another method of solving this ODE..

**ashes** · April 9th 2008, 01:23 AM

another method of solving this ODE..
$(3x^2y-y^3)dx-(3xy^2-x^3)dy=0$
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..

**roy_zhang** · April 9th 2008, 07:58 AM

Originally Posted by ashes

another method of solving this ODE..
$(3x^2y-y^3)dx-(3xy^2-x^3)dy=0$
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..

I think you could try to divide both sides of the D.E. by $x^3$ , this will transform the given D.E. into the form $\left(3\left(\frac{y}{x}\right)-\left(\frac{y}{x}\right)^3\right)dx-\left(3\left(\frac{y}{x}\right)^2-1\right)dy=0$
then you can introduce a new dependent variable $u(x)=\frac{y}{x}$ (or $y=xu(x)$ ), and express $\frac{dy}{dx}$ in terms of $x,\;u$ and $\frac{du}{dx}$ . Now you have a separable D.E.

Roy

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