Solution
= Solution
{c}
After division by $x^2$, the <linear ordinary differential equation> becomes
$$
y'-\frac{y}{x^2}=\frac{e^{-1/x}}{x^2}.
$$
The <integrating factor> is $e^{1/x}$. Thus
$$
\frac{d}{dx}\left(e^{1/x}y\right)=\frac1{x^2},\qquad e^{1/x}y=-\frac1x+C.
$$
The <initial condition> $y(1)=1$ gives $C=e+1$, so
$$
\boxed{y(x)=\left(e+1-\frac1x\right)e^{-1/x},\qquad x>0.}
$$