Method of operators

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 + ... + a_1,1dx₁/dt + x₁ + a_1,0 = f₁(t)

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dⁿx_n/dtⁿ + ... + a_n,1dx_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:

{ Dⁿ, a_1,n-1D^n-1, ... a_1,1D, a_1,0

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Dⁿ, a_n,n-1D^n-1, ... a_n,1D, a_n,0 }

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

f₁(t)

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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!