= Solution
On an <interval> with $x\ne0$, division by $x^2$ gives the <separable differential equation> $y'=(4/x^2-1/x)y^2$. For $y\ne0$, take <antiderivatives>:
$$
\int\frac{dy}{y^2}=\int\left(\frac4{x^2}-\frac1x\right)dx,\qquad -\frac1y=-\frac4x-\ln|x|+C.
$$
Renaming the constant yields the explicit nonzero family
$$
\boxed{y(x)=\frac1{4/x+\ln|x|+C}.}
$$
For $x\ne0$, the <interval> of definition must avoid zeros of the denominator. Division by $y^2$ excluded the additional solution $\boxed{y(x)\equiv0}$, which must also be included.
The original undivided <differential equation> also makes sense at $x=0$, where it requires $y(0)=0$. For <continuation through a zero of the leading ODE coefficient>, rewrite the nonzero family as
$$
y(x)=\frac{x}{4+x\ln|x|+Cx}.
$$
It extends differentiably with $y(0)=0$ and $y'(0)=1/4$, since $y(x)/x\to1/4$. One may choose the integration constants independently on the two sides of zero and obtain a differentiable solution through zero. The zero solution also extends through zero, with <derivative> $0$; joining a nonzero branch on just one side to a zero branch on the other would fail <differentiability>.
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