Differential Equations Solution Guide - Math is Fun

There is no magic way to solve all Differential Equations. ... For non-homogeneous equations the general solution is the sum of:. DifferentialEquationsSolutionGuide ADifferentialEquationis anequationwithafunctionand oneormoreofitsderivatives: Example:anequationwiththefunctionyanditsderivativedydx  Inourworldthingschange,anddescribinghowtheychangeoftenendsupasaDifferentialEquation. Realworldexampleswhere DifferentialEquationsareusedincludepopulationgrowth,electrodynamics,heat flow,planetarymovement,economicsystemsandmuchmore! Solving ADifferentialEquationcanbeaverynaturalwayofdescribingsomething. Example:PopulationGrowth Thisshortequationsaysthatapopulation"N"increases(atanyinstant)asthegrowthratetimesthepopulationatthatinstant: dNdt=rN Butitisnotveryusefulasitis. Weneedto solveit! Wesolveitwhenwediscoverthefunctiony(or setoffunctionsy)thatsatisfiestheequation,andthenitcanbeusedsuccessfully. Example:continued Ourexampleissolvedwiththisequation: N(t)=N0ert Whatdoesitsay?Let'suseittosee: Withtinmonths,apopulationthatstartsat1000(N0)andagrowthrateof10%permonth(r)weget: N(1month)=1000e0.1x1=1105 N(6months)=1000e0.1x6=1822 etc   ThereisnomagicwaytosolveallDifferentialEquations. Butoverthemillenniagreatmindshavebeenbuildingoneachothersworkandhavediscovereddifferentmethods(possiblylongandcomplicatedmethods!)ofsolvingsometypesofDifferentialEquations. Solet’stakea lookatsomedifferenttypesofDifferentialEquationsandhowtosolvethem: SeparationofVariables SeparationofVariablescanbeusedwhen: Alltheyterms(includingdy)canbemovedtooneside oftheequation,and Allthexterms(includingdx)totheotherside. Ifthatisthecase,wecanthenintegrateandsimplifytogetthethe solution. FirstOrderLinear FirstOrderLinearDifferentialEquationsareofthistype: dydx+P(x)y=Q(x) WhereP(x)andQ(x)arefunctionsofx. Theyare"FirstOrder"whenthereisonlydydx(notd2ydx2ord3ydx3,etc.) Note:anon-lineardifferentialequationisoftenhardtosolve,butwecansometimesapproximateitwithalineardifferentialequationto findaneasiersolution. HomogeneousEquations HomogeneousDifferentialEquationslooklikethis: dydx=F( yx) Wecansolvethembyusingachangeofvariables: v=yx whichcanthenbesolvedusingSeparationofVariables. BernoulliEquation BernoullEquationsareofthisgeneralform: dydx+P(x)y=Q(x)yn wherenisanyRealNumberbutnot0or1 Whenn=0theequationcanbesolvedasaFirstOrderLinear DifferentialEquation. Whenn=1theequationcanbesolvedusingSeparationof Variables. Forothervaluesofnwecansolveitbysubstituting u=y1−nandturningitintoalineardifferentialequation(andthensolvethat). SecondOrderEquation SecondOrder(homogeneous)areofthetype: d2ydx+P(x)dydx+Q(x)y=0 Noticethereisasecondderivative d2ydx2 The generalsecondorderequationlookslikethis  a(x)d2ydx2+b(x)dydx+c(x)y=Q(x) Therearemanydistinctivecasesamongthese equations. Theyareclassifiedashomogeneous(Q(x)=0),non-homogeneous, autonomous,constantcoefficients,undeterminedcoefficientsetc. Fornon-homogeneousequationsthegeneral solutionisthesumof: thesolutiontothecorrespondinghomogeneous equation,and theparticularsolutionofthe non-homogeneousequation UndeterminedCoefficients The Undetermined Coefficientsmethodworksforanon-homogeneousequationlikethis: d2ydx2+P(x)dydx+Q(x)y =f(x) wheref(x)isapolynomial,exponential,sine,cosineoralinearcombinationofthose.(ForamoregeneralversionseeVariationofParametersbelow)Thismethodalsoinvolvesmakingaguess! VariationofParameters Variation ofParametersisalittlemessierbutworksonawiderrangeoffunctionsthanthepreviousUndetermined Coefficients. ExactEquationsandIntegratingFactors ExactEquationsandIntegratingFactorscanbeusedforafirst-orderdifferentialequationlikethis: M(x,y)dx+N(x,y)dy=0 thatmusthavesomespecialfunctionI(x,y)whosepartialderivativescanbeputinplaceofMandNlikethis: ∂I∂xdx+∂I∂ydy=0 OurjobistofindthatmagicalfunctionI(x,y)ifitexists. OrdinaryDifferentialEquations(ODEs)vsPartialDifferentialEquations(PDEs) AllofthemethodssofarareknownasOrdinaryDifferentialEquations(ODE's). Thetermordinaryisusedincontrastwiththetermpartialtoindicatederivativeswithrespecttoonlyoneindependentvariable. DifferentialEquationswithunknownmulti-variablefunctionsandtheir partialderivativesareadifferenttypeandrequireseparatemethodsto solvethem. TheyarecalledPartialDifferentialEquations(PDE's),and sorry,butwedon'thaveanypageonthistopicyet.   CalculusIndex Copyright©2021MathsIsFun.com