nth ORDER ORDINARY LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
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Expansion of methods used for solving second order equations of this form may swiftly be expanded to encompass any order of equation.  The basic principal (that of finding the roots of the _auxiliary equation_) remains unchanged.  However, some further explanation of the general formula of the solution is required.

Firstly, it should be understood that ever root of the _auxiliary equation_ generates a term:

	Ci e^(RiX) , where Ri is the root in question

in the solution.  This includes complex roots, whither or not they accompany their conjugates.  Any cos or sin terms in the solution are formed when the complex root and its conjugate both appear and are combined according to Euler`s formula.  The solution is formed merely by summing each of the terms generated by the roots of the auxiliary equation.

	Y = C1 e^(R1 X) + C2 e^(R2 X) + ....


Repeated Roots
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If the auxiliary equation generates a repeated root, it must be handled with care.  This is slightly more difficult in general than with second order equations, since a root may be repeated several times.

The _multiplicity_ of a root is the number of times the root is repeated (thus a root of multiplicity 1 is not repeated at all).

For a root of multiplicity m, the terms generated in the solution are:

	C1 e^(R X) + C2 e^(R X) X + C3 e^(R X) X^2 + ... ... + Cm e^(R X) x^(m-1)

Often, due to cancellings, etc., these terms may be simplified and this is,indeed, the case when we memorise the second order general solution for a root of multiplicity 2:

	Y = (C1 X + C2) e^(R2 X)

Thus a root of multiplicity two gives:

	C1 e^(R X) + C2 e^(R X) X

etcetera.