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1、ChapterContinuousTime:First-OrderDifferentialEquationsIntheDomargrowthmodel,wehavesolvedasimpledifferentialequationbydirel~tint(;~gration,FormorecomplicateddilTcrcntialequations,therearevariousestablishedmethodsofsolution.Eveninthelattercases,however,thefundamentali
2、deaunderlyingthemethodsofSlliutionisstillthetedmiquesofintegralcalculus.Forthisreason,thesolutiontoadifferentialequationisoftenreferredtoastheintewalofth.tequation.Only,lir,t-orderdifferentialequationswillbediscussedInthepresentchapter.IIIthiscontext,thewordorderr
3、eferstothehighcl'torderofthederivatives(orditferentials)llppcarillginthedifferentialequation;thusalirst·orderdifferentialequationcancontainonlythefirf:!tderivative,say,dy/dr,15.1First-OrderLinearDifferentialEquationswithConstantCoefficientandConstantTermThefirstderi
4、vativedy/dti...theonly(lnCthateanappearinafirst-orderdilleremialequa•tion,butitmayenterinvariou.'>power~:dy/dt,(dy/dt)?,or(dy/df)3.ThehighestpowerattainedbythedcnvativeIIItheequa(i()Jli~rl'ferredtoa~thedegreeoftheditteremialequation.Incasethederivativedy/dtappcar~on
5、lyinthefirstdegree,andsodoesthedependentvariabley,andfurthermore,noproductoftheformy(dy/dl)occurs,then(heequationissaidtobelinear.Thusalirsh)rd..:rlineardifferentialequationwillgenerallytaketheforml'dv_.+u(r)v=I1-'(f)(15.1)dt..tNotethatthederivativetermdy/dtin(15,1)
6、hasaunitcoefficient.Thisi,nottoimplythatitCilnneveractuallyhaveacoefficientotherthanone,butwhensuchacoefficientappears,WI":'ctlnalways"normalize"theequationbydividingeilchtermbythesaidcoeffiCient.Forthisreason,thefOrmgivenin(15.1)m~ynonethelessberegardedasageneralre
7、presentation,475476PartFiveDynmn/('Analysiswhereuand}varetwofunctionsoft.asisy,Tnc.ontra~ttody/dtandy,however,nore::;tric.tionwhatsoeverisplacedontheindependentvariablef.ThusthefunctionsIIandwmayverywellrepresentsuehexpressionsasl(2andeorsomemorecomplicatedfunctions
8、oft;ontheotherhand,uandwmayalsobeconstants.Thislastpointleadsustoafurtherdassificati()n.Whenthefunctionu(theeoeffkicntofthedependentvariab
