Ordinary differential equation - Wikipedia

General definition Ordinarydifferentialequation FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Differentialequationcontainingderivativeswithrespecttoonlyonevariable DifferentialequationsNavier–Stokesdifferentialequationsusedtosimulateairflowaroundanobstruction Scope Fields NaturalsciencesEngineering Astronomy Physics Chemistry Biology Geology Appliedmathematics Continuummechanics Chaostheory Dynamicalsystems Socialsciences Economics Populationdynamics Classification Types Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear Byvariabletype Dependentandindependentvariables Autonomous Coupled /Decoupled Exact Homogeneous /Nonhomogeneous Features Order Operator Notation Relationtoprocesses Difference(discreteanalogue) Stochastic Stochasticpartial Delay Solution Existenceanduniqueness Picard–Lindelöftheorem Peanoexistencetheorem Carathéodory'sexistencetheorem Cauchy–Kowalevskitheorem Generaltopics Wronskian Phaseportrait Phasespace Lyapunov /Asymptotic /Exponentialstability Rateofconvergence Series /Integralsolutions Numericalintegration Diracdeltafunction Solutionmethods Inspection Methodofcharacteristics Euler Exponentialresponseformula Finitedifference (Crank–Nicolson) Finiteelement Infiniteelement Finitevolume Galerkin Petrov–Galerkin Integratingfactor Integraltransforms Perturbationtheory Runge–Kutta Separationofvariables Undeterminedcoefficients Variationofparameters People List IsaacNewton JosephFourier GottfriedLeibniz LeonhardEuler ÉmilePicard JózefMariaHoene-Wroński ErnstLindelöf RudolfLipschitz Augustin-LouisCauchy JohnCrank PhyllisNicolson CarlDavidTolméRunge MartinKutta vte Inmathematics,anordinarydifferentialequation(ODE)isadifferentialequationcontainingoneormorefunctionsofoneindependentvariableandthederivativesofthosefunctions.[1]Thetermordinaryisusedincontrastwiththetermpartialdifferentialequationwhichmaybewithrespecttomorethanoneindependentvariable.[2] Contents 1Differentialequations 2Background 3Definitions 3.1Generaldefinition 3.2SystemofODEs 3.3Solutions 3.4SolutionsofFiniteDuration 4Theories 4.1Singularsolutions 4.2Reductiontoquadratures 4.3Fuchsiantheory 4.4Lie'stheory 4.5Sturm–Liouvilletheory 5Existenceanduniquenessofsolutions 5.1Localexistenceanduniquenesstheoremsimplified 5.2Globaluniquenessandmaximumdomainofsolution 6Reductionoforder 6.1Reductiontoafirst-ordersystem 7Summaryofexactsolutions 7.1Separableequations 7.2Generalfirst-orderequations 7.3Generalsecond-orderequations 7.4Lineartothenthorderequations 8Theguessingmethod 9SoftwareforODEsolving 10Seealso 11Notes 12References 13Bibliography 14Externallinks Differentialequations[edit] Alineardifferentialequationisadifferentialequationthatisdefinedbyalinearpolynomialintheunknownfunctionanditsderivatives,thatisanequationoftheform a 0 ( x ) y + a 1 ( x ) y ′ + a 2 ( x ) y ″ + ⋯ + a n ( x ) y ( n ) + b ( x ) = 0 , {\displaystylea_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''+\cdots+a_{n}(x)y^{(n)}+b(x)=0,} where a 0 ( x ) {\displaystylea_{0}(x)} ,..., a n ( x ) {\displaystylea_{n}(x)} and b ( x ) {\displaystyleb(x)} arearbitrarydifferentiablefunctionsthatdonotneedtobelinear,and y ′ , … , y ( n ) {\displaystyley',\ldots,y^{(n)}} arethesuccessivederivativesoftheunknownfunctionyofthevariablex. Amongordinarydifferentialequations,lineardifferentialequationsplayaprominentroleforseveralreasons.Mostelementaryandspecialfunctionsthatareencounteredinphysicsandappliedmathematicsaresolutionsoflineardifferentialequations(seeHolonomicfunction).Whenphysicalphenomenaaremodeledwithnon-linearequations,theyaregenerallyapproximatedbylineardifferentialequationsforaneasiersolution.Thefewnon-linearODEsthatcanbesolvedexplicitlyaregenerallysolvedbytransformingtheequationintoanequivalentlinearODE(see,forexampleRiccatiequation). SomeODEscanbesolvedexplicitlyintermsofknownfunctionsandintegrals.Whenthatisnotpossible,theequationforcomputingtheTaylorseriesofthesolutionsmaybeuseful.Forappliedproblems,numericalmethodsforordinarydifferentialequationscansupplyanapproximationofthesolution. Background[edit] ThetrajectoryofaprojectilelaunchedfromacannonfollowsacurvedeterminedbyanordinarydifferentialequationthatisderivedfromNewton'ssecondlaw. Ordinarydifferentialequations(ODEs)ariseinmanycontextsofmathematicsandsocialandnaturalsciences.Mathematicaldescriptionsofchangeusedifferentialsandderivatives.Variousdifferentials,derivatives,andfunctionsbecomerelatedviaequations,suchthatadifferentialequationisaresultthatdescribesdynamicallychangingphenomena,evolution,andvariation.Often,quantitiesaredefinedastherateofchangeofotherquantities(forexample,derivativesofdisplacementwithrespecttotime),orgradientsofquantities,whichishowtheyenterdifferentialequations. Specificmathematicalfieldsincludegeometryandanalyticalmechanics.Scientificfieldsincludemuchofphysicsandastronomy(celestialmechanics),meteorology(weathermodeling),chemistry(reactionrates),[3]biology(infectiousdiseases,geneticvariation),ecologyandpopulationmodeling(populationcompetition),economics(stocktrends,interestratesandthemarketequilibriumpricechanges). Manymathematicianshavestudieddifferentialequationsandcontributedtothefield,includingNewton,Leibniz,theBernoullifamily,Riccati,Clairaut,d'Alembert,andEuler. AsimpleexampleisNewton'ssecondlawofmotion—therelationshipbetweenthedisplacementxandthetimetofanobjectundertheforceF,isgivenbythedifferentialequation m d 2 x ( t ) d t 2 = F ( x ( t ) ) {\displaystylem{\frac{\mathrm{d}^{2}x(t)}{\mathrm{d}t^{2}}}=F(x(t))\,} whichconstrainsthemotionofaparticleofconstantmassm.Ingeneral,Fisafunctionofthepositionx(t)oftheparticleattimet.Theunknownfunctionx(t)appearsonbothsidesofthedifferentialequation,andisindicatedinthenotationF(x(t)).[4][5][6][7] Definitions[edit] Inwhatfollows,letybeadependentvariableandxanindependentvariable,andy=f(x)isanunknownfunctionofx.Thenotationfordifferentiationvariesdependingupontheauthoranduponwhichnotationismostusefulforthetaskathand.Inthiscontext,theLeibniz'snotation(dy/dx,d2y/dx2,…,dny/dxn)ismoreusefulfordifferentiationandintegration,whereasLagrange'snotation(y′,y′′,…,y(n))ismoreusefulforrepresentingderivativesofanyordercompactly,andNewton'snotation ( y ˙ , y ¨ , y . . . ) {\displaystyle({\dot{y}},{\ddot{y}},{\overset{...}{y}})} isoftenusedinphysicsforrepresentingderivativesofloworderwithrespecttotime. Generaldefinition[edit] GivenF,afunctionofx,y,andderivativesofy.Thenanequationoftheform F ( x , y , y ′ , … , y ( n − 1 ) ) = y ( n ) {\displaystyleF\left(x,y,y',\ldots,y^{(n-1)}\right)=y^{(n)}} iscalledanexplicitordinarydifferentialequationofordern.[8][9] Moregenerally,animplicitordinarydifferentialequationoforderntakestheform:[10] F ( x , y , y ′ , y ″ ,   … ,   y ( n ) ) = 0 {\displaystyleF\left(x,y,y',y'',\\ldots,\y^{(n)}\right)=0} Therearefurtherclassifications: AutonomousAdifferentialequationnotdependingonxiscalledautonomous. Linear AdifferentialequationissaidtobelinearifFcanbewrittenasalinearcombinationofthederivativesofy: y ( n ) = ∑ i = 0 n − 1 a i ( x ) y ( i ) + r ( x ) {\displaystyley^{(n)}=\sum_{i=0}^{n-1}a_{i}(x)y^{(i)}+r(x)} wherea i (x)andr (x)arecontinuousfunctionsofx.[8][11][12] Thefunctionr(x)iscalledthesourceterm,leadingtotwofurtherimportantclassifications:[11][13] HomogeneousIfr(x)=0,andconsequentlyone"automatic"solutionisthetrivialsolution,y=0.Thesolutionofalinearhomogeneousequationisacomplementaryfunction,denotedherebyyc. Nonhomogeneous(orinhomogeneous)Ifr(x)≠0.Theadditionalsolutiontothecomplementaryfunctionistheparticularintegral,denotedherebyyp. Non-linearAdifferentialequationthatcannotbewrittenintheformofalinearcombination. SystemofODEs[edit] Mainarticle:Systemofdifferentialequations Anumberofcoupleddifferentialequationsformasystemofequations.Ifyisavectorwhoseelementsarefunctions;y(x)=[y1(x),y2(x),...,ym(x)],andFisavector-valuedfunctionofyanditsderivatives,then y ( n ) = F ( x , y , y ′ , y ″ , … , y ( n − 1 ) ) {\displaystyle\mathbf{y}^{(n)}=\mathbf{F}\left(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots,\mathbf{y}^{(n-1)}\right)} isanexplicitsystemofordinarydifferentialequationsofordernanddimensionm.Incolumnvectorform: ( y 1 ( n ) y 2 ( n ) ⋮ y m ( n ) ) = ( f 1 ( x , y , y ′ , y ″ , … , y ( n − 1 ) ) f 2 ( x , y , y ′ , y ″ , … , y ( n − 1 ) ) ⋮ f m ( x , y , y ′ , y ″ , … , y ( n − 1 ) ) ) {\displaystyle{\begin{pmatrix}y_{1}^{(n)}\\y_{2}^{(n)}\\\vdots\\y_{m}^{(n)}\end{pmatrix}}={\begin{pmatrix}f_{1}\left(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots,\mathbf{y}^{(n-1)}\right)\\f_{2}\left(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots,\mathbf{y}^{(n-1)}\right)\\\vdots\\f_{m}\left(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots,\mathbf{y}^{(n-1)}\right)\end{pmatrix}}} Thesearenotnecessarilylinear.Theimplicitanalogueis: F ( x , y , y ′ , y ″ , … , y ( n ) ) = 0 {\displaystyle\mathbf{F}\left(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots,\mathbf{y}^{(n)}\right)={\boldsymbol{0}}} where0=(0,0,...,0)isthezerovector.Inmatrixform ( f 1 ( x , y , y ′ , y ″ , … , y ( n ) ) f 2 ( x , y , y ′ , y ″ , … , y ( n ) ) ⋮ f m ( x , y , y ′ , y ″ , … , y ( n ) ) ) = ( 0 0 ⋮ 0 ) {\displaystyle{\begin{pmatrix}f_{1}(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots,\mathbf{y}^{(n)})\\f_{2}(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots,\mathbf{y}^{(n)})\\\vdots\\f_{m}(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots,\mathbf{y}^{(n)})\end{pmatrix}}={\begin{pmatrix}0\\0\\\vdots\\0\end{pmatrix}}} Forasystemoftheform F ( x , y , y ′ ) = 0 {\displaystyle\mathbf{F}\left(x,\mathbf{y},\mathbf{y}'\right)={\boldsymbol{0}}} ,somesourcesalsorequirethattheJacobianmatrix ∂ F ( x , u , v ) ∂ v {\displaystyle{\frac{\partial\mathbf{F}(x,\mathbf{u},\mathbf{v})}{\partial\mathbf{v}}}} benon-singularinordertocallthisanimplicitODE[system];animplicitODEsystemsatisfyingthisJacobiannon-singularityconditioncanbetransformedintoanexplicitODEsystem.Inthesamesources,implicitODEsystemswithasingularJacobianaretermeddifferentialalgebraicequations(DAEs).Thisdistinctionisnotmerelyoneofterminology;DAEshavefundamentallydifferentcharacteristicsandaregenerallymoreinvolvedtosolvethan(nonsingular)ODEsystems.[14][15][16]Presumablyforadditionalderivatives,theHessianmatrixandsofortharealsoassumednon-singularaccordingtothisscheme,[citationneeded]althoughnotethatanyODEofordergreaterthanonecanbe(andusuallyis)rewrittenassystemofODEsoffirstorder,[17]whichmakestheJacobiansingularitycriterionsufficientforthistaxonomytobecomprehensiveatallorders. ThebehaviorofasystemofODEscanbevisualizedthroughtheuseofaphaseportrait. Solutions[edit] Givenadifferentialequation F ( x , y , y ′ , … , y ( n ) ) = 0 {\displaystyleF\left(x,y,y',\ldots,y^{(n)}\right)=0} afunctionu:I⊂R→R,whereIisaninterval,iscalledasolutionorintegralcurveforF,ifuisn-timesdifferentiableonI,and F ( x , u , u ′ ,   … ,   u ( n ) ) = 0 x ∈ I . {\displaystyleF(x,u,u',\\ldots,\u^{(n)})=0\quadx\inI.} Giventwosolutionsu:J⊂R→Randv:I⊂R→R,uiscalledanextensionofvifI⊂Jand u ( x ) = v ( x ) x ∈ I . {\displaystyleu(x)=v(x)\quadx\inI.\,} Asolutionthathasnoextensioniscalledamaximalsolution.AsolutiondefinedonallofRiscalledaglobalsolution. Ageneralsolutionofannth-orderequationisasolutioncontainingnarbitraryindependentconstantsofintegration.Aparticularsolutionisderivedfromthegeneralsolutionbysettingtheconstantstoparticularvalues,oftenchosentofulfillset'initialconditionsorboundaryconditions'.[18]Asingularsolutionisasolutionthatcannotbeobtainedbyassigningdefinitevaluestothearbitraryconstantsinthegeneralsolution.[19] InthecontextoflinearODE,theterminologyparticularsolutioncanalsorefertoanysolutionoftheODE(notnecessarilysatisfyingtheinitialconditions),whichisthenaddedtothehomogeneoussolution(ageneralsolutionofthehomogeneousODE),whichthenformsageneralsolutionoftheoriginalODE.Thisistheterminologyusedintheguessingmethodsectioninthisarticle,andisfrequentlyusedwhendiscussingthemethodofundeterminedcoefficientsandvariationofparameters. SolutionsofFiniteDuration[edit] Fornon-linearautonomousODEsitispossibleundersomeconditionstodevelopsolutionsoffiniteduration,[20]meaningherethatfromitsowndynamics,thesystemwillreachthevaluezeroatanendingtimeandstaysthereinzeroforeverafter.Thesefinite-durationsolutionscan'tbeanalyticalfunctionsonthewholerealline,andbecausetheywillbeingnon-Lipschitzfunctionsattheirendingtime,theydon´tstanduniquenessofsolutionsofLipschitzdifferentialequations. Asexample,theequation: y ′ = − sgn ( y ) | y | , y ( 0 ) = 1 {\displaystyley'=-{\text{sgn}}(y){\sqrt{|y|}},\,\,y(0)=1} Admitsthefinitedurationsolution: y ( x ) = 1 4 ( 1 − x 2 + | 1 − x 2 | ) 2 {\displaystyley(x)={\frac{1}{4}}\left(1-{\frac{x}{2}}+\left|1-{\frac{x}{2}}\right|\right)^{2}} Theories[edit] Singularsolutions[edit] ThetheoryofsingularsolutionsofordinaryandpartialdifferentialequationswasasubjectofresearchfromthetimeofLeibniz,butonlysincethemiddleofthenineteenthcenturyhasitreceivedspecialattention.Avaluablebutlittle-knownworkonthesubjectisthatofHoutain(1854).Darboux(from1873)wasaleaderinthetheory,andinthegeometricinterpretationofthesesolutionsheopenedafieldworkedbyvariouswriters,notablyCasoratiandCayley.Tothelatterisdue(1872)thetheoryofsingularsolutionsofdifferentialequationsofthefirstorderasacceptedcirca1900. Reductiontoquadratures[edit] Theprimitiveattemptindealingwithdifferentialequationshadinviewareductiontoquadratures.Asithadbeenthehopeofeighteenth-centuryalgebraiststofindamethodforsolvingthegeneralequationofthenthdegree,soitwasthehopeofanalyststofindageneralmethodforintegratinganydifferentialequation.Gauss(1799)showed,however,thatcomplexdifferentialequationsrequirecomplexnumbers.Hence,analystsbegantosubstitutethestudyoffunctions,thusopeninganewandfertilefield.Cauchywasthefirsttoappreciatetheimportanceofthisview.Thereafter,therealquestionwasnolongerwhetherasolutionispossiblebymeansofknownfunctionsortheirintegrals,butwhetheragivendifferentialequationsufficesforthedefinitionofafunctionoftheindependentvariableorvariables,and,ifso,whatarethecharacteristicproperties. Fuchsiantheory[edit] Mainarticle:Frobeniusmethod TwomemoirsbyFuchs[21]inspiredanovelapproach,subsequentlyelaboratedbyThoméandFrobenius.Colletwasaprominentcontributorbeginningin1869.Hismethodforintegratinganon-linearsystemwascommunicatedtoBertrandin1868.Clebsch(1873)attackedthetheoryalonglinesparalleltothoseinhistheoryofAbelianintegrals.Asthelattercanbeclassifiedaccordingtothepropertiesofthefundamentalcurvethatremainsunchangedunderarationaltransformation,Clebschproposedtoclassifythetranscendentfunctionsdefinedbydifferentialequationsaccordingtotheinvariantpropertiesofthecorrespondingsurfacesf=0underrationalone-to-onetransformations. Lie'stheory[edit] From1870,SophusLie'sworkputthetheoryofdifferentialequationsonabetterfoundation.Heshowedthattheintegrationtheoriesoftheoldermathematicianscan,usingLiegroups,bereferredtoacommonsource,andthatordinarydifferentialequationsthatadmitthesameinfinitesimaltransformationspresentcomparableintegrationdifficulties.Healsoemphasizedthesubjectoftransformationsofcontact. Lie'sgrouptheoryofdifferentialequationshasbeencertified,namely:(1)thatitunifiesthemanyadhocmethodsknownforsolvingdifferentialequations,and(2)thatitprovidespowerfulnewwaystofindsolutions.Thetheoryhasapplicationstobothordinaryandpartialdifferentialequations.[22] Ageneralsolutionapproachusesthesymmetrypropertyofdifferentialequations,thecontinuousinfinitesimaltransformationsofsolutionstosolutions(Lietheory).Continuousgrouptheory,Liealgebras,anddifferentialgeometryareusedtounderstandthestructureoflinearandnon-linear(partial)differentialequationsforgeneratingintegrableequations,tofinditsLaxpairs,recursionoperators,Bäcklundtransform,andfinallyfindingexactanalyticsolutionstoDE. Symmetrymethodshavebeenappliedtodifferentialequationsthatariseinmathematics,physics,engineering,andotherdisciplines. Sturm–Liouvilletheory[edit] Mainarticle:Sturm–Liouvilletheory Sturm–Liouvilletheoryisatheoryofaspecialtypeofsecondorderlinearordinarydifferentialequation.Theirsolutionsarebasedoneigenvaluesandcorrespondingeigenfunctionsoflinearoperatorsdefinedviasecond-orderhomogeneouslinearequations.TheproblemsareidentifiedasSturm-LiouvilleProblems(SLP)andarenamedafterJ.C.F.SturmandJ.Liouville,whostudiedtheminthemid-1800s.SLPshaveaninfinitenumberofeigenvalues,andthecorrespondingeigenfunctionsformacomplete,orthogonalset,whichmakesorthogonalexpansionspossible.Thisisakeyideainappliedmathematics,physics,andengineering.[23]SLPsarealsousefulintheanalysisofcertainpartialdifferentialequations. Existenceanduniquenessofsolutions[edit] ThereareseveraltheoremsthatestablishexistenceanduniquenessofsolutionstoinitialvalueproblemsinvolvingODEsbothlocallyandglobally.Thetwomaintheoremsare Theorem Assumption Conclusion Peanoexistencetheorem Fcontinuous localexistenceonly Picard–Lindelöftheorem FLipschitzcontinuous localexistenceanduniqueness Intheirbasicformbothofthesetheoremsonlyguaranteelocalresults,thoughthelattercanbeextendedtogiveaglobalresult,forexample,iftheconditionsofGrönwall'sinequalityaremet. Also,uniquenesstheoremsliketheLipschitzoneabovedonotapplytoDAEsystems,whichmayhavemultiplesolutionsstemmingfromtheir(non-linear)algebraicpartalone.[24] Localexistenceanduniquenesstheoremsimplified[edit] Thetheoremcanbestatedsimplyasfollows.[25]Fortheequationandinitialvalueproblem: y ′ = F ( x , y ) , y 0 = y ( x 0 ) {\displaystyley'=F(x,y)\,,\quady_{0}=y(x_{0})} ifFand∂F/∂yarecontinuousinaclosedrectangle R = [ x 0 − a , x 0 + a ] × [ y 0 − b , y 0 + b ] {\displaystyleR=[x_{0}-a,x_{0}+a]\times[y_{0}-b,y_{0}+b]} inthex-yplane,whereaandbarereal(symbolically:a,b∈R)and×denotestheCartesianproduct,squarebracketsdenoteclosedintervals,thenthereisaninterval I = [ x 0 − h , x 0 + h ] ⊂ [ x 0 − a , x 0 + a ] {\displaystyleI=[x_{0}-h,x_{0}+h]\subset[x_{0}-a,x_{0}+a]} forsomeh∈Rwherethesolutiontotheaboveequationandinitialvalueproblemcanbefound.Thatis,thereisasolutionanditisunique.SincethereisnorestrictiononFtobelinear,thisappliestonon-linearequationsthattaketheformF(x,y),anditcanalsobeappliedtosystemsofequations. Globaluniquenessandmaximumdomainofsolution[edit] WhenthehypothesesofthePicard–Lindelöftheoremaresatisfied,thenlocalexistenceanduniquenesscanbeextendedtoaglobalresult.Moreprecisely:[26] Foreachinitialcondition(x0,y0)thereexistsauniquemaximum(possiblyinfinite)openinterval I max = ( x − , x + ) , x ± ∈ R ∪ { ± ∞ } , x 0 ∈ I max {\displaystyleI_{\max}=(x_{-},x_{+}),x_{\pm}\in\mathbb{R}\cup\{\pm\infty\},x_{0}\inI_{\max}} suchthatanysolutionthatsatisfiesthisinitialconditionisarestrictionofthesolutionthatsatisfiesthisinitialconditionwithdomain I max {\displaystyleI_{\max}} . Inthecasethat x ± ≠ ± ∞ {\displaystylex_{\pm}\neq\pm\infty} ,thereareexactlytwopossibilities explosioninfinitetime: lim sup x → x ± ‖ y ( x ) ‖ → ∞ {\displaystyle\limsup_{x\tox_{\pm}}\|y(x)\|\to\infty} leavesdomainofdefinition: lim x → x ± y ( x )   ∈ ∂ Ω ¯ {\displaystyle\lim_{x\tox_{\pm}}y(x)\\in\partial{\bar{\Omega}}} whereΩistheopensetinwhichFisdefined,and ∂ Ω ¯ {\displaystyle\partial{\bar{\Omega}}} isitsboundary. Notethatthemaximumdomainofthesolution isalwaysaninterval(tohaveuniqueness) maybesmallerthan R {\displaystyle\mathbb{R}} maydependonthespecificchoiceof(x0,y0). Example. y ′ = y 2 {\displaystyley'=y^{2}} ThismeansthatF(x,y)=y2,whichisC1andthereforelocallyLipschitzcontinuous,satisfyingthePicard–Lindelöftheorem. Eveninsuchasimplesetting,themaximumdomainofsolutioncannotbeall R {\displaystyle\mathbb{R}} sincethesolutionis y ( x ) = y 0 ( x 0 − x ) y 0 + 1 {\displaystyley(x)={\frac{y_{0}}{(x_{0}-x)y_{0}+1}}} whichhasmaximumdomain: { R y 0 = 0 ( − ∞ , x 0 + 1 y 0 ) y 0 > 0 ( x 0 + 1 y 0 , + ∞ ) y 0 < 0 {\displaystyle{\begin{cases}\mathbb{R}&y_{0}=0\\[4pt]\left(-\infty,x_{0}+{\frac{1}{y_{0}}}\right)&y_{0}>0\\[4pt]\left(x_{0}+{\frac{1}{y_{0}}},+\infty\right)&y_{0}<0\end{cases}}} Thisshowsclearlythatthemaximumintervalmaydependontheinitialconditions.Thedomainofycouldbetakenasbeing R ∖ ( x 0 + 1 / y 0 ) , {\displaystyle\mathbb{R}\setminus(x_{0}+1/y_{0}),} butthiswouldleadtoadomainthatisnotaninterval,sothatthesideoppositetotheinitialconditionwouldbedisconnectedfromtheinitialcondition,andthereforenotuniquelydeterminedbyit. Themaximumdomainisnot R {\displaystyle\mathbb{R}} because lim x → x ± ‖ y ( x ) ‖ → ∞ , {\displaystyle\lim_{x\tox_{\pm}}\|y(x)\|\to\infty,} whichisoneofthetwopossiblecasesaccordingtotheabovetheorem. Reductionoforder[edit] Differentialequationscanusuallybesolvedmoreeasilyiftheorderoftheequationcanbereduced. Reductiontoafirst-ordersystem[edit] Anyexplicitdifferentialequationofordern, F ( x , y , y ′ , y ″ ,   … ,   y ( n − 1 ) ) = y ( n ) {\displaystyleF\left(x,y,y',y'',\\ldots,\y^{(n-1)}\right)=y^{(n)}} canbewrittenasasystemofnfirst-orderdifferentialequationsbydefininganewfamilyofunknownfunctions y i = y ( i − 1 ) . {\displaystyley_{i}=y^{(i-1)}.\!} fori=1,2,...,n.Then-dimensionalsystemoffirst-ordercoupleddifferentialequationsisthen y 1 ′ = y 2 y 2 ′ = y 3 ⋮ y n − 1 ′ = y n y n ′ = F ( x , y 1 , … , y n ) . {\displaystyle{\begin{array}{rcl}y_{1}'&=&y_{2}\\y_{2}'&=&y_{3}\\&\vdots&\\y_{n-1}'&=&y_{n}\\y_{n}'&=&F(x,y_{1},\ldots,y_{n}).\end{array}}} morecompactlyinvectornotation: y ′ = F ( x , y ) {\displaystyle\mathbf{y}'=\mathbf{F}(x,\mathbf{y})} where y = ( y 1 , … , y n ) , F ( x , y 1 , … , y n ) = ( y 2 , … , y n , F ( x , y 1 , … , y n ) ) . {\displaystyle\mathbf{y}=(y_{1},\ldots,y_{n}),\quad\mathbf{F}(x,y_{1},\ldots,y_{n})=(y_{2},\ldots,y_{n},F(x,y_{1},\ldots,y_{n})).} Summaryofexactsolutions[edit] Somedifferentialequationshavesolutionsthatcanbewritteninanexactandclosedform.Severalimportantclassesaregivenhere. Inthetablebelow,P(x),Q(x),P(y),Q(y),andM(x,y),N(x,y)areanyintegrablefunctionsofx,y,andbandcarerealgivenconstants,andC1,C2,...arearbitraryconstants(complexingeneral).Thedifferentialequationsareintheirequivalentandalternativeformsthatleadtothesolutionthroughintegration. Intheintegralsolutions,λandεaredummyvariablesofintegration(thecontinuumanaloguesofindicesinsummation),andthenotation∫xF(λ)dλjustmeanstointegrateF(λ)withrespecttoλ,thenaftertheintegrationsubstituteλ=x,withoutaddingconstants(explicitlystated). Separableequations[edit] Differentialequation Solutionmethod Generalsolution First-order,separableinxandy(generalcase,seebelowforspecialcases)[27] P 1 ( x ) Q 1 ( y ) + P 2 ( x ) Q 2 ( y ) d y d x = 0 P 1 ( x ) Q 1 ( y ) d x + P 2 ( x ) Q 2 ( y ) d y = 0 {\displaystyle{\begin{aligned}P_{1}(x)Q_{1}(y)+P_{2}(x)Q_{2}(y)\,{\frac{dy}{dx}}&=0\\P_{1}(x)Q_{1}(y)\,dx+P_{2}(x)Q_{2}(y)\,dy&=0\end{aligned}}} Separationofvariables(dividebyP2Q1). ∫ x P 1 ( λ ) P 2 ( λ ) d λ + ∫ y Q 2 ( λ ) Q 1 ( λ ) d λ = C {\displaystyle\int^{x}{\frac{P_{1}(\lambda)}{P_{2}(\lambda)}}\,d\lambda+\int^{y}{\frac{Q_{2}(\lambda)}{Q_{1}(\lambda)}}\,d\lambda=C} First-order,separableinx[25] d y d x = F ( x ) d y = F ( x ) d x {\displaystyle{\begin{aligned}{\frac{dy}{dx}}&=F(x)\\dy&=F(x)\,dx\end{aligned}}} Directintegration. y = ∫ x F ( λ ) d λ + C {\displaystyley=\int^{x}F(\lambda)\,d\lambda+C} First-order,autonomous,separableiny[25] d y d x = F ( y ) d y = F ( y ) d x {\displaystyle{\begin{aligned}{\frac{dy}{dx}}&=F(y)\\dy&=F(y)\,dx\end{aligned}}} Separationofvariables(dividebyF). x = ∫ y d λ F ( λ ) + C {\displaystylex=\int^{y}{\frac{d\lambda}{F(\lambda)}}+C} First-order,separableinxandy[25] P ( y ) d y d x + Q ( x ) = 0 P ( y ) d y + Q ( x ) d x = 0 {\displaystyle{\begin{aligned}P(y){\frac{dy}{dx}}+Q(x)&=0\\P(y)\,dy+Q(x)\,dx&=0\end{aligned}}} Integratethroughout. ∫ y P ( λ ) d λ + ∫ x Q ( λ ) d λ = C {\displaystyle\int^{y}P(\lambda)\,d\lambda+\int^{x}Q(\lambda)\,d\lambda=C} Generalfirst-orderequations[edit] Differentialequation Solutionmethod Generalsolution First-order,homogeneous[25] d y d x = F ( y x ) {\displaystyle{\frac{dy}{dx}}=F\left({\frac{y}{x}}\right)} Sety=ux,thensolvebyseparationofvariablesinuandx. ln ⁡ ( C x ) = ∫ y / x d λ F ( λ ) − λ {\displaystyle\ln(Cx)=\int^{y/x}{\frac{d\lambda}{F(\lambda)-\lambda}}} First-order,separable[27] y M ( x y ) + x N ( x y ) d y d x = 0 y M ( x y ) d x + x N ( x y ) d y = 0 {\displaystyle{\begin{aligned}yM(xy)+xN(xy)\,{\frac{dy}{dx}}&=0\\yM(xy)\,dx+xN(xy)\,dy&=0\end{aligned}}} Separationofvariables(dividebyxy). ln ⁡ ( C x ) = ∫ x y N ( λ ) d λ λ [ N ( λ ) − M ( λ ) ] {\displaystyle\ln(Cx)=\int^{xy}{\frac{N(\lambda)\,d\lambda}{\lambda[N(\lambda)-M(\lambda)]}}} IfN=M,thesolutionisxy=C. Exactdifferential,first-order[25] M ( x , y ) d y d x + N ( x , y ) = 0 M ( x , y ) d y + N ( x , y ) d x = 0 {\displaystyle{\begin{aligned}M(x,y){\frac{dy}{dx}}+N(x,y)&=0\\M(x,y)\,dy+N(x,y)\,dx&=0\end{aligned}}} where ∂ M ∂ y = ∂ N ∂ x {\displaystyle{\frac{\partialM}{\partialy}}={\frac{\partialN}{\partialx}}} Integratethroughout. F ( x , y ) = ∫ x M ( λ , y ) d λ + ∫ y Y ( λ ) d λ = ∫ y N ( x , λ ) d λ + ∫ x X ( λ ) d λ = C {\displaystyle{\begin{aligned}F(x,y)&=\int^{x}M(\lambda,y)\,d\lambda+\int^{y}Y(\lambda)\,d\lambda\\&=\int^{y}N(x,\lambda)\,d\lambda+\int^{x}X(\lambda)\,d\lambda=C\end{aligned}}} where Y ( y ) = N ( x , y ) − ∂ ∂ y ∫ x M ( λ , y ) d λ {\displaystyleY(y)=N(x,y)-{\frac{\partial}{\partialy}}\int^{x}M(\lambda,y)\,d\lambda} and X ( x ) = M ( x , y ) − ∂ ∂ x ∫ y N ( x , λ ) d λ {\displaystyleX(x)=M(x,y)-{\frac{\partial}{\partialx}}\int^{y}N(x,\lambda)\,d\lambda} Inexactdifferential,first-order[25] M ( x , y ) d y d x + N ( x , y ) = 0 M ( x , y ) d y + N ( x , y ) d x = 0 {\displaystyle{\begin{aligned}M(x,y){\frac{dy}{dx}}+N(x,y)&=0\\M(x,y)\,dy+N(x,y)\,dx&=0\end{aligned}}} where ∂ M ∂ x ≠ ∂ N ∂ y {\displaystyle{\frac{\partialM}{\partialx}}\neq{\frac{\partialN}{\partialy}}} Integrationfactorμ(x,y)satisfying ∂ ( μ M ) ∂ y = ∂ ( μ N ) ∂ x {\displaystyle{\frac{\partial(\muM)}{\partialy}}={\frac{\partial(\muN)}{\partialx}}} Ifμ(x,y)canbefoundinasuitableway,then F ( x , y ) = ∫ x μ ( λ , y ) M ( λ , y ) d λ + ∫ y Y ( λ ) d λ = ∫ y μ ( x , λ ) N ( x , λ ) d λ + ∫ x X ( λ ) d λ = C {\displaystyle{\begin{aligned}F(x,y)=&\int^{x}\mu(\lambda,y)M(\lambda,y)\,d\lambda+\int^{y}Y(\lambda)\,d\lambda\\=&\int^{y}\mu(x,\lambda)N(x,\lambda)\,d\lambda+\int^{x}X(\lambda)\,d\lambda=C\end{aligned}}} where Y ( y ) = N ( x , y ) − ∂ ∂ y ∫ x μ ( λ , y ) M ( λ , y ) d λ {\displaystyleY(y)=N(x,y)-{\frac{\partial}{\partialy}}\int^{x}\mu(\lambda,y)M(\lambda,y)\,d\lambda} and X ( x ) = M ( x , y ) − ∂ ∂ x ∫ y μ ( x , λ ) N ( x , λ ) d λ {\displaystyleX(x)=M(x,y)-{\frac{\partial}{\partialx}}\int^{y}\mu(x,\lambda)N(x,\lambda)\,d\lambda} Generalsecond-orderequations[edit] Differentialequation Solutionmethod Generalsolution Second-order,autonomous[28] d 2 y d x 2 = F ( y ) {\displaystyle{\frac{d^{2}y}{dx^{2}}}=F(y)} Multiplybothsidesofequationby2dy/dx,substitute 2 d y d x d 2 y d x 2 = d d x ( d y d x ) 2 {\displaystyle2{\frac{dy}{dx}}{\frac{d^{2}y}{dx^{2}}}={\frac{d}{dx}}\left({\frac{dy}{dx}}\right)^{2}} ,thenintegratetwice. x = ± ∫ y d λ 2 ∫ λ F ( ε ) d ε + C 1 + C 2 {\displaystylex=\pm\int^{y}{\frac{d\lambda}{\sqrt{2\int^{\lambda}F(\varepsilon)\,d\varepsilon+C_{1}}}}+C_{2}} Lineartothenthorderequations[edit] Differentialequation Solutionmethod Generalsolution First-order,linear,inhomogeneous,functioncoefficients[25] d y d x + P ( x ) y = Q ( x ) {\displaystyle{\frac{dy}{dx}}+P(x)y=Q(x)} Integratingfactor: e ∫ x P ( λ ) d λ . {\displaystylee^{\int^{x}P(\lambda)\,d\lambda}.} Armourformula: y = e − ∫ x P ( λ ) d λ [ ∫ x e ∫ λ P ( ε ) d ε Q ( λ ) d λ + C ] {\displaystyley=e^{-\int^{x}P(\lambda)\,d\lambda}\left[\int^{x}e^{\int^{\lambda}P(\varepsilon)\,d\varepsilon}Q(\lambda)\,d\lambda+C\right]} Second-order,linear,inhomogeneous,functioncoefficients d 2 y d x 2 + 2 p ( x ) d y d x + ( p ( x ) 2 + p ′ ( x ) ) y = q ( x ) {\displaystyle{\frac{d^{2}y}{dx^{2}}}+2p(x){\frac{dy}{dx}}+\left(p(x)^{2}+p'(x)\right)y=q(x)} Integratingfactor: e ∫ x P ( λ ) d λ {\displaystylee^{\int^{x}P(\lambda)\,d\lambda}} y = e − ∫ x P ( λ ) d λ [ ∫ x ( ∫ ξ e ∫ λ P ( ε ) d ε Q ( λ ) d λ ) d ξ + C 1 x + C 2 ] {\displaystyley=e^{-\int^{x}P(\lambda)\,d\lambda}\left[\int^{x}\left(\int^{\xi}e^{\int^{\lambda}P(\varepsilon)\,d\varepsilon}Q(\lambda)\,d\lambda\right)d\xi+C_{1}x+C_{2}\right]} Second-order,linear,inhomogeneous,constantcoefficients[29] d 2 y d x 2 + b d y d x + c y = r ( x ) {\displaystyle{\frac{d^{2}y}{dx^{2}}}+b{\frac{dy}{dx}}+cy=r(x)} Complementaryfunctionyc:assumeyc=eαx,substituteandsolvepolynomialinα,tofindthelinearlyindependentfunctions e α j x {\displaystylee^{\alpha_{j}x}} . Particularintegralyp:ingeneralthemethodofvariationofparameters,thoughforverysimpler(x)inspectionmaywork.