常微分方程1.5一阶线性方程-Linear-first-order-equtions.ppt

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1、Section1.5:LinearFirst-OrderEquationsConceptsofintegratingfactor;积分因子；linearfirst-orderequation;一阶线性微分方程;Solutionoflinearfirst-orderequation;一阶线性微分方程的解法；Existenceanduniquenesstheoremforlinearfirst-orderequation;一阶线性微分方程解的存在唯一性定理；Mixtureproblems.溶液的混合问题。Section1.5:Li

2、nearFirst-OrderEquationsConceptsofintegratingfactor;积分因子；Anintegratingfactorforadifferentialequationisafunctionp(x,y)suchthatthemultiplicationofeachsideofthedifferentialequationbyp(x,y)yieldsanequationinwhicheachsideisrecognizableasaderivative.Forinstance:thatis,For

3、thisreason,thefunctionp(y)=l/yiscalledanintegratingfactorfortheoriginalequationin(1).Section1.5:LinearFirst-OrderEquationslinearfirst-orderequation;一阶线性微分方程;Method:Solutionoflinearfirst-orderequation:WemultiplyeachsideinEq.(3)bytheintegratingfactor:Finally,solvingfo

4、ry,weobtainthegeneralsolutionofthelinearfirst-orderequationin(3):Section1.5:METHOD:SOLUTIONOFFIRST-ORDEREQ.2.Thenmultiplybothsidesofthedifferentialequationbytheintegratingfactorp(x);1.Beginbycalculatingtheintegratingfactor;3.Next,recognizetheleft-handsideoftheresult

5、ingequationasthederivativeofaproduct:4.Finally,integratethisequationTheorem1:Existenceanduniquenesstheoremforlinearfirst-orderequation(P48)IfthefunctionsP(x)andQ(x)arecontinuousontheopenintervalIcontainingthepointXo,thentheinitialvalueproblemhasauniquesolutiony(x)on

6、I,givenbytheformulainEq.(6)withanappropriatevalueofC.Theorem1:Existenceanduniquenesstheoremforlinearfirst-orderequation(P48)constantCinEq.(6)-asneededtosolvetheinitialvalueprobleminEq.(11)-canbeselected"automatically"bywritingExample3SolvetheinitialvalueproblemMixtu

7、reProblems(P50)Asafirstapplicationoflinearfirst-orderequations,weconsideratankcontainingasolution-amixtureofsoluteandsolvent-suchassaltdissolvedinwater.Thereisbothinflowandoutflow,andwewanttocomputetheamountx(t)ofsoluteinthetankattimet,giventheamountx(0)=Xoattimet=0

8、.SupposethatsolutionwithaconcentrationofCigramsofsoluteperliterofsolutionflowsintothetankattheconstantrateofriliterspersecond,andthattheso