The solution of differential equation  $2x^{3}ydy+(1-y^2)(x^2{y}^2+y^2-1)dx=0$  is:

$2x^{3}ydy+(1-y^2)(x^2{y}^2+y^2-1)dx=0$

$\Rightarrow \frac{2y}{{(1-y^2)}^2}\frac{dy}{dx}+\frac{y^2}{1-y^2}\frac{1}{x}=\frac{1}{x^3}$

Put $\frac{y^2}{1-y^2}=u.$

Then $\frac{2y}{{(1-y^2)}^2}\frac{dy}{dx}=\frac{du}{dx}$

$\Rightarrow \frac{du}{dx}+\frac{u}{x}=\frac{1}{x^3}\Rightarrow$ Integrating factor $=e^{\int \frac{1}{x}dx}=e^{\ln x}=x$
Solution is  $u.x=\int \frac{1}{x^2}dx+c$

$\Rightarrow x^2{y}^2=(cx-1)(1-y^2)$