gen. soln. to diff. eqn: dx/dy= (xy^3)/(2y^4 + x^4)

differential equation dx/dy= (xy^3)/(2y^4 + x^4) is not separable.
use the subst. x=yu and simplify( y and u are separable). find the general solution.
?

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

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.