In mathematics, a linear differential equation is a differential equation, most generally in the form
(where D is the differential operator), is said to have order n.

To solve a linear differential equation one makes a substitution y=eλx in the homogeneous equation (ie., setting f(x)=0), to form the characteristic equation

to obtain the solutions
Where the solutions are distinct, we have immediately n solutions to the differential equation in the form
and we have that the general solution to the homogeneous equation can be formed from a linear combination of the yi, ie.,

Where the solutions are not distinct, it may be necessary to multiply them by some power of x to obtain linear dependence.

To obtain the solution to the inhomogeneous equation, find a particular solution yP(x) by the method of undetermined coefficients and the general solution to the linear differential equation is the sum of the homogeneous and the particular equation.

A linear differential equation can also refer to an equation in the form

where this equation can be solved by forming the integrating factor , multiplying throughout to obtain
which simplifies due to the product rule to
on integrating both sides yields

See also: