Differential Equations - Definitions - Pauls Online Math Notes

The general solution to a differential equation is the most general form that the solution can take and doesn't take any initial conditions into ... GoTo Notes PracticeandAssignmentproblemsarenotyetwritten.AstimepermitsIamworkingonthem,howeverIdon'thavetheamountoffreetimethatIusedtosoitwilltakeawhilebeforeanythingshowsuphere. Show/Hide ShowallSolutions/Steps/etc. HideallSolutions/Steps/etc. Sections BasicConceptsIntroduction DirectionFields Chapters FirstOrderDE's Classes Algebra CalculusI CalculusII CalculusIII DifferentialEquations Extras Algebra&TrigReview CommonMathErrors ComplexNumberPrimer HowToStudyMath CheatSheets&Tables Misc ContactMe MathJaxHelpandConfiguration NotesDownloads CompleteBook PracticeProblemsDownloads Problemsnotyetwritten. AssignmentProblemsDownloads Problemsnotyetwritten. 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Section1-1:Definitions DifferentialEquation Thefirstdefinitionthatweshouldcovershouldbethatofdifferentialequation.Adifferentialequationisanyequationwhichcontainsderivatives,eitherordinaryderivativesorpartialderivatives. Thereisonedifferentialequationthateverybodyprobablyknows,thatisNewton’sSecondLawofMotion.Ifanobjectofmass\(m\)ismovingwithacceleration\(a\)andbeingactedonwithforce\(F\)thenNewton’sSecondLawtellsus. \[\begin{equation}F=ma\label{eq:eq1}\end{equation}\] Toseethatthisisinfactadifferentialequationweneedtorewriteitalittle.First,rememberthatwecanrewritetheacceleration,\(a\),inoneoftwoways. \[\begin{equation}a=\frac{{dv}}{{dt}}\hspace{0.25in}{\mbox{OR}}\hspace{0.25in}\,\,\,\,\,\,a=\frac{{{d^2}u}}{{d{t^2}}}\label{eq:eq2}\end{equation}\] Where\(v\)isthevelocityoftheobjectand\(u\)isthepositionfunctionoftheobjectatanytime\(t\).Weshouldalsorememberatthispointthattheforce,\(F\)mayalsobeafunctionoftime,velocity,and/orposition. So,withallthesethingsinmindNewton’sSecondLawcannowbewrittenasadifferentialequationintermsofeitherthevelocity,\(v\),ortheposition,\(u\),oftheobjectasfollows. \[\begin{equation}m\frac{{dv}}{{dt}}=F\left({t,v}\right)\label{eq:eq3}\end{equation}\] \[\begin{equation}m\frac{{{d^2}u}}{{d{t^2}}}=F\left({t,u,\frac{{du}}{{dt}}}\right)\label{eq:eq4}\end{equation}\] So,hereisourfirstdifferentialequation.Wewillseebothformsofthisinlaterchapters. Hereareafewmoreexamplesofdifferentialequations. \[\begin{equation}ay''+by'+cy=g\left(t\right)\label{eq:eq5}\end{equation}\] \[\begin{equation}\sin\left(y\right)\frac{{{d^2}y}}{{d{x^2}}}=\left({1-y}\right)\frac{{dy}}{{dx}}+{y^2}{{\bf{e}}^{-5y}}\label{eq:eq6}\end{equation}\] \[\begin{equation}{y^{\left(4\right)}}+10y'''-4y'+2y=\cos\left(t\right)\label{eq:eq7}\end{equation}\] \[\begin{equation}{\alpha^2}\frac{{{\partial^2}u}}{{\partial{x^2}}}=\frac{{\partialu}}{{\partialt}}\label{eq:eq8}\end{equation}\] \[\begin{equation}{a^2}{u_{xx}}={u_{tt}}\label{eq:eq9}\end{equation}\] \[\begin{equation}\frac{{{\partial^3}u}}{{{\partial}x^2\partialt}}=1+\frac{{\partialu}}{{\partialy}}\label{eq:eq10}\end{equation}\] Order Theorderofadifferentialequationisthelargestderivativepresentinthedifferentialequation.Inthedifferentialequationslistedabove\(\eqref{eq:eq3}\)isafirstorderdifferentialequation,\(\eqref{eq:eq4}\),\(\eqref{eq:eq5}\),\(\eqref{eq:eq6}\),\(\eqref{eq:eq8}\),and\(\eqref{eq:eq9}\)aresecondorderdifferentialequations,\(\eqref{eq:eq10}\)isathirdorderdifferentialequationand\(\eqref{eq:eq7}\)isafourthorderdifferentialequation. Notethattheorderdoesnotdependonwhetherornotyou’vegotordinaryorpartialderivativesinthedifferentialequation. Wewillbelookingalmostexclusivelyatfirstandsecondorderdifferentialequationsinthesenotes.Asyouwillseemostofthesolutiontechniquesforsecondorderdifferentialequationscanbeeasily(andnaturally)extendedtohigherorderdifferentialequationsandwe’lldiscussthatidealateron. OrdinaryandPartialDifferentialEquations