$\displaystyle y\left( y^{3}\; -\; x \right)dx\; +\; x\left( y^{3}\; +\; x \right)dy\; =\; 0$

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

Any ideas?

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- Apr 4th 2010, 06:52 PMjameselmore91[SOLVED] Tricky First Order ODE
$\displaystyle y\left( y^{3}\; -\; x \right)dx\; +\; x\left( y^{3}\; +\; x \right)dy\; =\; 0$

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

Any ideas? - Apr 4th 2010, 07:11 PMjameselmore91
Finally got it!

Regroup the original equation to get:

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

or:

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

Divide through by [Math]y^{3}[/tex] and eventually get it into the form:

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

And a final family of solutions in the form of:

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