Chapter 3
Applications of the Initial Value
Problem y' = f(x, y) ; y(c) = d
In this chapter, we consider a variety of applications of the initial value
problem y' = f(x, y); y(c) = d. First, in the section titled Calculus Re-
visited, we show that the solut ion to the particular initia l value problem
y' = f(x); y(a) = 0 is equivalent to the definite integral l: f(t) dt. Then,
we show how to use computer software to calculate an approximation to the
definite integral l: f(x) dx. This will a llow us to solve problems from calcu-
lus numerically. In the sections titled Learning Theory Models, Population
Models, Simple Epidemic Models, Falling Bodies, Mixture Problems, Curves
of Pursuit, and Chemical Reactions, we examine some physical problems from
a number of d iverse disciplines which can be written as initial value problems
and then solved using numerical integration software. Finally, we present a
few additiona l applications in the Miscellaneous Exercises which appear at
the end of this chapter.
3.1 Calculus Revisited
Many calculus problems involve computing a value for the definite integrall: f(x) dx. Examples of such problems include finding the area under a curve,
finding the area between two curves, finding the length of an arc of a curve,
finding the area of a surface generated by revolving a curve about an axis,
finding the volume of a solid generated by revolving a region about an axis,
and computing physical quantities such as work, force, pressure, moments,
center of mass, and centroids.
A function F(x) is called an antiderivative of the function f(x) on aninterval [a, b], if F'(x) = f(x) for a ll x E [a, b]. The fundamental theorem of
integral calculus stated below tells us how to compute the value of the definite
integral of f(x) on [a, b], l: f(x) dx, provided we can find an antiderivative of
f(x) on [a , b].
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