In this chapter we continue the discussion started in Chapter 3 by looking at a special type of first order differential equation, namely, y'=f(y)g(x), where the righthand side is the product of a function of y and a function of x. Such equations are called separable. They are important for three reasons: they can be used to model various situations, they can be solved analytically (up to an integration), and they can be used for solving other types of differential equations. In the first part of the chapter we focus on obtaining analytical solutions and developing a technique for graphing such solutions if we cannot solve explicitly for one of the variables. We introduce a new type of differential equation—one with homogeneous coefficients—that reduces to a separable one by an appropriate change of variable. We also give examples on how to proceed from a data set to a differential equation to model the process giving rise to this data set. Other applications include models of sky diving, orthogonal trajectories, and the population of Ireland during the potato famine.
