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In mathematics, the annihilator method is a procedure used to find a particular solution to certain types of inhomogeneous ordinary differential equations. 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.
The annihilator method is used as follows. Given the ODE , find another differential operator such that . This operator is called the annihilator, thus giving the method its name. Applying to both sides of the ODE gives a homogeneous ODE for which we find a solution basis 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.
Given , . The simplest annihilator of is . The zeros of are , so the solution basis of is
Setting we find
giving the system
which has solutions
giving the solution set
This solution can be broken down into the homogeneous and nonhomogeneous parts. In particular, is a particular integral for the nonhomogeneous differential equation, and is a complementary solution to the corresponding homogeneous equation. The values of and 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.
The fundamental solutions and can be further rewritten using Euler's formula:
Then , and a suitable reassignment of the constants gives a simpler and more understandable form of the complementary solution, .