VARIATION OF PARAMETERS
=======================

In order to solve a nonhomogeneous linear differential equation with constant coefficients, it is necessary to first calculate the _Kernel_, that is, the solution of the homogeneous version of the equation.  Thus, if the equation in question is:

	(D^2 + a D + b)Y = f(X)                            (1)

then the kernel Yk is the solution of:

	(D^2 + a D + b)Y = 0                               (2)

Which may be found by the usual method.

Linear mathematics tells us that the solution to the nonhomogeneous equation is:

	Y = Yp + Yk                                        (3)

Where Yp is a particular solution to (1).  This may be found by guesswork, but a more reliable method is the method of _Variation of Parameters_.  We will here exammine the second order case, as an example.

If the kernel is of the form:

	Yk = C1 U1(X) + C2 U2(X)                           (4)

we first replace the constants in (4) with undefined functions of X, V1(X) & V2(X).  Our requirements are:

	V1` U1 + V2` U2 = 0                                (5)

	V1` U1` + V2` U2` = f(X)                           (6)

The solutions of V1` & V2` may be extracted using Cramer`s Rule and integrated.  The overall solution to equation (1) is then:

	Y = V1 U1 + V2 U2                                  (7)

Summary of Technique
--------------------

	1 - Find Kernel
	2 - Construct (5) & (6)
	3 - Apply Cramer`s Rule and integrate
	4 - Construct solution



As usual, no guarantees are made as to the accuracy or obsolescence of this information.  Use at your own risk.