apuntes 4 - 7 Ordinary dierential equations A dierential...

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7 Ordinary di ff erential equations A di ff erential equation is an equation involving one or more derivatives of an unknown function. If all derivatives are taken with respect to a single independent variable we call it an ordinary di ff erential equation , whereas we have a partial di ff erential equation when partial derivatives are present. The di ff erential equation (ordinary or partial) has order p if p is the maximum order of di ff erentiation that is present. The next chapter will be devoted to the study of partial di ff erential equations, whereas in the present chapter we will deal with ordinary di ff erential equations of first order. Ordinary di ff erential 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 internal temperature T which is set in an environment with constant temperature T e . Assume that its mass m is concentrated in a single point. Then the heat transfer between the body and the external environment can be described by the Stefan-Boltzmann law v ( t ) = γ S ( T 4 ( t ) - T 4 e ) , where t is the time variable, the Boltzmann constant (equal to 5 . 6 · 10 - 8 J / m 2 K 4 s where J stands for Joule, K for Kelvin and, obviously, m for meter, s for second), γ is the emissivity constant of the body, S the area of its surface and v is the rate of the heat transfer. The rate of variation of the energy E ( t ) = mCT ( t ) (where C denotes the specific heat of the material constituting the body) equals, in absolute value, the rate v . Consequently, setting T (0) = T 0 , the computation of T ( t ) requires the solution of the ordinary di ff erential equation dT dt = - v ( t ) mC . (7.1)
188 7 Ordinary di ff erential equations See Exercise 7.15. Problem 7.2 (Population dynamics) Consider a population of bac- teria in a confined environment in which no more than B elements can coexist. Assume that, at the initial time, the number of individuals is equal to y 0 B and the growth rate of the bacteria is a positive con- stant C . In this case the rate of change of the population is proportional to the number of existing bacteria, under the restriction that the total number cannot exceed B . This is expressed by the di ff erential equation d y d t = Cy 1 - y B , (7.2) whose solution y = y ( t ) denotes the number of bacteria at time t . Assuming that two populations y 1 and y 2 be in competition, instead of (7.2) we would have d y 1 d t = C 1 y 1 (1 - b 1 y 1 - d 2 y 2 ) , d y 2 d t = - C 2 y 2 (1 - b 2 y 2 - d 1 y 1 ) , (7.3) where C 1 and C 2 represent the growth rates of the two populations. The coe ffi cients d 1 and d 2 govern the type of interaction between the two populations, while b 1 and b 2 are related to the available quantity of nutrients. The above equations (7.3) are called the Lotka-Volterra equations and form the basis of various applications. For their numerical solution, see Example 7.7.

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