# [SOLVED] Tricky First Order ODE

• Apr 4th 2010, 06:52 PM
jameselmore91
[SOLVED] Tricky First Order ODE
$y\left( y^{3}\; -\; x \right)dx\; +\; x\left( y^{3}\; +\; x \right)dy\; =\; 0$

Can't seem to find the family of solutions...

Any ideas?
• Apr 4th 2010, 07:11 PM
jameselmore91
Finally got it!

Regroup the original equation to get:

$y^{3}\left( ydx\; +\; xdy \right)\; +\; x^{2}dy\; -\; xydx\; =\; 0$
or:
$y^{3}\left( xy \right)'\; +\; x^{2}dy\; -\; xydx\; =\; 0$

Divide through by $$y^{3}$$ and eventually get it into the form:

$2\left( xy \right)'\; -\; \left( \frac{x^{2}}{y^{2}} \right)'\; =\; 0$

And a final family of solutions in the form of:

$2xy^{3}\; -\; x^{2}\; =\; cy^{2}$