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 Revisited, we show that the solution to the particular initial value problem y′ = f(x); y(a) = 0 is equivalent to the definite integral

∫ x a f(t) dt. Then,

we show how to use computer software to calculate an approximation to the

definite integral ∫ b a f(x) dx. This will allow 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 diverse disciplines which can be written as initial value problems and then solved using numerical integration software. Finally, we present a few additional applications in the Miscellaneous Exercises which appear at the end of this chapter.