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
y ' + 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.