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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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.