7Ordinary differential equationsA differential equation is an equation involving one or more derivativesof an unknown function. If all derivatives are taken with respect to asingle independent variable we call it anordinary differential equation,whereas we have apartial differential equationwhen partial derivativesare present.The differential equation (ordinary or partial) hasorder pifpis themaximum order of differentiation that is present. The next chapter willbe devoted to the study of partial differential equations, whereas in thepresent chapter we will deal with ordinary differential equations of firstorder.Ordinary differential equations describe the evolution of many phe-nomena in various fields, as we can see from the following four examples.Problem 7.1 (Thermodynamics)Consider a body having internaltemperatureTwhich is set in an environment with constant temperatureTe. Assume that its massmis concentrated in a single point. Then theheat transfer between the body and the external environment can bedescribed by the Stefan-Boltzmann lawv(t) =γS(T4(t)-T4e),wheretis the time variable,the Boltzmann constant (equal to 5.6·10-8J/m2K4s where J stands for Joule, K for Kelvin and, obviously, mfor meter, s for second),γis the emissivity constant of the body,Sthearea of its surface andvis the rate of the heat transfer. The rate ofvariation of the energyE(t) =mCT(t) (whereCdenotes the specificheat of the material constituting the body) equals, in absolute value,the ratev. Consequently, settingT(0) =T0, the computation ofT(t)requires the solution of the ordinary differential equationdTdt=-v(t)mC.(7.1)
1887 Ordinary differential equationsSee Exercise 7.15.Problem 7.2 (Population dynamics)Consider a population of bac-teria in a confined environment in which no more thanBelements cancoexist. Assume that, at the initial time, the number of individuals isequal toy0Band the growth rate of the bacteria is a positive con-stantC. In this case the rate of change of the population is proportionalto the number of existing bacteria, under the restriction that the totalnumber cannot exceedB. This is expressed by the differential equationdydt=Cy1-yB,(7.2)whose solutiony=y(t) denotes the number of bacteria at timet.Assuming that two populationsy1andy2be in competition, insteadof (7.2) we would havedy1dt=C1y1(1-b1y1-d2y2),dy2dt=-C2y2(1-b2y2-d1y1),(7.3)whereC1andC2represent the growth rates of the two populations.The coefficientsd1andd2govern the type of interaction between thetwo populations, whileb1andb2are related to the available quantityof nutrients. The above equations (7.3) are called the Lotka-Volterraequations and form the basis of various applications. For their numericalsolution, see Example 7.7.