Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 23
Method of Annihilators
In this chapter we will return to using the D operator to represent the
derivative operator. In particular, we will be interested the general n th
order linear equation with constant coefficients
a n D n y + a n−1 D n−1 y + · · · + a 1 Dy + a 0 y = g(t) (23.1)
which we will represent as
where L is the operator
Ly = g(t) (23.2)
L = a n D n + a n−1 D n−1 + · · · + a 1 D + a 0 (23.3)
As we have seen earlier (see chapter 15) the L operator has the useful
property that it can be factored
L = (D − r 1 )(D − r 2 ) · · · (D − r n ) (23.4)
where r 1 , . . . , r n are the roots of the characteristic equation
a n r n + a n−1 r n−1 + · · · + a 1 r + a 0 = 0 (23.5)
Definition 23.1. An operator L is said to be an annihilator of a function
f(t) if Lf = 0.
A solution of a linear homogeneous equation is then any function that can
be annihilated by the corresponding differential operator.
Theorem 23.2. D n annihilates t n−1 .
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