 Solution

On either connected half-line, the homogeneous linear differential equation isIntegration gives $\boxed{y_h=C{x}^{-2}{e}^{-1/x}}$, including the zero solution. For the inhomogeneous equation choose the positive integrating factorIt satisfies ${\mu }^{\prime }/\mu =2/x-1/x^2$. Multiplication gives the exact derivativeHence all solutions on either half-line areFor $a\ne 0$, impose $y(a)=b$ to obtain $C=a^2{e}^{1/a}b-\arctan a$. The complete initial value problem solution is thereforeIt is smooth throughout the maximal connected interval $(0,\infty )$ if $a>0$, and $(-\infty ,0)$ if $a<0$.