Page 67 - Schaum's Outline of Differential Equations

CHAPTER 7

Applications

of First-Order

Differential Equations

GROWTH AND DECAY PROBLEMS
Let N(t) denote ihe amount of substance {or population) that is either grow ing or deca\ ing. It' we assume
that dN/dt. the lime rale of change of this amount of substance, is proportional to the amount of substance
present. Ihen ilNldt = kN. or

where k is the constant of proportionality. (See Problems 7.1-7.7.}
We are assuming that N(n is a dilTcrenliabie, hence continuous, function of time. For population
(
problems, where N(t) is actually discrete and integer-valued, this assumption is incorrect. Nonetheless, 7.1)
still provides a good approNi million io she physical laws governing such a system. (.See Problem 7.5.)

TEMPERATURE PROBLEMS
Newton's law of cooling, "hieh is equally applicable lo healing, stales lhal ihe lime rate of change of ihe
temperature of a body is proportional to the temperature difference between the body and iis surrounding
medium. Let T denote ihe temperature of the body and lei T, H denote the temperature of the surrounding
medium. Then the time rate of change of Ihe temperature of the body \sdT/di, and Newton's law of cooling can
be formulated as ilT/di = -k(T- T m). or as

where k is a positive constant of proportionality Once k is chosen positive, the minus sign is required in
Newton's law to make dT/di negative in a cooling process, when T is greater than T m. and positive in a heating
process, when TMs less than T,,,. (See Problems 7.8-7.10.1

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