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=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:
