In mathematics, the annihilator method is a procedure used to find a particular solution to certain types of non-homogeneous ordinary differential equations (ODEs). It is similar to the method of undetermined coefficients, but instead of guessing the particular solution in the method of undetermined coefficients, the particular solution is determined systematically in this technique. The phrase undetermined coefficients can also be used to refer to the step in the annihilator method in which the coefficients are calculated.

, find another differential operator. This operator is called the annihilator as before. Then the original inhomogeneous ODE is used to construct a system of equations restricting the coefficients of the linear combination to satisfy the ODE.

This method is not as general as variation of parameters in the sense that an annihilator does not always exist.

Annihilator table

is in the natural numbers are in the real numbers.

consists of the sum of the expressions given in the table, the annihilator is the product of the corresponding annihilators.

Example

we find

giving the system

which has solutions

giving the solution set

is a particular integral are determined usually through a set of initial conditions. Since this is a second-order equation, two such conditions are necessary to determine these values.

can be further rewritten using Euler's formula:

.

References