Simultaneous Systems of Differential and Difference Equations

It is common in economic models for two or more variables to be determined simultaneously. When the model is dynamic and involves two or more variables, a system of differential or difference equations arises. The purpose of this chapter is to extend our single equation techniques to solve systems of autonomous differential and difference equations.

24.1 Linear Differential Equation Systems

We begin with the simplest case—a system of two linear differential equations— and solve it using the substitution method. We then proceed to a more general method, known as the direct method, that can be used to solve a system of linear differential equations with more than two equations.

The Substitution Method

This method is suited to solving a differential equation system consisting of exactly two linear differential equations.

A linear system of two autonomous differential equations is expressed as

The system is linear because it contains only linear differential equations which, as usual, means that y, and vf arc not raised to any power other than one. The system is autonomous because the coefficients. an. and the terms, b/, are constants. The equations must be solved simultaneously because y, depends on the solution for y2 and y: depends on the solution for y(.

As in previous chapters on linear differential and difference equations, we separate the problem of finding the complete solutions into two parts We first y, = «ii.vi +any2 +

find the homogeneous solutions and then find particular solutions. The complete solutions are the sum of the homogeneous and particular solutions. In symbols.

where .v, is the complete solution, yj' is the general homogeneous solution for y,. and yf is the particular solution for y,.

Definition 24.2

The General Solution to the Homogeneous Forms

The first step in obtaining a complete solution is to put lite differential equation system into its homogeneous form. This is done by setting the terms in each equation equal to zero.

The homogeneous form of the system of two linear, first-order differential equations (24.1) and (24.2) is

It is possible to convert this system of two first-order differential equations into a single second-order differential equation using a combination of differentiation and substitution. Since we already know how to solve a linear, second-order differential equation, this procedure provides an easy way to rind the solution.

To transform the system into a second-order differential equation, differentiate equation (24.3) to get y, = «11 v | -F any.

This gives a second-order equation but one that still depends on y2. Therefore use equation (24.4) to substitute for y2. This gives yj = flityj +ai2(«2i.vi +«22.v:)

Now use equation (24.3) to get the following expression that can be used to substitute for v2

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