Solution
= Solution
After division by $x^2$, Abel's identity gives $W=C/x$. <Reduction of order> from $y_1=x^{-1/2}e^{-x}$ allows the convenient choices
$$
y_2=x^{-1/2}e^x,\qquad W=\frac2x.
$$
The normalized forcing is $f=x^{3/2}$, so
$$
u'=-\frac12x^2e^x,\qquad v'=\frac12x^2e^{-x}.
$$
Taking convenient antiderivatives and simplifying gives the particular solution $-x^{3/2}-2x^{-1/2}$. Hence
$$
\boxed{y=C_1x^{-1/2}e^{-x}+C_2x^{-1/2}e^x-x^{3/2}-2x^{-1/2}.}
$$