Mathematical methods for economic theory

Martin J. Osborne

8.1 Differential equations: introduction

Many economic models with a temporal dimension involve relationships between the rate of change of a variable and its value at a given point in time. For example, a model of price dynamics might assume that the rate of change of a price is proportional to the difference between the demand and supply at that price, and a macroeconomic model of economic growth might assume that the rate of change of the capital stock is a constant fraction of the value of output. When time is modeled as a discrete variable (taking the values 1, 2, 3, ..., for example), relationships like these may be modeled as difference equations, studied in the next section. When time is modeled as a continuous variable, they may be modeled as differential equations.

Ordinary differential equations

Definition
An ordinary differential equation is a relationship between a variable t, the value of a function x of a single variable at t, and the derivatives of x of all orders up to some number n at t. It can be written as
G(t, x(t), x'(t), x"(t), ..., x(n)(t)) = 0,
where G is a function of n + 2 variables and x'(t), x"(t), ..., x(n)(t) are the first, second, ..., nth derivatives of x at t.
The unknown in an ordinary differential equation is the function x; we wish to find a function x that satisfies the equation for all values of t.

In many applications, t represents time, and it is often convenient to think of it as time when interpreting equations. However, it is simply a variable, and none of the theory requires it to have any property specific to time.

In expositions of the theory of differential equations, derivatives are sometimes denoted by dots over the variable rather than primes beside the variable; I use primes because they are easier to show on a webpage.

The word “ordinary” indicates that the unknown function x has only one argument. An equation involving the partial derivatives of a function of more than one variable is known as a “partial differential equation”.

Example
Here are some examples of ordinary differential equations.
x'(t) − 1  =  0
x"(t) − 1  =  0
x"(t) − 2tx(t)  =  0
x"(t)(x'(t))1/2 − t/x(t)  =  0.

Solutions

Definition
The function x of a single variable is a solution of the ordinary differential equation
G(t, x(t), x'(t), x"(t), ..., x(n)(t)) = 0
if the equation is satisfied for all values of t.
A solution of the equation
x'(t) − 1 = 0,
for example, is the function x given by x(t) = t for all t, because the derivative x'(t) is equal to 1 for all values of t. You will immediately notice that the equation has many other solutions: for any number C, the derivative of the function x defined by x(t) = t + C for all t is also 1 for all values of t, and is hence a solution.

This multiplicity of solutions is normal: most differential equations with at least one solution have many solutions. It reflects the fact that a differential equation specifies a relationship among the derivatives of the unknown function x of various orders but not the level of this function for any given value of t. Adding to a differential equation the requirement that the function x or its derivatives of various orders take specific values for specific values of t may narrow down the set of solutions to a single function.

For example, suppose that we add the requirement

x(0) = 1
to the differential equation x'(t) − 1 = 0. For a solution x(t) = t + C to satisfy this additional requirement, we need C = 1, so the only solution of the differential equation that satisfies the requirement is x(t) = t + 1.

A function that includes parameters, like C in the previous example, is the “general solution” of a differential equation if every solution is equal to the function for some value of the parameters.

Definition
The function f of the variable t and a vector C of numbers is the general solution of an ordinary differential equation if for every solution x of the equation there is a value of C such that x(t) = f(t, C) for all t.
Example
The general solution of the differential equation x'(t) − 1 = 0 is the function f defined by f(t, C) = t + C for all t, where C is a scalar.
A common though somewhat informal way to express the conclusion of this example is that “the general solution of the equation x'(t) − 1 = 0 is x(t) = t + C.” That is, the symbol C in a solution is taken to be a constant that may take any value. The same is true for the symbols C1, C2, and so on.
Example
Consider the differential equation x"(t) − 1 = 0. From the previous example we know every solution x has the property that x'(t) = t + C1 for all t. Thus the general solution of the equation is x(t) = t2/2 + C1t + C2.