Detailed Instructions Download | oreceivefullcredityoumustshowallwork.ThereareFIVEproblemsworthatotalof100points.Youhavethreedays(72hours)todothisexam.Inthisexam,youwillmodifytheprovidedcodesinthisfile.YoumaydownloadthefileandworkonitusingAnacondaDistribution(localhost)orCoCalc.Inordertobeaccepted,youMUSTsubmitthetwofollowingfiles:a.AsinglePDFfileshowingallyoursupportingworkwiththename:b.AfileonJupyternotebookincludingallcodingassignmentswiththename:Thepowerseriesmethodappearstobeoneoftheefficientmethodtosolvedifferentialequationswithvariablecoefficients.Thiskindoftechniquecanbeusedtosolvemanyofthenonelementarydifferentialequationsthatappearmostfrequentlyinapplicationssuchasacoustic,healflow,andelectromagnetic,radiation.(a)(2)Brieflydiscusstheideaofthepowerseriesmethodtosolvedifferentialequationsoftheformy′′(x)+p(x)y′(x)+q(x)y(x)=f(x)assumingthatp(x),q(x),andf(x)haveapowerseriesexpansion.(b)Findgeneralsolutionsinpowersofxofthedifferentialequations.Statetherecur–rencerelationandtheguaranteedradiusofconvergenceineachcase.i.(4)y′′−x2y′−3xy=0ii.(4)y′′+xy=0(anAiryequation)(c)Findathree–termrecurrencerelationforsolutionsoftheformy=!cnxn.Thenfindthefirstthreenonzerotermsineachoftwolinearlyindependentsolutions.i.(4)(x2−1)y′′+2xy′+2xy=0ii.(4)(1+x3)y′′+x4y=0(d)(6)Solvetheinitialvalueproblem“y′′+xy′+(2×2+1)y=0y(0)=1,y′(0)=−12.TheHermiteequationoforderαisy′′−2xy′+2αy=0.(a)(8)Derivethetwopowerseriessolutionsy1=1+∞#m=1(−1)m2mα(α−2)…(α−2m+2)(2m)!x2mandy2=x+∞#m=1(−1)m2m(α−1)(α−3)…(α−2m+1)x2m+1(2m+1)!.Showthaty1isapolynomialifαisaneveninteger,whereasy2isasolutionifαisanoddinteger.(b)(4)TheHermitepolynomialofdegreenisdenotedbyHn(x).Itisthenth–degreepolynomialsolutionofHermite’sequation,multipliedbyasuitableconstantsothatthecoefficientofxnis2n.ShowthatthefirstsixHermitepolynomialsareH0(x)≡1,H1(x)=2x,H2(x)=4×2−2,H3(x)=8×3−12x,H4(x)=16×4−48×2+12H5(x)=32×5−160×3+120x.(c)AgeneralformulafortheHermitepolynomialsisHn(x)=(−1)nex2dndxn$e−x2%(1)Verifytheformula(1)byfollowingthestepsoutlinedbelow.i.(2)Showthatv:=e−x2satisfiesthedifferentialequationv′+2xv=0.ii.(2)Differentiateeachsideofthisequationtoobtainv′′+2xv′+2v=0.iii.(2)Differentiateeachsideofthelastequationntimesinsuccessiontoobtainv(n+2)+2xv(n+1)+2(n+1)v(n)=0.iv.(2)Definey=(−1)nex2v(n).Showthaty′′−2xy′+2ny=(−1)nex2&v(n+2)+2xv(n+1)+2(n+1)v(n)‘.ThenexplainwhyyisasolutionoftheHermiteequationofordern.(d)(2)WriteacodeinSageMathtogeneratethefirstsixHermitepolynomialsusingtheformulain(1).(e)(2)GraphthesefirstsixHermitepolynomialsusingSageMathtoinvestigatethecon–jecturethat(foreachn)thezerosoftheHermitepolynomialsHnandHn+1are“interlaced”–thatis,thenzerosofHnlieinthenboundedopenintervalswhoseendpointsaresuccessivepairsofzerosofHn+1 Click here to request for this assignment help Get Asnwer
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