Mathematical Modeling in the Social and Life Sciences

APPENDIX V

Differential Equations

In this text we deal with a number of differential equations of a fairly simple type. The following paragraphs will serve as an introduction for those who have not studied this topic before.

By a first-order differential equation, we will mean an equation of the form

where F is a given function of two variables defined on some region R of the (x, y)-plane. Eq. (1) asserts that y is a differentiable function of x over some interval [a, b] and that the derivative satisfies (1) for all values of x in that interval. A more precise statement would be that there is a function y = f(x) such that

1. f is a differentiable function of x on [a, b]
2. The graph {(x, y) : y = f(x), a ≤ x ≤ b} is contained in R
3. f′(x*) = F(x*, f(x*)) for all x* in [a, b]

EXAMPLE 1

Consider the differential equation

Here the function F(x, y) is given by F(x, y) = 3x²y + .

This function is defined everywhere on the region R of the plane consisting of all points with positive first coordinate, the “open right half-plane.” If a and b are any two positive numbers with a < b, then the function

satisfies (A), (B), and (C). You should verify this (immediately).

Note that the function

also ...