<s>[INST] $\frac{dy}{dx} - {8} -{2}x^2+{4}y^2+y^2x^2 = 0.$ how should I procced from here having the equation $$\frac{dy}{dx} - {8} -{2}x^2+{4}y^2+y^2x^2 = 0.$$
I am getting to the following $$\frac{1}{2^{\frac{3}{2}}}\ln \left(y+\sqrt{2}\right)-\frac{1}{2^{\frac{3}{2}}}\ln \left(y-\sqrt{2}\right)=\frac{x^3}{3}+4x+c$$ from here I can do $\frac{1}{2^{\frac{3}{2}}}(ln\frac{y+\sqrt(2)}{y-\sqrt(2)})=\frac{x^3}{3}+4x+c$
how should I get none implicit function of $y$?
 [/INST] $$\frac{y+\sqrt{2}}{y-\sqrt{2}}=\exp\left[2^{3/2}\left(\frac{x^3}{3}+4x+c\right)\right]$$
to make a few steps easier I will call the RHS $f(x)$:
$$y+\sqrt{2}=(y-\sqrt{2})f(x)$$
$$y(1-f(x))=-\sqrt{2}(1+f(x))$$
and so:
$$y=-\sqrt{2}\frac{1+f(x)}{1-f(x)}$$
</s>