Linear differential equation

This article is about linear differential equations with one variable by itself. For similar equations with two or more variables by itself, see Partial differential equation.

In mathematics, a linear differential equation is a differential equation that is stated by a linear polynomial in the unknown function and its derivatives. It is an equation of the form:

$a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''\cdots +a_{n}(x)y^{(n)}=b(x)$

where a₀(x), ..., a_n(x) and b(x) are random differential functions that do not need to be linear. And y′, ..., y⁽ⁿ⁾ are the derivatives of an unknown function y that follow after the variable x.

This equation is an ordinary differential equation (ODE). A linear differential equation may also be a linear partial differential equation (PDE), if the unknown function depends on several variables. And the derivatives that show in the equation are partial derivatives.