Undetermined coefficients—
Annhilator Approach

• If L is a linear differential operator with constant coefficients and f is a sufficiently differential function such that L(f(x))=0 then L is said to be an annihilator of the function.

n
The Differential operator D annihilates each of
2

1 the functions: 1, x, x ,..., x n
( D
 
) n The Differential operator annihilates each of the functions e
 x
, xe
 x
,..., x n

1 e
 x
The Differential operator [ D
2 
2

D

(

2  
2
)] n annihilates each of the functions e
 x cos
 x , xe
 x cos
 x ,...
x n

1 e
 x cos
 x and e
 x sin
 x , xe
 x sin
 x ,...
x n

1 e
 x sin
 x

The differential operator that annihilates a function is not unique. When we are trying to determine an appropriate annihilator, we want the operator of lowest possible order that does the trick.

Assuming that our linear differential equation has constant coefficients and the function g(x) consists of finite sums and products of constants, polynomials, natural exponential functions, sines and cosines, the annihilator approach goes something like this….

1) Find the complementary solution for the homogeneous equations L(y)=0
2) Operate on both sides of the nonhomogeneous equation with a differential operator L1 that annihilates the function g(x).
3) Find the general solution of the higher order homogeneous differential equation
L1(L(y))=0

• 4) Delete from the solution in step 3 all those terms that are duplicated n the complementary solution y . Form a linear c combination of the terms that remain. That is c the form of a particular solution.
• 5) Substitute and equate coefficients