Solution
= Solution
Put
$$
Q(x)=-q(x)=\frac14+\frac2x+\frac2{x^2}>0
$$
for sufficiently large $x$. The <Liouville–Green approximation> gives
$$
u_\pm(x)\sim Q(x)^{-1/4}\exp\left(\pm\int^x\sqrt{Q(t)}\,dt\right).
$$
Since
$$
\sqrt{Q(x)}=\frac12+\frac2x+O(x^{-2}),
\qquad
Q(x)^{-1/4}=\sqrt2\,[1+O(x^{-1})],
$$
we obtain, after absorbing nonzero constants into the two solutions,
$$
\boxed{u_+(x)\sim x^2e^{x/2},\qquad u_-(x)\sim x^{-2}e^{-x/2}}.
$$