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 dependence.
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


See also: