《高数双语》课件section 6.1

Section7.1BasicConceptsofDifferentialEquations2ExamplesofDifferentialEquationsExample1Supposethataplanecurvepassedthroughthepoint(1,2)inthexOyplane.Theslopeofthetangentatanypoint(x,y)tothecurveis2x.Findtheequationofthecurve.SolutionBythegeometricmeaningofderivatives,thedesiredcurvey=f(x)shouldsatisfy.Integratingonbothsidesofthe1stequationwithrespecttox,weobtaindifferentialequationInitialcondition3ExamplesofDifferentialEquationsSolution(continued)SubstitutingtheinitialconditiontothelastTherefore,theequationofthedesiredcurveisequation,wehaveFinish.whereCisanarbitraryconstant.Itcanbeevaluatedbytheinitialcondition.Example1Supposethataplanecurvepassedthroughthepoint(1,2)inthexOyplane.Theslopeofthetangentatanypoint(x,y)tothecurveis2x.Findtheequationofthecurve.Example2SupposethataparticlewithmassmfallsfreelyfromapositionofheightH,withinitialvelocityV0.Ifweneglecttheresistanceofair,findtherelationshipbetweentheheightH

andtimet

whiletheparticleisfalling.4ExamplesofDifferentialEquationsHh(t)SolutionDenotetheinitialtimewhentheparticlestartstofallbyt=0,anddenotetheheightoftheparticleatanytimetintheprocessoffallingbyh=h(t).ByNewton’ssecondlaw,hshouldsatisfythefollowingequationdifferentialequationInitialconditionsExample2SupposethataparticlewithmassmfallsfreelyfromapositionofheightH,withinitialvelocityV0.Ifweneglecttheresistanceofair,findtherelationshipbetweentheheightH

andtimet

whiletheparticleisfalling.5ExamplesofDifferentialEquationsHh(t)Solution(continued)

Integratingbothsidesofthelastequationtwice,wehaveFinish.Substitutingtheinitialconditiontotheaboveequationyields6BasicConceptsDefinition(Differentialequation)Anequationiscalledadifferentialequation(微分方程)ifitcontainsthederivativeordifferentialofanunknownfunction.andarebothdifferentialequations.DefinitionAdifferentialequationinwhichtheunknownfunctionyisaunivariate

function,iscalledanordinarydifferentialequation(常微分方程)

andwillbereferredasdifferentialequation(微分方程).Example:Theorderofthehighestorderderivativeoftheunknownfunctionintheequationiscalledtheorder(阶)

oftheequation.7BasicConceptsFirst-OrderSecond-OrderExample:Thegeneralformofafirstorderdifferentialequationmaybe

expressedbyandF(x,y)isafunctionwhichdependsontheindependentvariablexandthedependentvariabley.8BasicConceptsDefinition(Solution,GeneralSolution,InitialConditionsandParticularSolution)Ifthesolutioncontainsarbitraryconstantsandthenumberoftheindependentconstantsjustequalstheorderoftheequation,thenthissolutioniscalledthegeneralsolution(通解)

oftheequation.Ifallthearbitraryconstantsinasolutionhavebeendetermined,thenthesolutioniscalledaparticularsolution(特解)

oftheequation.Ifafunctiony=f(x)satisfiesagivendifferentialequation,thenthefunctiony=f(x)iscalledasolution(解)

oftheequation.Theadditionalconditionsarecalledtheinitialconditions(初始条件)

oftheequation.9FundamentalConceptsofDifferentialEquationsNotethatthegeneralsolutionmaynotbethetotalsolutions.Example10GeometricInterpretationoftheFirstOrderDifferentialEquation11GeometricInterpretationoftheFirstOrderDifferentialEquationSlopeFields:ViewingSolutionCurvesEachtimewespecifyaninitialconditiony(x0)=y0forthesolutionofadifferentialequation,thesolutioncurve(graphofthesolution)isrequiredtopassthroughthepoint(x0,y0)andtohaveslope

f(x0,y0)there.12GeometricInterpretationoftheFirstOrderDifferentialEquationSlopeFields:ViewingSolutionCurvesWecanpicturetheslopesgraphicallybydrawingshortlinesegmentsofslopef(x,y)atselectedpoints(x,y)intheregionofthexy