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1、Chapter1DifferentialEquations1.1BasicIdeasWestartbyconsideringafirst-order(ordinary)differentialequationoftheformdx=F(x;t)withtheinitialconditionx(τ)=ξ.(1.1)dtWesaythatx=f(t)isasolutionof(1.1)ifitissatisfiedidenticallywhenwesubstitutef(t)forx.Thatisdf(t)≡F(f(t);t)andf(τ)=ξ.(1.2)dtIfweplot(t,f(t))inth
2、eCartesiant–xplanethenwegenerateasolutioncurveof(1.1)whichpassesthroughthepoint(τ,ξ)determinedbytheinitialcondition.Iftheinitialconditionisvariedthenafamilyofsolutioncurvesisgenerated.Thisideacorrespondstothinkingofthegeneralsolutionx=f(t,c)withf(t,c0)=f(t)andf(τ,c0)=ξ.Thefamilyofsolutionsareobta
3、inedbyplotting(t,f(t,c))fordifferentvaluesofc.Throughoutthiscoursethevariabletcanbethoughtofastime.Whenconvenientweshallusethe‘dot’notationtosignifydifferentiationwithrespecttot.Thusdxd2x=˙x(t),=¨x(t),dtdt2with(1.1)expressedintheform˙x(t)=F(x;t).1Weshallalsosometimesdenotethesolutionof(1.1)simplyas
4、x(t)ratherthanusingthedifferentletterf(t).Inpracticeitisnotpossible,inmostcases,toobtainacompletesolutiontoadifferentialequationintermsofelementaryfunctions.Toseewhythisisthecaseconsiderthesimplecaseofaseparableequationdx=T(t)/X(x)withtheinitialconditionx(τ)=ξ.(1.3)dt1Forderivativesofhigherthanseco
5、ndorderthisnotationbecomescumbersomeandwillnotbeused.12CHAPTER1.DIFFERENTIALEQUATIONSThiscanberearrangedtogivetxT(u)du=X(y)dy.(1.4)τξSotocompletethesolutionwemustbeabletoperformbothintegralsandinvertthesolutionforminxtogetanexplicitexpressionforx=f(t).Boththesetasksarenotnecessarypossible.Unlik
6、edifferentiation,integrationisaskillratherthanascience.Ifyouknowtherulesandcanapplythemyoucandifferentiateanything.Youcansolveanintegralonlyifyoucanspotthatitbelongstoaclassofeasilysolvableforms(bysubstitution,integrationbypartsetc.).Soevenaseparableequationisnotnecessarilysolvableandtheproblemsinc
7、reaseformorecomplicatedequations.Itis,however,importanttoknowwhetheragivenequationpossessesasolutionand,ifso,whetherthesolutionisunique.ThisinformationisgivenbyPicard’sTheorem:Theorem1.1.1Considerthe(square)setA={(x,t)
