Higherorder Differential Equations

In the last chapter, we discussed the methods of solving a first-order differential equation, one in which there appears no derivative (or differential) of orders higher than 1. At times, however, the specification of a model may involve the second derivative or a derivative of an even higher order. We may, for instance, be given a function describing " the rate of change of the rate of change" of the income variable Y, say, d2Y

from which we are supposed to find the time path of Y. In this event, the given function constitutes a second-order differential equation, and the task of finding the time path Y(t) is that of solving the second-order differential equation. The present chapter is concerned with the methods of solution and the economic applications of such higher-order differential equations, but we shall confine our discussion to the linear case only.

A simple variety of linear differential equations of order n is of the following form:

(15.1) -rv + a,-f + • • ■ + an_,-~ + a v = b v 7 dt" dt"~ dt nJ

or, in an alternative notation,

(15.1') y(n>(t) + a]y{"-]'(t) + ■■■ +a„. , v'(i) + a„y = b

This equation is of order n, because the n th derivative (the first term on the left) is the highest derivative present. It is linear, since all the derivatives, as well as the dependent variable y, appear only in the first degree, and moreover, no product term occurs in which y and any of its derivatives are multiplied together. You will note, in addition, that this differential equation is characterized by constant coefficients (the a's) and a constant term (b). The constancy of the coefficients is an assumption we shall retain throughout this chapter. The constant term b, on the other hand, is adopted here as a first approach; later, in Sec. 15.5, we shall drop it in favor of a variable term.

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