In mathematics the method of operators is a method for finding the solution to a system of linear differential equations be it homogenous or inhomogenous.
Given a system of n nth order differential equations:
dnx1/dtn + ... + a1,1dx1/dt + x1 + a1,0 = f1(t)
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dnxn/dtn + ... + an,1dxn/dt + xn + an,0 = fn(t)
We can rewrite the system as a matrix using the differential operator, D, as:
AX = F(t)
Where A is the n x n matrix:
{ Dn, a1,n-1Dn-1, ... a1,1D, a1,0
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Dn, an,n-1Dn-1, ... an,1D, an,0 }
and F(t) is the n x 1 matrix:
f1(t)
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fn(t)
Let Bj be the matrix A with the jth column replaced by the matrix F(t). Then by Cramer's Rule:
|A|xj = |Bj|, where |Y| denotes the determinant of the matrix.
Solving for each xj gives n solutions each with identical null spaces. (Remember that the general solution of a differential equation is given by the superposition of the solution to the homogenous equation, where f(t) = 0, and the particular solution.)
It is important to remember when using this method that operators must come before the functions they operate on (i.e. tD ≠ Dt) and that unlike the usual version of Cramer's Rule, we cannot divide by the |A|. Also remember that the method of operators provides a method of turning a system of differential equation into n decoupled differential equations that can be easily solved, you will still need to solve n differential equations. Also remember that by taking the determinant of the matrix A, each differential equation may be of the order n!