常微分方程1.3 坡地和解曲线Slope fields and solution curve.ppt

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1、Section1.3:SlopeFieldsandSolutionCurvesConceptsofsolutioncurve（orintegralcurve）；一阶微分方程的解曲线(积分曲线)；slopefield(ordirectionfield);斜率场(方向场)；Graphicalmethodforconstructingapproximatesolutionsofthefirst-orderequation;求其近似解的图像方法；ApplicationsofSlopefields;斜率场的应用；ExistenceandUniquenessofSolutions.一阶微分方程解的存在

2、唯一性定理。Section1.3:SlopeFieldsandSolutionCurvesConceptsofsolutioncurve（orintegralcurve）:一阶微分方程的解曲线(积分曲线):Asolutioncurveofthedifferentialequationy'=f(x,y)--thegraphofasolutionoftheequation--issimplyacurveinthexy-planewhosetangentlineateachpoint(x,y)hasslopem=f(x,y).FIGURE1.3.1.Asolutioncurveforthedif

3、ferentialequationy'=x-ytogetherwithtangentlineshavingslopex-yatthepoint(x,y).Objective:Thegraphicalandnumericalmethodsofthisandlatersectionscanbeusedtoconstructapproximatesolutionsofthedifferentialequationsy'=f(x,y);Agraphicalmethod:Througheachofarepresentativecollectionofpointsintheplanewedrawash

4、ortlinesegmenthavingtheproperslopem=f(x,y).Alltheselinesegmentsconstituteaslopefield(oradirectionfield)fortheequationy'=f(x,y).Section1.3:SlopeFieldsandGraphicalSolutionsSection1.3:Example1FIGURE1.3.2(a)Slopefieldandsolutioncurvesfory'=2y.Constructaslopefieldforthedifferentialequationy'=x-yandusei

5、ttosketchanapproximatesolutioncurvethatpassesthroughthepoint(-4,4).Section1.3:Example2FIGURE1.3.5.Thesolutioncurvethrough(-4,4).Supposeyouthrowabaseballstraightdownwardfromahelicopterhoveringatanaltitudeof3000feet.Youwonderwhethersomeonestandingonthegroundbelowcouldconceivablycatchit.Inordertoesti

6、matethespeedwithwhichtheballwillland,weconstructaslopefieldforthedifferentialequationApplicationofSlopefields:Example3FIGURE1.3.7.Slopefieldandtypicalsolutioncurvesfor(4).Example4ApplicationofSlopefieldsExample5(a)[Failureofexistence]Theinitialvalueproblem,(b)[Failureofuniqueness]Theinitialvaluepr

7、oblem,Example4ApplicationofSlopefieldsExample5(a)[Failureofexistence]Theinitialvalueproblem,(b)[Failureofuniqueness]Theinitialvalueproblem,Supposethatboththefunctionf(x,y)anditspartialderivativeDyf(x,y)arecontinu