Annihilator method

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the annihilator method is a procedure used to find a particular solution to certain types of <a href="/facts/Non-homogeneous_differential_equation/ygKGQ8kD">non-homogeneous</a> <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equations</a> (ODEs). It is similar to the <a href="/facts/Method_of_undetermined_coefficients/L4jKFpPE">method of undetermined coefficients</a>, but instead of guessing the particular solution in the <a href="/facts/Method_of_undetermined_coefficients/L4jKFpPE">method of undetermined coefficients</a>, 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 
 
 
 
 P
 (
 D
 )
 y
 =
 f
 (
 x
 )
 
 
 {\displaystyle P(D)y=f(x)}
 
, find another <a href="/facts/Differential_operator/Nr15J0IE">differential operator</a> 
 
 
 
 A
 (
 D
 )
 
 
 {\displaystyle A(D)}
 
 such that 
 
 
 
 A
 (
 D
 )
 f
 (
 x
 )
 =
 0
 
 
 {\displaystyle A(D)f(x)=0}
 
. This operator is called the annihilator, hence the name of the method. Applying 
 
 
 
 A
 (
 D
 )
 
 
 {\displaystyle A(D)}
 
 to both sides of the ODE gives a homogeneous ODE 
 
 
 
 
 
 (
 
 
 A
 (
 D
 )
 P
 (
 D
 )
 
 
 )
 
 
 y
 =
 0
 
 
 {\displaystyle {\big (}A(D)P(D){\big )}y=0}
 
 for which we find a solution basis 
 
 
 
 {
 
 y
 
 1
 
 
 ,
 …
 ,
 
 y
 
 n
 
 
 }
 
 
 {\displaystyle \{y_{1},\ldots ,y_{n}\}}
 
 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 <a href="/facts/Variation_of_parameters/3Ca8uY6A">variation of parameters</a> in the sense that an annihilator does not always exist.

Annihilator method open-in-new

Annihilator method