\(\displaystyle \L\\\frac{dx}{dy}=\frac{xy^{3}}{2y^{4}+x^{4}}\)

First, rewrite like this:

\(\displaystyle \L\\(2y^{4}+x^{4})dx=(xy^{3})dy\)

\(\displaystyle \L\\=>(2y^{4}+x^{4})dx-(xy^{3})dy=0\)

\(\displaystyle \L\\(2y^{4}+y^{4}u^{4})\underbrace{(ydu+udy)}+(uy^{4})dy=0\)

The underbrace is dx, the product rule with x=yu.

Now, try grouping terms and separating variables.