Ordinary differential equations
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.
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”.
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.
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)) = 0if the equation is satisfied for all values of t.
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
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.
- 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.
- 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.