another method of solving this ODE..

$\displaystyle (3x^2y-y^3)dx-(3xy^2-x^3)dy=0$

I ve proved that it is an exact differential equation ..

I need to state another method with reasons of solving this ODE...

Thanks..mate..whoever u are..

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- Apr 9th 2008, 01:23 AMashesanother method of solving this ODE..
another method of solving this ODE..

$\displaystyle (3x^2y-y^3)dx-(3xy^2-x^3)dy=0$

I ve proved that it is an exact differential equation ..

I need to state another method with reasons of solving this ODE...

Thanks..mate..whoever u are.. - Apr 9th 2008, 07:58 AMroy_zhang
I think you could try to divide both sides of the D.E. by $\displaystyle x^3$, this will transform the given D.E. into the form $\displaystyle \left(3\left(\frac{y}{x}\right)-\left(\frac{y}{x}\right)^3\right)dx-\left(3\left(\frac{y}{x}\right)^2-1\right)dy=0$

then you can introduce a new dependent variable $\displaystyle u(x)=\frac{y}{x}$ (or $\displaystyle y=xu(x)$), and express $\displaystyle \frac{dy}{dx}$ in terms of $\displaystyle x,\;u$ and $\displaystyle \frac{du}{dx}$. Now you have a separable D.E.

Roy