first-order ordinary differential equations

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1、First-OrderOrdinaryDifferential2EquationsWewilldevoteaconsiderableamountoftimeinthistexttodevelopingexplicit,implicit,numerical,andgraphicalsolutionsofdifferentialequations.Inthischap-terweintroducefrequentlyencounteredformsof?rst-orderordinarydifferentialequationsandmethodstoconstructe

2、xplicit,numerical,andgraphicalsolutionsofthem.Severaloftheequationsalongwiththemethodsofsolutiondiscussedherewillbeusedinsubsequentchaptersofthetext.2.1TheoryofFirst-OrderEquations:ABriefDiscussionInordertounderstandthetypesof?rst-orderinitial-valueproblemsthathaveauniquesolution,thefol

3、lowingtheoremisstated.Theorem1(ExistenceandUniqueness).Considertheinitial-valueproblemdy/dxfx,y(2.1)yx0y0.Iffandf/yarecontinuousfunctionsontherectangularregionR,Seetextslike[6],[7],or[3].Rx,ya

4、x0,y0,thereexistsanintervalxx0

5、o?ndageneralsolutionofyfx,y.EXAMPLE2.1.1:Solvetheinitial-valueproblemdy/dxx/yy00.DoesthisresultcontradicttheExistenceandUniquenessTheorem?SOLUTION:ThisequationissolvedwithDSolvetodeterminethefamilyofsolutionsy2x2C.In[74]:=Clearx,yxDSolveyx,yx,xyxOut[74]=yxx

6、22C1,yxx22C1Wenotethatthegraphofy2x2CforvariousvaluesofCisthesameasthegraphofthelevelcurvesoffx,yy2x2.MembersofthisfamilycorrespondingtoC40,38,...,38,40aregraphedwithContourPlotinFigure2-1.In[75]:=cvalsTablei,i,40,40,2 22In[76]:=ContourPlotyx,x,6,6,y,6,6,Plo

7、tPoints120,Contours>cvals,ContourShadingFalseApplicationoftheinitialconditionyields0202C,soC0.There-fore,solutionsthatpassthrough0,0,satisfyy2x20,sothere2.1TheoryofFirst-OrderEquations:ABriefDiscussion436420-2-4-6-6-4-20246Figure2-1Plotoffx,yCforvariousvaluesofCaret