Adifferentialequationiscalledanordinarydifferentialequation,abbreviatedbyode,ifithasordinaryderivativesinit.Likewise,adifferentialequationiscalledapartialdifferentialequation,abbreviatedbypde,ifithaspartialderivativesinit.Inthedifferentialequationsabove\(\eqref{eq:eq3}\)-\(\eqref{eq:eq7}\)areode’sand\(\eqref{eq:eq8}\)-\(\eqref{eq:eq10}\)arepde’s. Thevastmajorityofthesenoteswilldealwithode’s.Theonlyexceptiontothiswillbethelastchapterinwhichwe’lltakeabrieflookatacommonandbasicsolutiontechniqueforsolvingpde’s. LinearDifferentialEquations Alineardifferentialequationisanydifferentialequationthatcanbewritteninthefollowingform. \[\begin{equation}{a_n}\left(t\right){y^{\left(n\right)}}\left(t\right)+{a_{n-1}}\left(t\right){y^{\left({n-1}\right)}}\left(t\right)+\cdots+{a_1}\left(t\right)y'\left(t\right)+{a_0}\left(t\right)y\left(t\right)=g\left(t\right)\label{eq:eq11}\end{equation}\] Theimportantthingtonoteaboutlineardifferentialequationsisthattherearenoproductsofthefunction,\(y\left(t\right)\),anditsderivativesandneitherthefunctionoritsderivativesoccurtoanypowerotherthanthefirstpower.Alsonotethatneitherthefunctionoritsderivativesare“inside”anotherfunction,forexample,\(\sqrt{y'}\)or\({{\bf{e}}^y}\). Thecoefficients\({a_0}\left(t\right),\,\,\ldots\,\,,{a_n}\left(t\right)\)and\(g\left(t\right)\)canbezeroornon-zerofunctions,constantornon-constantfunctions,linearornon-linearfunctions.Onlythefunction,\(y\left(t\right)\),anditsderivativesareusedindeterminingifadifferentialequationislinear. Ifadifferentialequationcannotbewrittenintheform,\(\eqref{eq:eq11}\)thenitiscalledanon-lineardifferentialequation. In\(\eqref{eq:eq5}\)-\(\eqref{eq:eq7}\)aboveonly\(\eqref{eq:eq6}\)isnon-linear,theothertwoarelineardifferentialequations.Wecan’tclassify\(\eqref{eq:eq3}\)and\(\eqref{eq:eq4}\)sincewedonotknowwhatformthefunction\(F\)has.Thesecouldbeeitherlinearornon-lineardependingon\(F\). Solution Asolutiontoadifferentialequationonaninterval\(\alpha0\). ShowSolution We’llneedthefirstandsecondderivativetodothis. \[y'\left(x\right)=-\frac{3}{2}{x^{-\frac{5}{2}}}\hspace{0.25in}y''\left(x\right)=\frac{{15}}{4}{x^{-\frac{7}{2}}}\] Plugtheseaswellasthefunctionintothedifferentialequation. \[\begin{align*}4{x^2}\left({\frac{{15}}{4}{x^{-\,\frac{7}{2}}}}\right)+12x\left({-\frac{3}{2}{x^{-\,\frac{5}{2}}}}\right)+3\left({{x^{-\,\frac{3}{2}}}}\right)&=0\\15{x^{-\,\frac{3}{2}}}-18{x^{-\,\frac{3}{2}}}+3{x^{-\,\frac{3}{2}}}&=0\\0&=0\end{align*}\] So,\(y\left(x\right)={x^{-\frac{3}{2}}}\)doessatisfythedifferentialequationandhenceisasolution.Whythendidweincludetheconditionthat\(x>0\)?Wedidnotusethisconditionanywhereintheworkshowingthatthefunctionwouldsatisfythedifferentialequation. Toseewhyrecallthat \[y\left(x\right)={x^{-\frac{3}{2}}}=\frac{1}{{\sqrt{{x^3}}}}\] Inthisformitisclearthatwe’llneedtoavoid\(x=0\)attheleastasthiswouldgivedivisionbyzero. Also,thereisageneralruleofthumbthatwe’regoingtorunwithinthisclass.Thisruleofthumbis:Startwithrealnumbers,endwithrealnumbers.Inotherwords,ifourdifferentialequationonlycontainsrealnumbersthenwedon’twantsolutionsthatgivecomplexnumbers.So,inordertoavoidcomplexnumberswewillalsoneedtoavoidnegativevaluesof\(x\). So,wesawinthelastexamplethateventhoughafunctionmaysymbolicallysatisfyadifferentialequation,becauseofcertainrestrictionsbroughtaboutbythesolutionwecannotuseallvaluesoftheindependentvariableandhence,mustmakearestrictionontheindependentvariable.Thiswillbethecasewithmanysolutionstodifferentialequations. Inthelastexample,notethatthereareinfactmanymorepossiblesolutionstothedifferentialequationgiven.Forinstance,allofthefollowingarealsosolutions \[\begin{align*}y\left(x\right)&={x^{-\frac{1}{2}}}\\y\left(x\right)&=-9{x^{-\frac{3}{2}}}\\y\left(x\right)&=7{x^{-\frac{1}{2}}}\\y\left(x\right)&=-9{x^{-\frac{3}{2}}}+7{x^{-\frac{1}{2}}}\end{align*}\] We’llleavethedetailstoyoutocheckthattheseareinfactsolutions.Giventheseexamplescanyoucomeupwithanyothersolutionstothedifferentialequation?Thereareinfactaninfinitenumberofsolutionstothisdifferentialequation. So,giventhatthereareaninfinitenumberofsolutionstothedifferentialequationinthelastexample(providedyoubelieveuswhenwesaythatanyway….)wecanaskanaturalquestion.Whichisthesolutionthatwewantordoesitmatterwhichsolutionweuse?Thisquestionleadsustothenextdefinitioninthissection. InitialCondition(s) InitialCondition(s)areacondition,orsetofconditions,onthesolutionthatwillallowustodeterminewhichsolutionthatweareafter.Initialconditions(oftenabbreviatedi.c.’swhenwe’refeelinglazy…)areoftheform, \[y\left({{t_0}}\right)={y_0}\hspace{0.25in}{\mbox{and/or}}{y^{\left(k\right)}}\left({{t_0}}\right)={y_k}\] So,inotherwords,initialconditionsarevaluesofthesolutionand/oritsderivative(s)atspecificpoints.Aswewillseeeventually,solutionsto“niceenough”differentialequationsareuniqueandhenceonlyonesolutionwillmeetthegiveninitialconditions. Thenumberofinitialconditionsthatarerequiredforagivendifferentialequationwilldependupontheorderofthedifferentialequationaswewillsee. Example2\(y\left(x\right)={x^{-\frac{3}{2}}}\)isasolutionto\(4{x^2}y''+12xy'+3y=0\),\(y\left(4\right)=\frac{1}{8}\),and\(y'\left(4\right)=-\frac{3}{{64}}\). ShowSolution Aswesawinpreviousexamplethefunctionisasolutionandwecanthennotethat \[\begin{align*}y\left(4\right)&={4^{-\frac{3}{2}}}=\frac{1}{{{{\left({\sqrt4}\right)}^3}}}=\frac{1}{8}\\y'\left(4\right)&=-\frac{3}{2}{4^{-\frac{5}{2}}}=-\frac{3}{2}\frac{1}{{{{\left({\sqrt4}\right)}^5}}}=-\frac{3}{{64}}\end{align*}\] andsothissolutionalsomeetstheinitialconditionsof\(y\left(4\right)=\frac{1}{8}\)and\(y'\left(4\right)=-\frac{3}{{64}}\).Infact,\(y\left(x\right)={x^{-\frac{3}{2}}}\)istheonlysolutiontothisdifferentialequationthatsatisfiesthesetwoinitialconditions. InitialValueProblem AnInitialValueProblem(orIVP)isadifferentialequationalongwithanappropriatenumberofinitialconditions. Example3ThefollowingisanIVP. \[4{x^2}y''+12xy'+3y=0\hspace{0.25in}y\left(4\right)=\frac{1}{8},\,\,\,\,y'\left(4\right)=-\frac{3}{{64}}\] Example4Here’sanotherIVP. \[2t\,y'+4y=3\hspace{0.25in}\,\,\,\,\,\,y\left(1\right)=-4\] Aswenotedearlierthenumberofinitialconditionsrequiredwilldependontheorderofthedifferentialequation. IntervalofValidity TheintervalofvalidityforanIVPwithinitialcondition(s) \[y\left({{t_0}}\right)={y_0}\hspace{0.25in}{\mbox{and/or}}{y^{\left(k\right)}}\left({{t_0}}\right)={y_k}\] isthelargestpossibleintervalonwhichthesolutionisvalidandcontains\({t_0}\).Theseareeasytodefine,butcanbedifficulttofind,sowe’regoingtoputoffsayinganythingmoreabouttheseuntilwegetintoactuallysolvingdifferentialequationsandneedtheintervalofvalidity. GeneralSolution