# Differential Equations and Their Solutions

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.

*x*^{2} + *y*^{2} = 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

d*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.