[25] y = y c + y p {\displaystyley=y_{c}+y_{p}} Ifb2>4c,then y c = C 1 e − x 2 ( b + b 2 − 4 c ) + C 2 e − x 2 ( b − b 2 − 4 c ) {\displaystyley_{c}=C_{1}e^{-{\frac{x}{2}}\,\left(b+{\sqrt{b^{2}-4c}}\right)}+C_{2}e^{-{\frac{x}{2}}\,\left(b-{\sqrt{b^{2}-4c}}\right)}} Ifb2=4c,then y c = ( C 1 x + C 2 ) e − b x 2 {\displaystyley_{c}=(C_{1}x+C_{2})e^{-{\frac{bx}{2}}}} Ifb2<4c,then y c = e − b x 2 [ C 1 sin ⁡ ( x 4 c − b 2 2 ) + C 2 cos ⁡ ( x 4 c − b 2 2 ) ] {\displaystyley_{c}=e^{-{\frac{bx}{2}}}\left[C_{1}\sin\left(x\,{\frac{\sqrt{4c-b^{2}}}{2}}\right)+C_{2}\cos\left(x\,{\frac{\sqrt{4c-b^{2}}}{2}}\right)\right]} nth-order,linear,inhomogeneous,constantcoefficients[29] ∑ j = 0 n b j d j y d x j = r ( x ) {\displaystyle\sum_{j=0}^{n}b_{j}{\frac{d^{j}y}{dx^{j}}}=r(x)} Complementaryfunctionyc:assumeyc=eαx,substituteandsolvepolynomialinα,tofindthelinearlyindependentfunctions e α j x {\displaystylee^{\alpha_{j}x}} . Particularintegralyp:ingeneralthemethodofvariationofparameters,thoughforverysimpler(x)inspectionmaywork.[25] y = y c + y p {\displaystyley=y_{c}+y_{p}} Sinceαjarethesolutionsofthepolynomialofdegreen: ∏ j = 1 n ( α − α j ) = 0 {\textstyle\prod_{j=1}^{n}(\alpha-\alpha_{j})=0} ,then: forαjalldifferent, y c = ∑ j = 1 n C j e α j x {\displaystyley_{c}=\sum_{j=1}^{n}C_{j}e^{\alpha_{j}x}} foreachrootαjrepeatedkjtimes, y c = ∑ j = 1 n ( ∑ ℓ = 1 k j C j , ℓ x ℓ − 1 ) e α j x {\displaystyley_{c}=\sum_{j=1}^{n}\left(\sum_{\ell=1}^{k_{j}}C_{j,\ell}x^{\ell-1}\right)e^{\alpha_{j}x}} forsomeαjcomplex,thensettingα=χj+iγj,andusingEuler'sformula,allowssometermsinthepreviousresultstobewrittenintheform C j e α j x = C j e χ j x cos ⁡ ( γ j x + φ j ) {\displaystyleC_{j}e^{\alpha_{j}x}=C_{j}e^{\chi_{j}x}\cos(\gamma_{j}x+\varphi_{j})} whereϕjisanarbitraryconstant(phaseshift). Theguessingmethod[edit] Thissectiondoesnotciteanysources.Pleasehelpimprovethissectionbyaddingcitationstoreliablesources.Unsourcedmaterialmaybechallengedandremoved.(January2020)(Learnhowandwhentoremovethistemplatemessage) WhenallothermethodsforsolvinganODEfail,orinthecaseswherewehavesomeintuitionaboutwhatthesolutiontoaDEmightlooklike,itissometimespossibletosolveaDEsimplybyguessingthesolutionandvalidatingitiscorrect.Tousethismethod,wesimplyguessasolutiontothedifferentialequation,andthenplugthesolutionintothedifferentialequationtovalidateifitsatisfiestheequation.IfitdoesthenwehaveaparticularsolutiontotheDE,otherwisewestartoveragainandtryanotherguess.ForinstancewecouldguessthatthesolutiontoaDEhastheform: y = A e α t {\displaystyley=Ae^{\alphat}} sincethisisaverycommonsolutionthatphysicallybehavesinasinusoidalway. InthecaseofafirstorderODEthatisnon-homogeneousweneedtofirstfindaDEsolutiontothehomogeneousportionoftheDE,otherwiseknownasthecharacteristicequation,andthenfindasolutiontotheentirenon-homogeneousequationbyguessing.Finally,weaddbothofthesesolutionstogethertoobtainthetotalsolutiontotheODE,thatis: totalsolution = homogeneoussolution + particularsolution {\displaystyle{\text{totalsolution}}={\text{homogeneoussolution}}+{\text{particularsolution}}} SoftwareforODEsolving[edit] Maxima,anopen-sourcecomputeralgebrasystem. COPASI,afree(ArtisticLicense2.0)softwarepackagefortheintegrationandanalysisofODEs. MATLAB,atechnicalcomputingapplication(MATrixLABoratory) GNUOctave,ahigh-levellanguage,primarilyintendedfornumericalcomputations. Scilab,anopensourceapplicationfornumericalcomputation. Maple,aproprietaryapplicationforsymboliccalculations. Mathematica,aproprietaryapplicationprimarilyintendedforsymboliccalculations. SymPy,aPythonpackagethatcansolveODEssymbolically Julia(programminglanguage),ahigh-levellanguageprimarilyintendedfornumericalcomputations. SageMath,anopen-sourceapplicationthatusesaPython-likesyntaxwithawiderangeofcapabilitiesspanningseveralbranchesofmathematics. SciPy,aPythonpackagethatincludesanODEintegrationmodule. Chebfun,anopen-sourcepackage,writteninMATLAB,forcomputingwithfunctionsto15-digitaccuracy. GNUR,anopensourcecomputationalenvironmentprimarilyintendedforstatistics,whichincludespackagesforODEsolving. Seealso[edit] Boundaryvalueproblem Examplesofdifferentialequations Laplacetransformappliedtodifferentialequations Listofdynamicalsystemsanddifferentialequationstopics Matrixdifferentialequation Methodofundeterminedcoefficients Recurrencerelation Notes[edit] ^DennisG.Zill(15March2012).AFirstCourseinDifferentialEquationswithModelingApplications.CengageLearning.ISBN 