Linearvs.NonlinearFirstOrderDifferentialEquations

Linear Nonlinear

Form y′+p(t)y=g(t),y(t0)=y0 y′=F(t,y),y(t0)=y0

Theorem IfpandgarecontinuousonanopenintervalI:α<t<βcontainingthepointt0,thenthereexistsauniquefunctiony=ϕ(t)thatsatisfiesthedifferentialequationforeachtinIandthatalsosatisfiestheinitialcondition.

Iffand∂f/∂yarecontinuousinsomerectangleα<t<β,γ<y<δcontainingthepoint(t0,y0),theninsomeintervalt0–h<t<t0+hcontainedinα<t<βthereisauniquesolutiony=ϕ(t)oftheinitialvalueproblem.

Weakerresult

None Iffiscontinuous,thenasolutionexists,butitmaynotbeunique

Method Integratingfactor Ifseparable,separate.Otherwise…?

Discontinuity Thesolutioncanonlybediscontinuouswherepandgarediscontinuous.Eventhen,itstillmayexistevenatpointswereeitherporgarediscontinuous.

Thesolutioncanbediscontinuousatpointsotherthanwherefand∂f/∂yarediscontinuous.Moreover,theremaybenothinginthedifferentialequationthatindicateswheretheseadditionaldiscontinuitiesexist.

Completenessofsolutions

Allsolutionscanbegeneratedbyvaryingaconstantinasingleexpression

Ageneralsolutionmaybeproducedintheformofanalgebraicexpression,buteventhenitispossiblethatnotallsolutionsmaybeproducedbyvaryingtheconstant

Explicitsolution

Thereisalwaysawaytosolveforyexplicitly

Itmaynotbepossibletosolveforyexplicitly