# Linear differential equation

## Differential equations that are linear with respect to the unknown function and its derivatives / From Wikipedia, the free encyclopedia

#### Dear Wikiwand AI, let's keep it short by simply answering these key questions:

Can you list the top facts and stats about Linear differential equation?

Summarize this article for a 10 year old

In mathematics, a **linear differential equation** is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that 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 *a*_{0}(*x*), ..., *a*_{n}(*x*) and *b*(*x*) are arbitrary differentiable functions that do not need to be linear, and *y*′, ..., *y*^{(n)} are the successive derivatives of an unknown function y of the variable x.

Such an 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 appear in the equation are partial derivatives.