= Solution
The dependent variable itself does not occur, so reduce order by putting $p=y'$. Then $p'+2p=3x$, a <first-order linear differential equation>. Its <integrating factor> is $e^{2x}$:
$$
(e^{2x}p)'=3xe^{2x},\qquad
p=\frac32x-\frac34+Ce^{-2x}.
$$
Integrating once more and renaming the constants gives
$$
\boxed{y(x)=\frac34x^2-\frac34x+A+Be^{-2x}.}
$$
The two homogeneous terms are the constant solution and the exponential associated with the roots $0,-2$ of the <characteristic equation>.
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