17Calculus Differential Equations - Singular Points

If both of these limits are finite, then is a regular singular point of the differential equation. If one (or both) of these limits is infinite or undefined ... \(\newcommand{\abs}[1]{\left|\,{#1}\,\right|}\) \(\newcommand{\cm}{\mathrm{cm}}\) \(\newcommand{\sec}{\,\mathrm{sec}\,}\) \(\newcommand{\units}[1]{\,\text{#1}}\) \(\newcommand{\vhat}[1]{\,\hat{#1}}\) \(\newcommand{\vhati}{\,\hat{i}}\) \(\newcommand{\vhatj}{\,\hat{j}}\) \(\newcommand{\vhatk}{\,\hat{k}}\) \(\newcommand{\vect}[1]{\boldsymbol{\vec{#1}}}\) \(\newcommand{\norm}[1]{\|{#1}\|}\) \(\newcommand{\arccot}{\,\mathrm{arccot}\,}\) \(\newcommand{\arcsec}{\,\mathrm{arcsec}\,}\) \(\newcommand{\arccsc}{\,\mathrm{arccsc}\,}\) \(\newcommand{\sech}{\,\mathrm{sech}\,}\) \(\newcommand{\csch}{\,\mathrm{csch}\,}\) \(\newcommand{\arcsinh}{\,\mathrm{arcsinh}\,}\) \(\newcommand{\arccosh}{\,\mathrm{arccosh}\,}\) \(\newcommand{\arctanh}{\,\mathrm{arctanh}\,}\) \(\newcommand{\arccoth}{\,\mathrm{arccoth}\,}\) \(\newcommand{\arcsech}{\,\mathrm{arcsech}\,}\) \(\newcommand{\arccsch}{\,\mathrm{arccsch}\,}\) SVC MVC ODE Precalc Search MyAccount 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RecommendedBooksonAmazon(affiliatelinks) Complete17CalculusRecommendedBooksList PrimeStudent6-monthTrial SingularPoints Wearegoingtolookatthegeneraldifferentialequation \[ P(x)y''+Q(x)y'+R(x)y=0 \] Noticethatthisishomogeneoussincetherightsideiszero.Also,\(P(x),Q(x)\)and\(R(x)\)arefunctionsof\(x\)only,i.e.therearenotytermsorderivativesofx. Singularpointsoccurwhere\(P(x)=0\).Atthesepoints,the\(y''\)termdisappears,whichwedonotwanttohappen.Also,wewilloftenwritethisas \[ y''+\frac{Q(x)}{P(x)}y'+\frac{R(x)}{P(x)}y=0 \] and,ofcourse,wedonotwantzerointhedenominatorsofthesecondandthirdterms. Sotofindthesingularpoints,itisprobablybesttowritethedifferentialequationinsecondformaboveanddeterminethevaluesofxwherethedenominatorsofeachofthefractionsontheleftarezero. Wealsorequirethatatleastoneof\(Q(x)\)and\(R(x)\)isnotzeroatthesingularpoints.Hereishowwedeterminethis.Let'scall\(x_0\)apointwhere\(P(x_0)=0\).Usingequations,wecalculatethetwolimits \[ \lim_{x-x_0}{(x-x_0)\frac{Q(x)}{P(x)}}~\text{and}~\lim_{x-x_0}{(x-x_0)^2\frac{R(x)}{P(x)}} \] Ifbothoftheselimitsarefinite,then\(x=x_0\)isaregularsingularpointofthedifferentialequation. Ifone(orboth)oftheselimitsisinfiniteorundefined,then\(x=x_0\)isanirregularsingularpoint. AnalyticFunctions Ifthefunctions,\(P(x),Q(x)\)and\(R(x)\)arepolynomials,thenwedon'tneedanyconstraintsotherthantheabovelimitswhencheckingsingularpoints.However,formorecomplicatedfunctions,werequirethatboth \[ (x-x_0)\frac{Q(x)}{P(x)}~\text{and}~(x-x_0)^2\frac{R(x)}{P(x)} \] haveconvergentTaylorSeriesabout\(x_0\).Thetermweusetodescribethistosaythatthetwofunctionsareanalytic. Fortunately,aslongaswehavepolynomials,exponentials,logarithms,sinesorcosines,wearegood.TheyareallanalyticandhaveconvergentTaylorSeries. BookRecommendation Ifyouwantmoredetailonpowerseriessolutionandradiusconvergence,thisbookseemstocoverthematerialinmoredetailthanmostothertextbookswehavelookedat. ReallyUNDERSTANDDifferentialEquations Logintoratethispageandtoseeit'scurrentrating. TopicsYouNeedToUnderstandForThisPage powerseries basicsofdifferentialequations RelatedTopicsandLinks externalsitesyoumayfindhelpful SeriesSolutionstoDifferentialEquations KSU-RegularSingularPoints Tobookmarkthispageandpracticeproblems,logintoyouraccountorsetupafreeaccount. SearchPracticeProblems Doyouhaveapracticeproblemnumberbutdonotknowonwhichpageitisfound?Ifso,enterthenumberbelowandclick'page'togotothepageonwhichitisfoundorclick'practice'tobetakentothepracticeproblem. howtotakegoodnotes TryAudiblePremiumPlusandGetUptoTwoFreeAudiobooks AsanAmazonAssociateIearnfromqualifyingpurchases. IrecentlystartedaPatreonaccounttohelpdefraytheexpensesassociatedwiththissite.Tokeepthissitefree,pleaseconsidersupportingme. 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