series solutions of second order linear equations

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1、August7,201221:04c05Sheetnumber1Pagenumber247cyanblackCHAPTER5SeriesSolutionsofSecondOrderLinearEquationsFindingthegeneralsolutionofalineardifferentialequationdependsondetermin-ingafundamentalsetofsolutionsofthehomogeneousequation.Sofar,wehavegivenasystematicprocedureforconstructingfundamentalsoluti

2、onsonlyiftheequationhasconstantcoefficients.Todealwiththemuchlargerclassofequationsthathavevariablecoefficients,itisnecessarytoextendoursearchforsolutionsbeyondthefamiliarelementaryfunctionsofcalculus.Theprincipaltoolthatweneedistherep-resentationofagivenfunctionbyapowerseries.Thebasicideaissimilartot

3、hatinthemethodofundeterminedcoefficients:weassumethatthesolutionsofagivendif-ferentialequationhavepowerseriesexpansions,andthenweattempttodeterminethecoefficientssoastosatisfythedifferentialequation.5.1ReviewofPowerSeriesInthischapterwediscusstheuseofpowerseriestoconstructfundamentalsetsofsolutionsofs

4、econdorderlineardifferentialequationswhosecoefficientsarefunc-tionsoftheindependentvariable.Webeginbysummarizingverybrieflythepertinentresultsaboutpowerseriesthatweneed.ReaderswhoarefamiliarwithpowerseriesmaygoontoSection5.2.Thosewhoneedmoredetailsthanarepresentedhereshouldconsultabookoncalculus.∞1.A

5、powerseriesa(x−x)nissaidtoconvergeatapointxifn0n=0mnliman(x−x0)m→∞n=0247August7,201221:04c05Sheetnumber2Pagenumber248cyanblack248Chapter5.SeriesSolutionsofSecondOrderLinearEquationsexistsforthatx.Theseriescertainlyconvergesforx=x0;itmayconvergeforallx,oritmayconvergeforsomevaluesofxandnotforothers.

6、∞2.Theseriesa(x−x)nissaidtoconvergeabsolutelyatapointxiftheseriesn0n=0∞∞nn

7、an(x−x0)

8、=

9、an

10、

11、x−x0

12、n=0n=0converges.Itcanbeshownthatiftheseriesconvergesabsolutely,thentheseriesalsoconverges;however,theconverseisnotnecessarilytrue.3.Oneofthemostusefultestsfortheabsoluteconvergenceofapowerseriesistherat

13、iotest.Ifan=0,andif,forafixedvalueofx,a(x−x)n+1alimn+10=

14、x−x

15、limn+1=

16、x−x

17、L,n→∞a(x−x)n0n→∞a0n0nthenthepowerseriesconvergesabsolutelyatthatvalueofxif

18、x−x0

19、L<1anddivergesif

20、x−x0

21、L>1.If

22、x−