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:

d^{n}x_{1}/dt^{n} + ... + a_{1,1}dx_{1}/dt + x_{1} + a_{1,0} = f_{1}(t)

.

.

.

d^{n}x_{n}/dt^{n} + ... + a_{n,1}dx_{n}/dt + x_{n} + a_{n,0} = f_{n}(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:

{ D^{n}, a_{1,n-1}D^{n-1}, ... a_{1,1}D, a_{1,0}

.

.

.

D^{n}, a_{n,n-1}D^{n-1}, ... a_{n,1}D, a_{n,0} }

and F(t) is the n x 1 matrix:

f_{1}(t)

.

.

.

f_{n}(t)

Let B_{j} be the matrix A with the jth column replaced by the matrix F(t). Then by Cramer's Rule:

|A|x_{j} = |B_{j}|, where |Y| denotes the determinant of the matrix.

Solving for each x_{j} 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!