N) kilogram(kg) meter(m) second(s) 9.8m/s2

Thelastlineofthistablegivesvaluesforthegravitationalacceleration g at thesurfaceoftheearth.Althoughtheseapproximatevalueswillsufficeformost examplesandproblems,moreprecisevaluesare 9:7805 m=s2 and 32:088 ft=s2 (at sealevelattheequator).

BothsystemsarecompatiblewithNewton’ssecondlaw F D ma .Thus1Nis (bydefinition)theforcerequiredtoimpartanaccelerationof1m=s2 toamassof1 kg.Similarly, 1 slugis(bydefinition)themassthatexperiencesanaccelerationof 1ft=s2 underaforceof 1 lb.(Wewillusemksunitsinallproblemsrequiringmass unitsandthuswillrarelyneedslugstomeasuremass.)

Inchesandcentimeters(aswellasmilesandkilometers)alsoarecommonly usedindescribingdistances.Forconversionsbetweenfpsandmksunitsithelpsto rememberthat

anditfollowsthat

ThusapostedU.S.speedlimitof50mi=hmeansthat—ininternationalterms—the legalspeedlimitisabout 50 1:609 80:45 km=h.

VerticalMotionwithGravitationalAcceleration

The weight W ofabodyistheforceexertedonthebodybygravity.Substitutionof a D g and F D W inNewton’ssecondlaw F D ma gives

a, 0)

-axis

Example3

fortheweight W ofthemass m atthesurfaceoftheearth(where g 32 ft=s2 9:8 m=s2 ).Forinstance,amassof m D 20 kghasaweightof W D (20kg)(9.8m=s2 ) D 196N.Similarly,amass m weighing100poundshasmksweight

soitsmassis

Todiscussverticalmotionitisnaturaltochoosethe y -axisasthecoordinate systemforposition,frequentlywith y D 0 correspondingto“groundlevel.”Ifwe choosethe upward directionasthepositivedirection,thentheeffectofgravityona verticallymovingbodyistodecreaseitsheightandalsotodecreaseitsvelocity v D dy=dt .Consequently,ifweignoreairresistance,thentheacceleration a D dv=dt of thebodyisgivenby

Thisaccelerationequationprovidesastartingpointinmanyproblemsinvolving verticalmotion.Successiveintegrations(asinEqs.(10)and(11))yieldthevelocity andheightformulas

and

a, 0)

-axis

Here, y0 denotestheinitial(t D 0)heightofthebodyand v0 itsinitialvelocity.

(a) Supposethataballisthrownstraightupwardfromtheground(y0 D 0)withinitial velocity v0 D 96 (ft=s,soweuse g D 32 ft=s2 infpsunits).Thenitreachesitsmaximum heightwhenitsvelocity(Eq.(16))iszero, v.t/ D 32t C 96 D 0; andthuswhen t D 3 s.Hencethemaximumheightthattheballattainsis y.3/ D 1 2 32

144(ft) (withtheaidofEq.(17)).

(b) Ifanarrowisshotstraightupwardfromthegroundwithinitialvelocity v0 D 49 (m=s, soweuse g D 9:8 m=s2 inmksunits),thenitreturnstothegroundwhen

andthusafter 10 sintheair.

ASwimmer’sProblem

Figure1.2.5showsanorthward-flowingriverofwidth w D 2a .Thelines x D˙a representthebanksoftheriverandthe y -axisitscenter.Supposethatthevelocity vR atwhichthewaterflowsincreasesasoneapproachesthecenteroftheriver,and indeedisgivenintermsofdistance x fromthecenterby vR D v0 1 x 2 a 2 (18)

FIGURE1.2.5. Aswimmer’s problem(Example4).

YoucanuseEq.(18)toverifythatthewaterdoesflowthefastestatthecenter, where vR D v0 ,andthat vR D 0 ateachriverbank.

Example4

Supposethataswimmerstartsatthepoint . a;0/ onthewestbankandswims dueeast(relativetothewater)withconstantspeed vS .AsindicatedinFig.1.2.5,his velocityvector(relativetotheriverbed)hashorizontalcomponent vS andvertical component vR .Hencetheswimmer’sdirectionangle ˛ isgivenby tan ˛ D vR vS :

Becausetan ˛ D dy=dx ,substitutionusing(18)givesthedifferentialequation

dx D v0 vS 1 x 2 a 2 (19)

fortheswimmer’strajectory y D y.x/ ashecrossestheriver.

Supposethattheriveris 1 milewideandthatitsmidstreamvelocityis v0 D 9 mi=h.Ifthe swimmer’svelocityis vS D 3 mi=h,thenEq.(19)takestheform

Integrationyields

fortheswimmer’strajectory.Theinitialcondition y

,so

sotheswimmerdrifts2milesdownstreamwhileheswims 1 mileacrosstheriver.

