ANNIHILATING FACTORS
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Application of Annihilating Factors to Differential Equations
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When attempting to solve an equation of the form:

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

much effort may be saved by finding a differential equation L, such that:

	L f(X) = 0                                         (2)

Such an equation is known as an _Annihilating Factor_, in that it annihilates the right hand side of the original equation, leaving us with a homogeneous equation with a greater order than the original:

	L (a D^2 + b D + c) Y = 0                          (3)

This equation may be solved using normal methods.  The solution thus obtained will contain the _Kernel_ (the original equation`s homogeneous version), summed with one or more further terms.  By removing the solution of

	(a D^2 + b D +c) Y = 0                             (4)

we are left with an equation of the form:

	Yp = C1 e^(S1) + C2 e^(S2) + ....                  (5)

with the exact number of terms dependent on the order of the equation.  By substituting this into the original equation, we obtain values for the constants, providing a particular solution to the original equation.  It should be noted that the solution of equation (3) contains all solutions of equation (1).  It is by this means by which we calculate the constants (in equation (5)), ie, by use of equation (1), that we constrain the final result to give a solution of equation (1) and filter out the unesssesary solutions of equation (3).

Our final solution is:

	Y = Yp + Yk     (as ever)

Summary of Technique
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	1 - Calculate Yk of original equation (ie the homogeneous solution)
	2 - Find L such that L f(X) =0
	3 - Solve equation formed by multiplying both sides by L
	4 - Subtract Yk from result of step 3
	5 - Calculate constants by back substitution into original
	6 - Form solution from results of steps 5 & 1



Determination of Annihilating Factors
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In order to find the annihilating factors previously mentioned, where the annihilating factor is L and:

	L f(X) = 0

We must, if the factor is to lead us to a solution of the nonhomogeneous equation, use differentiation.  It is clear that such factors as L = 0 will satisfy the above equation, but it would be somewhat unhelpful.

If we differentiate once and the result is zero:

	D f(X) = 0

then it is clear that the annihilating factor is D.  If the result is not zero, then we must use further differentiation and/or algebraic manipulation to introduce a zero term and isolate L and f(X).  For example, with:

	f(X) = e^X

	D e^X = e^X

	De^X - e^X = 0

	(D - 1) e^X = 0

Which is of the form L f(X) = 0.  Thus, we may state that the annihilating factor in the above example is (D - 1).  In another vein:

	f(X) = 4 X^2

	D 4 X^2 = 8 X

	D^2 4 X^2 = 8

	D^3 4 X^2 =0

Thus, L = D^3.  As can be seen, it is very much a case of practice & experience.



Annihilating Factors for Multiple Term Functions
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In the case where the right hand side of the nonhomogeneous equation is formed of a sum of terms, then it is necessary only to determine the annihilating factor for each term and multiply these together to provide the annihilating factor for the whole sum.

For example:

	f(X) = 3 + 4X

	L of 3 is D
	Lof 4X is D^2

	thus, L of f(X) = D D^2

However, such a factor is more complicated than necessary, since D^2 is a legitimate annihilating factor of 3.  Thus, we may disregard the D term as superfluous, leaving L = D^2.

Put more elegantly, this means that:

Annihilating factors of a higher order override those of a lower order.







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