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# The particular integral of the differential equation $f(D)y=e^{ax}$ where $\;.f(D)=(D-a)g(D).g(a)\neq 0$ is

$\begin{array}{1 1}(1)me^{ax}&(2)\frac{e^{ax}}{g(a)}\\(3)g(a)e^{ax}&(4)\frac{xe^{ax}}{g(a)}\end{array}$

$f(D)y =e^{ax}$
$P.I= \large\frac{e^{ax}}{f(D)}$
$\qquad= \large\frac{e^{ax}}{(D-a)g(D)}$
$\qquad = \large\frac{1}{g(a)}. \frac{e^{ax}}{D-a}$
$\qquad=\large\frac{xe^{ax}}{g(a)}$