We consider another way of arriving at the proper form of for use in
the method of undetermined coefficients. The procedure is based on the
observation that exponential, polynomial, or sinusoidal terms (or sums and
products of such terms) can be viewed as solutions of certain linear
homogeneous differential equations with constant coefficients. It is
convenient to use the symbol for . Then, for example, is
a solution of the differential operator is said to
annihilate, or to be an annihilator of, . Similarly, is an
annihilator of or is an
annihilator of or and so forth.
Show that linear differential operators with constant coefficients obey the
commutative law, that is,
for any twice differentiable function and any constants and The
result extends at once to any finite number of factors.