الفهرس 
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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.