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Abstract Differentialequationshavealongandrichhistory,datingbacktothe17th century with theworkofSirIsaacNewtonandGottfriedWilhelmLeibniz,whoindepen- dentlydevelopedcalculusasameansofsolvingphysicalproblemsinvolvingrates of change.ThefirstpublishedworkondifferentialequationsisattributedtoJames Bernoulli, whopublished”ArsConjectandi”in1713.Inthiswork,hesolvedthe differentialequation y′ + P (x) y = Q(x) , whichisknownasafirst-orderlin- ear differentialequation.Later,LeonhardEulermadesignificantcontributions to thefieldofdifferentialequations,developingthetheoryofordinarydifferen- tial equationsdescribethebehaviorofasinglevariableovertime.Theycanbe classified basedontheirorder,whichisthehighestderivativepresentintheequa- tion. First-orderordinarydifferentialequationsinvolveonlythefirstderivative, while second-orderordinarydifferentialequationsinvolvethesecondderivative, and soon.Oneoftheearliestexamplesofanordinarydifferentialequationisthe lawofcooling,whichdescribestherateatwhichanobjectcoolsdowninagiven environment.ThisequationwasfirstformulatedbySirIsaacNewtoninthe17th centuryandmost18th-centurydevelopmentsconsolidatedtheLeibniziantradition, extending itsmulti-variateform,thusleadingtopartialdifferentialequationsde- scribethebehaviorofmultiplevariablessimultaneously,suchasthetemperature at everypointinaroom. |