978-1-285-40110-2.Archivedfromtheoriginalon17January2020.Retrieved11July2019. ^"Whatistheoriginoftheterm"ordinarydifferentialequations"?".hsm.stackexchange.com.StackExchange.Retrieved2016-07-28. ^MathematicsforChemists,D.M.Hirst,MacmillanPress,1976,(NoISBN)SBN:333-18172-7 ^Kreyszig(1972,p. 64) ^Simmons(1972,pp. 1,2) ^Halliday&Resnick(1977,p. 78) ^Tipler(1991,pp. 78–83) ^abHarper(1976,p. 127) ^Kreyszig(1972,p. 2) ^Simmons(1972,p. 3) ^abKreyszig(1972,p. 24) ^Simmons(1972,p. 47) ^Harper(1976,p. 128) ^Kreyszig(1972,p. 12) ^Ascher(1998,p. 12)harvtxterror:notarget:CITEREFAscher1998(help) ^AchimIlchmann;TimoReis(2014).SurveysinDifferential-AlgebraicEquationsII.Springer.pp. 104–105.ISBN 978-3-319-11050-9. ^Ascher(1998,p. 5)harvtxterror:notarget:CITEREFAscher1998(help) ^Kreyszig(1972,p. 78) ^Kreyszig(1972,p. 4) ^VardiaT.Haimo(1985)."FiniteTimeDifferentialEquations".198524thIEEEConferenceonDecisionandControl.pp. 1729–1733.doi:10.1109/CDC.1985.268832.S2CID 45426376. ^Crelle,1866,1868 ^Lawrence(1999,p. 9)harvtxterror:notarget:CITEREFLawrence1999(help) ^Logan,J.(2013).Appliedmathematics(Fourthed.). ^Ascher(1998,p. 13)harvtxterror:notarget:CITEREFAscher1998(help) ^abcdefghijElementaryDifferentialEquationsandBoundaryValueProblems(4thEdition),W.E.Boyce,R.C.Diprima,WileyInternational,JohnWiley&Sons,1986,ISBN 0-471-83824-1 ^Boscain;Chitour2011,p. 21 ^abMathematicalHandbookofFormulasandTables(3rdedition),S.Lipschutz,M.R.Spiegel,J.Liu,Schaum'sOutlineSeries,2009,ISC_2N978-0-07-154855-7 ^FurtherElementaryAnalysis,R.Porter,G.Bell&Sons(London),1978,ISBN 0-7135-1594-5 ^abMathematicalmethodsforphysicsandengineering,K.F.Riley,M.P.Hobson,S.J.Bence,CambridgeUniversityPress,2010,ISC_2N978-0-521-86153-3 References[edit] Halliday,David;Resnick,Robert(1977),Physics(3rd ed.),NewYork:Wiley,ISBN 0-471-71716-9 Harper,Charlie(1976),IntroductiontoMathematicalPhysics,NewJersey:Prentice-Hall,ISBN 0-13-487538-9 Kreyszig,Erwin(1972),AdvancedEngineeringMathematics(3rd ed.),NewYork:Wiley,ISBN 0-471-50728-8. Polyanin,A.D.andV.F.Zaitsev,HandbookofExactSolutionsforOrdinaryDifferentialEquations(2ndedition),Chapman&Hall/CRCPress,BocaRaton,2003.ISBN 1-58488-297-2 Simmons,GeorgeF.(1972),DifferentialEquationswithApplicationsandHistoricalNotes,NewYork:McGraw-Hill,LCCN 75173716 Tipler,PaulA.(1991),PhysicsforScientistsandEngineers:Extendedversion(3rd ed.),NewYork:WorthPublishers,ISBN 0-87901-432-6 Boscain,Ugo;Chitour,Yacine(2011),Introductionàl'automatique(PDF)(inFrench) Dresner,Lawrence(1999),ApplicationsofLie'sTheoryofOrdinaryandPartialDifferentialEquations,BristolandPhiladelphia:InstituteofPhysicsPublishing,ISBN 978-0750305303 Ascher,Uri;Petzold,Linda(1998),ComputerMethodsforOrdinaryDifferentialEquationsandDifferential-AlgebraicEquations,SIAM,ISBN 978-1-61197-139-2 Bibliography[edit] Coddington,EarlA.;Levinson,Norman(1955).TheoryofOrdinaryDifferentialEquations.NewYork:McGraw-Hill. Hartman,Philip(2002)[1964],Ordinarydifferentialequations,ClassicsinAppliedMathematics,vol. 38,Philadelphia:SocietyforIndustrialandAppliedMathematics,doi:10.1137/1.9780898719222,ISBN 978-0-89871-510-1,MR 1929104 W.Johnson,ATreatiseonOrdinaryandPartialDifferentialEquations,JohnWileyandSons,1913,inUniversityofMichiganHistoricalMathCollection Ince,EdwardL.(1944)[1926],OrdinaryDifferentialEquations,DoverPublications,NewYork,ISBN 978-0-486-60349-0,MR 0010757 WitoldHurewicz,LecturesonOrdinaryDifferentialEquations,DoverPublications,ISBN 0-486-49510-8 Ibragimov,NailH.(1993).CRCHandbookofLieGroupAnalysisofDifferentialEquationsVol.1-3.Providence:CRC-Press.ISBN 0-8493-4488-3.. Teschl,Gerald(2012).OrdinaryDifferentialEquationsandDynamicalSystems.Providence:AmericanMathematicalSociety.ISBN 978-0-8218-8328-0. A.D.Polyanin,V.F.Zaitsev,andA.Moussiaux,HandbookofFirstOrderPartialDifferentialEquations,Taylor&Francis,London,2002.ISBN 0-415-27267-X D.Zwillinger,HandbookofDifferentialEquations(3rdedition),AcademicPress,Boston,1997. Externallinks[edit] Wikibookshasabookonthetopicof:Calculus/Ordinarydifferentialequations WikimediaCommonshasmediarelatedtoOrdinarydifferentialequations. "Differentialequation,ordinary",EncyclopediaofMathematics,EMSPress,2001[1994] EqWorld:TheWorldofMathematicalEquations,containingalistofordinarydifferentialequationswiththeirsolutions. OnlineNotes/DifferentialEquationsbyPaulDawkins,LamarUniversity. DifferentialEquations,S.O.S.Mathematics. AprimeronanalyticalsolutionofdifferentialequationsfromtheHolisticNumericalMethodsInstitute,UniversityofSouthFlorida. OrdinaryDifferentialEquationsandDynamicalSystemslecturenotesbyGeraldTeschl. NotesonDiffyQs:DifferentialEquationsforEngineersAnintroductorytextbookondifferentialequationsbyJiriLeblofUIUC. ModelingwithODEsusingScilabAtutorialonhowtomodelaphysicalsystemdescribedbyODEusingScilabstandardprogramminglanguagebyOpeneeringteam. 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