Where do I begin to solve this ODE?

Question

I've a problem:

$ye^{xy}+4y^3+(xe^{xy}+12xy^2-2y)y'=0 $

I can't begin to solve it because I don't know where to start.

Help me, please. Thanks!

Answer 1

Note that $$ \frac{d}{dx} (e^{xy}) = ye^{xy}+xy'e^{xy}, $$ so that simplifies two terms. Next, $$ \frac{d}{dx}(4xy^3) = 4y^3+12xy^2 y'. $$ Lastly, $$ \frac{d}{dx}(y^2) = 2yy', $$ and so the whole thing can be written as $$ \frac{d}{dx}f(x,y) = 0, $$ which is easy to solve. Solving the result of this for $y$, on the other hand, is not possible using elementary functions.