Linear differential equation

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.