Linear First Order Differential Equations

In the next three chapters we explain techniques for solving and analyzing ordinary differential equations. We do not attempt to provide exhaustive coverage of the topic but instead focus on the types of differential equations and techniques of analysis that are most common in economics. We begin in this chapter with linear, first-order differential equations. In the next chapter we turn to an examination of nonlinear, first-order differential equations, and in the chapter after that we examine linear, second-order differential equations. In this chapter and throughout, we will solve a large number of examples and economic applications to illustrate the uses of ordinary differential equations in economics.

In this section we explain how to solve linear, first-order differential equations that are autonomous, meaning ones in which the variable r does not enter the equation explicitly.

The general form of the linear, autonomous, first-order diff erential equation is where a and b are known constants.

The differential equation is linear because y and v are not raised to any power other than I. It is of the first-order because dial is the highest-order derivative in the equation. It is autonomous because the coefficient a and the term b are constant. If à or h vary with t (i.e., are explicit functions of t), the equation is nonautonomous—this case is taken up in section 21.2.

The solution method we use in this section relies on the technique of separating the problem of finding the general solution for the complete differential equation into two simpler subproblems. If >'/, denotes the general solution to the homogeneous form (obtained by setting b = 0) and y,. denotes a particular

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