I'm working through a question where the differential equation is

${y}^{2}({y}^{\prime 2}-1)(3{y}^{\prime 2}+1)=c,\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}y(0)=0$

and the answer proceeds with two cases

$c=0\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}y(x)=0\vee y(x)=\pm x$

$c\ne 0\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}\underset{x\to 0}{lim}{y}^{2}{y}^{\prime 4}=c/3$

I'm fine with (1), but I can't see how to get (2). I've tried multiplying out the brackets and eliminating terms involving y only, but that doesn't get the answer so far as I can see. I'm not very familiar with how to manipulate limits, especially in the context of differential equations, and would appreciate some help.