A solution of $y=2x \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)+x^2 {\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}^4$ is

Writing $p=\frac{\mathrm{d}y}{\mathrm{d}x}$ and differentiating w.r.t. $x$,

 we have 

$\begin{array}{l}p=2p+2x \frac{\mathrm{d}p}{\mathrm{d}x}+2x p^4+4p^3 x^2\frac{\mathrm{d}p}{\mathrm{d}x}\\ \Rightarrow 0=p\left(1+2x p^3\right)+2x \frac{\mathrm{d}p}{\mathrm{d}x}\left(1+2p^3 x\right)\\ \Rightarrow p+2x \frac{\mathrm{d}p}{\mathrm{d}x}=0\Rightarrow 2 \frac{\mathrm{d}p}{p}=-\frac{\mathrm{d}x}{x}\\ \Rightarrow 2 \log p+\log x=\text{ const }\Rightarrow p^{2}x=c\text{ or }p=\sqrt{\frac{c}{x}}\end{array}$

Substituting this value in the given equation, we get

$y=2\sqrt{c x}+c^2$