Linear differential equation

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=e^λx 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 y_i, ie.,

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

To obtain the solution to the inhomogeneous equation, find a particular solution y_P(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