1.2 Problems

InProblems1through10,findafunction y D f.x/ satisfyingthegivendifferentialequationandtheprescribedinitial condition.

1. dy dx D 2x C 1; y.0/ D 3

2. dy dx D .x 2/2 ; y.2/ D 1

3. dy dx D px ; y.4/ D 0

4. dy dx D 1 x 2 ; y.1/ D 5

5. dy dx D 1 px C 2 ; y.2/ D 1

6. dy dx D x px 2 C 9; y. 4/ D 0

7. dy dx D 10 x 2 C 1 ; y.0/ D 0 8. dy dx D cos 2x ; y.0/ D 1

9. dy dx D 1 p1 x 2 ; y.0/ D 0

InProblems11through18,findthepositionfunction x.t/ ofa movingparticlewiththegivenacceleration a.t/,initialposition x0 D x.0/,andinitialvelocity v0 D v.0/.

11. a.t/ D 50, v0 D 10, x0 D 20

12. a.t/ D 20, v0 D 15, x0 D 5

13. a.t/ D 3t , v0 D 5, x0 D 0

14. a.t/ D 2t C 1, v0 D 7, x0 D 4

15. a.t/ D 4.t C 3/2 , v0 D 1, x0 D 1

16. a.t/ D 1 pt C 4 , v0 D 1, x0 D 1

17. a.t/ D 1 .t C 1/3 , v0 D 0, x0 D 0

18. a.t/ D 50 sin 5t , v0 D 10, x0 D 8

InProblems19through22,aparticlestartsattheoriginand travelsalongthe x -axiswiththevelocityfunction v.t/ whose graphisshowninFigs.1.2.6through1.2.9.Sketchthegraph oftheresultingpositionfunction x.t/ for 0 5 t 5 10

23. Whatisthemaximumheightattainedbythearrowofpart (b)ofExample3?

24. Aballisdroppedfromthetopofabuilding 400 fthigh. Howlongdoesittaketoreachtheground?Withwhat speeddoestheballstriketheground?

25. Thebrakesofacarareappliedwhenitismovingat 100 km=handprovideaconstantdecelerationof 10 metersper secondpersecond(m=s2 ).Howfardoesthecartravelbeforecomingtoastop?

26. Aprojectileisfiredstraightupwardwithaninitialvelocityof 100 m=sfromthetopofabuilding 20 mhighand fallstothegroundatthebaseofthebuilding.Find(a)its maximumheightabovetheground;(b)whenitpassesthe topofthebuilding;(c)itstotaltimeintheair.

27. Aballisthrownstraightdownwardfromthetopofatall building.Theinitialspeedoftheballis10m=s.Itstrikes thegroundwithaspeedof60m=s.Howtallisthebuilding?

28. Abaseballisthrownstraightdownwardwithaninitial speedof40ft=sfromthetopoftheWashingtonMonument(555fthigh).Howlongdoesittaketoreachthe ground,andwithwhatspeeddoesthebaseballstrikethe ground?

29. Adieselcargraduallyspeedsupsothatforthefirst10s itsaccelerationisgivenby dv dt D .0:12/t 2 C .0:6/t (ft=s2 ).

Ifthecarstartsfromrest(x0 D 0, v0 D 0),findthedistance ithastraveledattheendofthefirst10sanditsvelocityat thattime.

30. Acartravelingat60mi=h(88ft=s)skids176ftafterits brakesaresuddenlyapplied.Undertheassumptionthat thebrakingsystemprovidesconstantdeceleration,what isthatdeceleration?Forhowlongdoestheskidcontinue?

31. Theskidmarksmadebyanautomobileindicatedthatits brakeswerefullyappliedforadistanceof75mbefore itcametoastop.Thecarinquestionisknowntohave aconstantdecelerationof20m=s2 undertheseconditions.Howfast—inkm=h—wasthecartravelingwhen thebrakeswerefirstapplied?

32. Supposethatacarskids15mifitismovingat50km=h whenthebrakesareapplied.Assumingthatthecarhas thesameconstantdeceleration,howfarwillitskidifitis movingat100km=hwhenthebrakesareapplied?

33. OntheplanetGzyx,aballdroppedfromaheightof20ft hitsthegroundin2s.Ifaballisdroppedfromthetopof a200-ft-tallbuildingonGzyx,howlongwillittaketohit theground?Withwhatspeedwillithit?

34. Apersoncanthrowaballstraightupwardfromthesurfaceoftheearthtoamaximumheightof144ft.How highcouldthispersonthrowtheballontheplanetGzyx ofProblem33?

35. Astoneisdroppedfromrestataninitialheight h above thesurfaceoftheearth.Showthatthespeedwithwhichit strikesthegroundis v D p2gh

36. Supposeawomanhasenough“spring”inherlegstojump (onearth)fromthegroundtoaheightof2.25feet.If shejumpsstraightupwardwiththesameinitialvelocity onthemoon—wherethesurfacegravitationalacceleration is(approximately)5.3ft/s2 —howhighabovethesurface willsherise?

37. Atnoonacarstartsfromrestatpoint A andproceedsat constantaccelerationalongastraightroadtowardpoint B .Ifthecarreaches B at12:50 P. M .withavelocityof 60mi=h,whatisthedistancefrom A to B ?

38. Atnoonacarstartsfromrestatpoint A andproceedswith constantaccelerationalongastraightroadtowardpoint C , 35milesaway.Iftheconstantlyacceleratedcararrivesat C withavelocityof 60 mi=h,atwhattimedoesitarrive at C ?

39. If a D 0:5 miand v0 D 9 mi=hasinExample4,whatmust theswimmer’sspeed vS beinorderthathedriftsonly1 miledownstreamashecrossestheriver?

40. Supposethat a D 0:5 mi, v0 D 9 mi=h,and vS D 3 mi=h asinExample4,butthatthevelocityoftheriverisgiven bythefourth-degreefunction

R D v0 1 x 4 a 4 !

ratherthanthequadraticfunctioninEq.(18).Nowfind howfardownstreamtheswimmerdriftsashecrossesthe river.

1.3 SlopeFieldsandSolutionCurves

41. Abombisdroppedfromahelicopterhoveringatanaltitudeof800feetabovetheground.Fromthegrounddirectlybeneaththehelicopter,aprojectileisfiredstraight upwardtowardthebomb,exactly2secondsafterthebomb isreleased.Withwhatinitialvelocityshouldtheprojectile befiredinordertohitthebombatanaltitudeofexactly 400feet?

42. Aspacecraftisinfreefalltowardthesurfaceofthemoon ataspeedof1000mph(mi/h).Itsretrorockets,when fired,provideaconstantdecelerationof20,000mi/h2 .At whatheightabovethelunarsurfaceshouldtheastronauts firetheretrorocketstoinsureasofttouchdown?(Asin Example2,ignorethemoon’sgravitationalfield.)

43. ArthurClarke’s TheWindfromtheSun (1963)describes Diana,aspacecraftpropelledbythesolarwind.Itsaluminizedsailprovidesitwithaconstantaccelerationof 0:001g D 0:0098 m/s2 .Supposethisspacecraftstartsfrom restattime t D 0 andsimultaneouslyfiresaprojectile (straightaheadinthesamedirection)thattravelsatonetenthofthespeed c D 3 108 m/soflight.Howlongwill ittakethespacecrafttocatchupwiththeprojectile,and howfarwillithavetraveledbythen?

44. Adriverinvolvedinanaccidentclaimshewasgoingonly 25mph.Whenpolicetestedhiscar,theyfoundthatwhen itsbrakeswereappliedat25mph,thecarskiddedonly 45feetbeforecomingtoastop.Butthedriver’sskid marksattheaccidentscenemeasured210feet.Assumingthesame(constant)deceleration,determinethespeed hewasactuallytravelingjustpriortotheaccident.

Consideradifferentialequationoftheform

wheretheright-handfunction f.x;y/ involvesboththeindependentvariable x and thedependentvariable y .Wemightthinkofintegratingbothsidesin(1)withrespectto x ,andhencewrite y.x/ D R f.x;y.x//dx C C .However,thisapproach doesnotleadtoasolutionofthedifferentialequation,becausetheindicatedintegral involvesthe unknown function y.x/ itself,andthereforecannotbeevaluatedexplicitly.Actually,thereexists no straightforwardprocedurebywhichageneraldifferentialequationcanbesolvedexplicitly.Indeed,thesolutionsofsuchasimple-looking differentialequationas y 0 D x 2 C y 2 cannotbeexpressedintermsoftheordinary elementaryfunctionsstudiedincalculustextbooks.Nevertheless,thegraphicaland numericalmethodsofthisandlatersectionscanbeusedtoconstruct approximate solutionsofdifferentialequationsthatsufficeformanypracticalpurposes.

SlopeFieldsandGraphicalSolutions

Thereisasimplegeometricwaytothinkaboutsolutionsofagivendifferential equation y 0 D f.x;y/.Ateachpoint .x;y/ ofthe xy-plane,thevalueof f.x;y/ determinesaslope m D f.x;y/.Asolutionofthedifferentialequationissimply adifferentiablefunctionwhosegraph y D y.x/ hasthis“correctslope”ateach

Example1

point .x;y.x// throughwhichitpasses—thatis, y 0 .x/ D f.x;y.x//.Thusa solutioncurve ofthedifferentialequation y 0 D f.x;y/—thegraphofasolutionof theequation—issimplyacurveinthe xy-planewhosetangentlineateachpoint .x;y/ hasslope m D f.x;y/.Forinstance,Fig.1.3.1showsasolutioncurveof thedifferentialequation y 0 D x y togetherwithitstangentlinesatthreetypical points. x y (x1, y1) (x2, y2) (x3, y3)

FIGURE1.3.1. Asolutioncurveforthedifferentialequation y 0 D x y togetherwithtangentlineshaving slope m1 D x1 y1 atthepoint .x1 ;y1 /; slope m2 D x2 y2 atthepoint .x2 ;y2 /;and slope m3 D x3 y3 atthepoint .x3 ;y3 /

Thisgeometricviewpointsuggestsa graphicalmethod forconstructing approximate solutionsofthedifferentialequation y 0 D f.x;y/.Througheachofa representativecollectionofpoints .x;y/ intheplanewedrawashortlinesegment havingtheproperslope m D f.x;y/.Alltheselinesegmentsconstitutea slope field (ora directionfield)fortheequation y 0 D f.x;y/

Figures1.3.2(a)–(d)showslopefieldsandsolutioncurvesforthedifferentialequation dy dx D ky (2)

withthevalues k D 2, 0:5, 1,and 3 oftheparameter k inEq.(2).Notethateachslope fieldyieldsimportantqualitativeinformationaboutthesetofallsolutionsofthedifferential equation.Forinstance,Figs.1.3.2(a)and(b)suggestthateachsolution y.x/ approaches ˙1 as x !C1 if k>0,whereasFigs.1.3.2(c)and(d)suggestthat y.x/ ! 0 as x !C1 if k<0.Moreover,althoughthesignof k determinesthe direction ofincreaseordecrease of y.x/,itsabsolutevalue jk j appearstodeterminethe rateofchange of y.x/.Allthisis apparentfromslopefieldslikethoseinFig.1.3.2,evenwithoutknowingthatthegeneral solutionofEq.(2)isgivenexplicitlyby y.x/ D Ce kx

Aslopefieldsuggestsvisuallythegeneralshapesofsolutioncurvesofthe differentialequation.Througheachpointasolutioncurveshouldproceedinsuch adirectionthatitstangentlineisnearlyparalleltothenearbylinesegmentsofthe slopefield.Startingatanyinitialpoint .a;b/,wecanattempttosketchfreehandan approximatesolutioncurvethatthreadsitswaythroughtheslopefield,following thevisiblelinesegmentsascloselyaspossible.

Example2

Constructaslopefieldforthedifferentialequation y 0 D x y anduseittosketchanapproximatesolutioncurvethatpassesthroughthepoint . 4;4/

Solution Figure1.3.3showsatableofslopesforthegivenequation.Thenumericalslope m D x y appearsattheintersectionofthehorizontal x -rowandthevertical y -columnofthetable.If youinspectthepatternofupper-lefttolower-rightdiagonalsinthistable,youcanseethatit

FIGURE1.3.2(a) Slopefieldand solutioncurvesfor y 0 D 2y

FIGURE1.3.2(b) Slopefieldand solutioncurvesfor y 0 D .0:5/y

FIGURE1.3.2(c) Slopefieldand solutioncurvesfor y 0 D y

FIGURE1.3.2(d) Slopefield andsolutioncurvesfor

FIGURE1.3.3. Valuesoftheslope y 0 D x y for 4 x;y 4

waseasilyandquicklyconstructed.(Ofcourse,amorecomplicatedfunction f.x;y/ onthe right-handsideofthedifferentialequationwouldnecessitatemorecomplicatedcalculations.)

Figure1.3.4showsthecorrespondingslopefield,andFig.1.3.5showsanapproximatesolutioncurvesketchedthroughthepoint . 4;4/ soastofollowthisslopefieldascloselyas possible.Ateachpointitappearstoproceedinthedirectionindicatedbythenearbyline segmentsoftheslopefield.

Althoughaspreadsheetprogram(forinstance)readilyconstructsatableof slopesasinFig.1.3.3,itcanbequitetedioustoplotbyhandasufficientnumber

0 5 x y –5 –5

FIGURE1.3.4. Slopefieldfor y 0 D x y correspondingtothetableofslopesinFig.1.3.3.

FIGURE1.3.5. Thesolutioncurve through . 4;4/.

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