(a) Show that
\(\begin{aligned}(D-a) e^{c x} &=(c-a) e^{c x} \\\\\left(D^{2}+5 D-3\right)
e^{c x} &=\left(c^{2}+5 c-3\right) e^{c x} \end{aligned}\)
\(L(D) e^{c x}=L(c) e^{c x},\) where \(L(D)\) is any polynomial in \(D\) \((D-c) x
e^{c x}=e^{c x}\)
\((D-c)^{2} x^{2} e^{c x}=2 e^{c x}\)
(b) Define the expression \(y=[1 / L(D)] u(x)\) to mean a solution of the
differential equation \(L(D) y=u .\) Using part (a), show that
$$\begin{aligned}
\frac{1}{D-a} e^{c x} &=\frac{e^{c x}}{c-a}, \quad c \neq a \\
\frac{1}{D^{2}+5 D-3} e^{c x} &=\frac{e^{c x}}{c^{2}+5 c-3} \\
\frac{1}{L(D)} e^{c x} &=\frac{e^{c x}}{L(c)}, \quad L(c) \neq 0 \\
\frac{1}{D-c} e^{c x} &=x e^{c x} \\
\frac{1}{(D-c)^{2}} e^{c x} &=\frac{1}{2} x^{2} e^{c x}
\end{aligned}$$
(c) The expressions \(1 / L(D)\) in (b) are called inverse operators. They can
be used to find particular solutions of differential equations. As an example
consider Problem 3. We write
$$\begin{array}{c}
\left(D^{2}+D-2\right) y=e^{2 x} \\
y=\frac{1}{D^{2}+D-2} e^{2 x}=\frac{e^{2 x}}{2^{2}+2-2}=\frac{e^{2 x}}{4}
\end{array}$$
Using inverse operators, find particular solutions of Problems 4 to \(20 .\) Be
careful to use parts 4 or 5 of \((\mathrm{b})\) if \(c\) is a root of the
auxiliary equation. For example,
$$\frac{1}{(D-a)(D-c)} e^{c x}=\frac{1}{D-c} \frac{1}{D-a} e^{c
x}=\frac{1}{D-c} \frac{e^{c x}}{c-a}=\frac{x e^{c x}}{c-a}$$