- Annihilator method
mathematics, the annihilator method is a procedure used to find a particular solution to certain types of inhomogeneous ordinary differential equations. It is equivalent to the method of undetermined coefficients, and the two names are sometimes used interchangeably, although they can describe different techniques. 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 operatorsuch 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 combinations to satisfy the ODE.
This method is not as general as
variation of parametersin 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 nonhomogenous parts. In particular, is a
particular solutionto the nonhomogeneous differential equation, and is a complementary solutionto the corresponding homogeneous equation. The values of and are determined usually through a set of initial conditions. Since this is a second order equation, there would be two such conditions necessary in order to fully determine these values.
The fundamental solutions and can be further rewritten using
Then , and a suitable reassignment of the constants gives a simpler and more understandable form of the complementary solution: .
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