A differential equation (d.e.) is any equation that has one or more derivative in it. These can be first derivatives, second derivatives...whatever.

Sample Problem

The following are differential equations.

y ' + y " + xy = 0

Sample Problem

The following are not differential equations, because they don't contain any derivatives.

x² + y² = 8

x + xy – y + 9 = 0

x = 9

The order of a differential equation is the highest derivative that occurs in that differential equation.

Sample Problem

The differential equation

dy ' + y " + y "' + x = 0

has order 3 because that's the highest derivative in the equation:

y ' + y " + y "' + x = 0.

Sample Problem

The differential equation

has order 1 because it only contains a first derivative.

A d.e. of order 1 is called a first-order differential equation, and a d.e. of order 2 is called a second-order differential equation. These are the kinds of differential equations that you'll probably see most often.

Practice: