chapter  16
32 Pages

Differential equations

In this chapter, dynamic models are considered where some variable varies con-

tinuously as a function of some other variable (such as time). If y is a function of

x, then assuming differentiability, the derivative is written dy dx

or y′(x). Knowing

y, it is easy to determine the derivative y′(x). However, in a variety of problems,

the situation is reversed – the derivative, y′(x) is known, but not the actual func-

tion y(x). And in this class of problem, the task is to determine y(x) from knowl-

edge of y′(x), called solving the differential equation. Section 16.1.1 provides a

few motivating examples to illustrate how such dynamics may arise. Generally,

it is difficult to determine y(x) from knowledge of x. For example, given y(x) = ln(1+ x2)+ xex, differentiation gives y′(x)= 2x

1+x2 + e x+ xex = 2x

1+x2 + (1+ x)e x. But if

given y′(x)= 2x 1+x2 + (1+ x)e

x, it is more difficult to see that y(x)= ln(1+ x2)+ xex. The general approach to this problem is to categorize differential equations

into classes of problems of varying difficulty. In a first-order differential equation,

the dynamic relation is expressed by a function f with y′ = f (y,x). If this can be written in the form a1(x)y

′(x)+ a0(x)y = b(x) it is called a first-order linear differential equation. Similarly, a second-order differential equation has the form

y′′(x) = f (y, y′,x), where y′′ is the second derivative of the (unknown) function y. Again, the task is to find the function y(x) that satisfies this equation. Depending

on the function f , this problemmay be extremely difficult to solve. In Section 16.2,

first-order linear differential equations are considered. These arise in a number

of economic models – such as growth models. Turning to non-linear differential

equations, these are typically difficult to solve but there are some specific func-

tional forms for which a solution may easily be found. Two non-linear first-order

differential equations (the logistic and Bernoulli equations) are introduced in Sec-

tion 16.3. Given a first-order differential equation, y′(x) = f (y,x), a fundamental question concerns the existence of a solution: is there a unique function y that sat-

isfies this equation? Section 16.4 provides sufficient conditions for the existence

of a unique solution. Second-order linear differential equations are considered in

Section 16.5. Finally, Section 16.6 introduces linear systems of differential equa-

tions and uses that framework to study price dynamics in a multimarket system.