mixed variable integration coefficient

**transgalactic** · November 11th 2010, 01:25 PM

$(y^{4}-4xy)dx+(2xy^{3}-3x^{2})dy=0$
prove that this equation has an xy dependant integration coefficient

**Krizalid** · November 11th 2010, 03:29 PM

show that exist constants that verify $\displaystyle\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}=m\frac{N}{x}-n\frac{M}{y}.$

**transgalactic** · November 13th 2010, 05:03 AM

why??
if the left side equals zero then we have potential and could be solved without multilying by integration coefficient.
what the right side says ?
why this equation says about the coefficient
?

**Krizalid** · November 13th 2010, 06:29 AM

the condition i gave you gives you the integrating factor, if those constants exist, then the integrating factor is $u(x,y)=x^my^n,$ but on your question you were asked to show that your equation has a integrating factor which depends of $xy,$ so that will force that $m=n$ in order to have $u(x,y)=h(xy).$