# Is this an exact differential equation or a first order non-linear ordinary differential equation?

I was trying to solve this : $$\frac{dy}{dx}\left(\frac{y^2}{x^3}-x\right)=\frac{y^3}{x^4}+y$$ using the exact equations method, but the final answer was getting very ugly with this form:

$$\frac{y^3}{3x^3}-xy=c$$

is there any other way that is simpler to solve this equation? I checked this on wolfram and it gave me that it is a first order non-linear ordinary differential equation.

Any hints are appreciated!

• It is a nonlinear first order ODE: the highest order derivative is of first order and it has terms with $y^2$ and $y^3$. However, it's also an exact ODE (as the very link you provided from WolframAlpha states). Your solution is a cubic equation on $y$, that seems to be solvable as an explicit solution for $y$, as WolframAlpha shows. If it's a homework assignment, I think that the solution you found is perfectly fine. – rafa11111 Dec 2 '18 at 22:14