Thegeneralsolutiontoadifferentialequationisthemostgeneralformthatthesolutioncantakeanddoesn’ttakeanyinitialconditionsintoaccount. Example5\(\displaystyley\left(t\right)=\frac{3}{4}+\frac{c}{{{t^2}}}\)isthegeneralsolutionto \[2t\,y'+4y=3\] We’llleaveittoyoutocheckthatthisfunctionisinfactasolutiontothegivendifferentialequation.Infact,allsolutionstothisdifferentialequationwillbeinthisform.Thisisoneofthefirstdifferentialequationsthatyouwilllearnhowtosolveandyouwillbeabletoverifythisshortlyforyourself. ActualSolution Theactualsolutiontoadifferentialequationisthespecificsolutionthatnotonlysatisfiesthedifferentialequation,butalsosatisfiesthegiveninitialcondition(s). Example6WhatistheactualsolutiontothefollowingIVP? \[2t\,y'+4y=3\hspace{0.25in}\,\,\,\,\,\,y\left(1\right)=-4\] ShowSolution Thisisactuallyeasiertodothanitmightatfirstappear.Fromthepreviousexamplewealreadyknow(wellthatisprovidedyoubelieveoursolutiontothisexample…)thatallsolutionstothedifferentialequationareoftheform. \[y\left(t\right)=\frac{3}{4}+\frac{c}{{{t^2}}}\] Allthatweneedtodoisdeterminethevalueof\(c\)thatwillgiveusthesolutionthatwe’reafter.Tofindthisallweneeddoisuseourinitialconditionasfollows. \[-4=y\left(1\right)=\frac{3}{4}+\frac{c}{{{1^2}}}\hspace{0.25in}\Rightarrow\hspace{0.25in}c=-4-\frac{3}{4}=-\frac{{19}}{4}\] So,theactualsolutiontotheIVPis. \[y\left(t\right)=\frac{3}{4}-\frac{{19}}{{4{t^2}}}\] Fromthislastexamplewecanseethatoncewehavethegeneralsolutiontoadifferentialequationfindingtheactualsolutionisnothingmorethanapplyingtheinitialcondition(s)andsolvingfortheconstant(s)thatareinthegeneralsolution. Implicit/ExplicitSolution Inthiscaseit’seasiertodefineanexplicitsolution,thentellyouwhatanimplicitsolutionisn’t,andthengiveyouanexampletoshowyouthedifference.So,that’swhatwe’lldo. Anexplicitsolutionisanysolutionthatisgivenintheform\(y=y\left(t\right)\).Inotherwords,theonlyplacethat\(y\)actuallyshowsupisonceontheleftsideandonlyraisedtothefirstpower.Animplicitsolutionisanysolutionthatisn’tinexplicitform.Notethatitispossibletohaveeithergeneralimplicit/explicitsolutionsandactualimplicit/explicitsolutions. Example7\({y^2}={t^2}-3\)istheactualimplicitsolutionto\(y'=\frac{t}{y},\,\,\,\,\,y\left(2\right)=-1\) Atthispointwewillaskthatyoutrustusthatthisisinfactasolutiontothedifferentialequation.Youwilllearnhowtogetthissolutioninalatersection.Thepointofthisexampleisthatsincethereisa\({y^2}\)ontheleftsideinsteadofasingle\(y\left(t\right)\)thisisnotanexplicitsolution! Example8Findanactualexplicitsolutionto\(y'=\frac{t}{y},\,\,\,\,\,y\left(2\right)=-1\). ShowSolution WealreadyknowfromthepreviousexamplethatanimplicitsolutiontothisIVPis\({y^2}={t^2}-3\).Tofindtheexplicitsolutionallweneedtodoissolvefor\(y\left(t\right)\). \[y\left(t\right)=\pm\sqrt{{t^2}-3}\] Now,we’vegotaproblemhere.Therearetwofunctionshereandweonlywantoneandinfactonlyonewillbecorrect!Wecandeterminethecorrectfunctionbyreapplyingtheinitialcondition.Onlyoneofthemwillsatisfytheinitialcondition. Inthiscasewecanseethatthe“-“solutionwillbethecorrectone.Theactualexplicitsolutionisthen \[y\left(t\right)=-\sqrt{{t^2}-3}\] Inthiscasewewereabletofindanexplicitsolutiontothedifferentialequation.Itshouldbenotedhoweverthatitwillnotalwaysbepossibletofindanexplicitsolution. Also,notethatinthiscasewewereonlyabletogettheexplicitactualsolutionbecausewehadtheinitialconditiontohelpusdeterminewhichofthetwofunctionswouldbethecorrectsolution. We’venowgottenmostofthebasicdefinitionsoutofthewayandsowecanmoveontoothertopics.