Why does the method of separation of variables fail to produce solution for this first order differential equation $dy/dx=-4xy^2$ when $y(0)=0?$ $\frac{dy}{dx}=-4xy^2$
Basically, the method just says:
Rewrite and integrate both side: $$\int{\frac{1}{y^2}}dy=\int{-4x}dx$$
And get: $$y=\frac{1}{2x^2-C}$$
The only reason my book gives is that by putting y in the denominator we assume y cannot be 0 and that the method fails because the integral becomes divergent when point x=0 is included in the integral.
Can someone elaborate (maybe correct the second reason because it's nothing about x but instead the problem is y)?

The fact that $y(0)=0$ means you are working in some square $[-a,a]\times [-b,b]$ and on this square $\int{\frac{1}{y^2}}dy$ s not defined, simply because $g(y)=1/y^2$ is not defined if $y=0$
On the other hand, $y=0$ satisfies the differential equation and the initial condition, so by existence/uniqueness, it is $the$ solution.
