= Solution
The homogeneous <characteristic equation> is $r^2-r-2=(r-2)(r+1)=0$, so its solutions are $e^{2x}$ and $e^{-x}$. Because the forcing contains the latter exponential, set $y=e^{-x}v$. Direct differentiation reduces the <inhomogeneous linear differential equation> to
$$
v''-3v'=18x.
$$
A polynomial particular solution is $v=-3x^2-2x$, since $v''-3v'=-6-3(-6x-2)=18x$. Therefore
$$
y=C_1e^{2x}+e^{-x}(C_2-2x-3x^2).
$$
The exponentially growing first term cannot be canceled by the decaying second term, so boundedness at positive infinity requires $C_1=0$. The <initial condition> $y(0)=1$ sets $C_2=1$. Hence
$$
\boxed{y(x)=(1-2x-3x^2)e^{-x}.}
$$
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