1.1 | Background

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A differential equation only involving ordinary derivatives is called an ordinary differential equation. A partial differential equation involves partial derivatives with respect to more than one independent variable. The order of a differential equation is given by the highest order derivative that appears.
A linear differential equation is any equation that can be written in the form

$a_n(x) \frac{d^ny}{dx^n} + a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_1(x) \frac{dy}{dx} + a_0(x)y = F(x).$
$a_n(x) \frac{d^ny}{dx^n} + a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_1(x) \frac{dy}{dx} + a_0(x)y = F(x).$
$a_n(x) \frac{d^ny}{dx^n} + \cdots + a_1(x) \frac{dy}{dx} + a_0(x)y = F(x).$

Otherwise the equation is said to be nonlinear.