We consider another way of arriving at the proper form of \(Y(t)\) 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 \(D\) for \(d / d t\). Then, for example, \(e^{-t}\) is
a solution of \((D+1) y=0 ;\) the differential operator \(D+1\) is said to
annihilate, or to be an annihilator of, \(e^{-t}\). Similarly, \(D^{2}+4\) is an
annihilator of \(\sin 2 t\) or \(\cos 2 t,\) \((D-3)^{2}=D^{2}-6 D+9\) is an
annihilator of \(e^{3 t}\) or \(t e^{3 t},\) and so forth.
Consider the problem of finding the form of the particular solution \(Y(t)\) of
$$
(D-2)^{3}(D+1) Y=3 e^{2 t}-t e^{-t}
$$
where the left side of the equation is written in a form corresponding to the
factorization of the characteristic polynomial.
(a) Show that \(D-2\) and \((D+1)^{2},\) respectively, are annihilators of the
terms on the right side of Eq. and that the combined operator \((D-2)(D+1)^{2}\)
annihilates both terms on the right side of Eq. (i) simultaneously.
(b) Apply the operator \((D-2)(D+1)^{2}\) to Eq. (i) and use the result of
Problem 20 to obtain
$$
(D-2)^{4}(D+1)^{3} Y=0
$$
Thus \(Y\) is a solution of the homogeneous equation (ii). By solving Eq. (ii),
show that
$$
\begin{aligned} Y(t)=c_{1} e^{2 t} &+c_{2} t e^{2 t}+c_{3} t^{2} e^{2 t}+c_{4}
t^{3} e^{2 t} \\ &+c_{5} e^{-t}+c_{6} t e^{-t}+c_{7} t^{2} e^{-t}
\end{aligned}
$$
where \(c_{1}, \ldots, c_{7}\) are constants, as yet undetermined.
(c) Observe that \(e^{2 t}, t e^{2 t}, t^{2} e^{2 t},\) and \(e^{-t}\) are
solutions of the homogeneous equation corresponding to Eq. (i); hence these
terms are not useful in solving the nonhomogeneous equation. Therefore, choose
\(c_{1}, c_{2}, c_{3},\) and \(c_{5}\) to be zero in Eq. (iii), so that
$$
Y(t)=c_{4} t^{3} e^{2 t}+c_{6} t e^{-t}+c_{7} t^{2} e^{-t}
$$
This is the form of the particular solution \(Y\) of Eq. (i). The values of the
coefficients \(c_{4}, c_{6},\) and \(c_{7}\) can be found by substituting from Eq.
(iv) in the differential equation (i).
