= Solution
Since $y=x^{-1/2}u$, the two linearly independent solutions of the original equation have the <asymptotic expansion>[asymptotic expansions]
$$
\boxed{
\begin{aligned}
y_{\rm grow}(x)
&\sim x^{1/2}e^{1/x}
\left(1+\frac x8+\frac{9x^2}{128}+\cdots\right),\\
y_{\rm decay}(x)
&\sim x^{1/2}e^{-1/x}
\left(1-\frac x8+\frac{9x^2}{128}+\cdots\right).
\end{aligned}}
$$
As a check, the substitution $z=1/x$ converts the equation into the modified Bessel equation of order zero, whose solutions $I_0(z)$ and $K_0(z)$ have exactly these large-$z$ expansions.
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