# Thread: mixed variable integration coefficient

1. ## mixed variable integration coefficient

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

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

3. 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
?

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

5. how you got this formula?
so if u(x,y)=xy then m=n=1
$\displaystyle 4y^3-4x-(2y^3-6x)=m\frac{2xy^3-3x^2}{x}-n\frac{y^4-4xy}{y}$
what to do now?

6. c'mon, simplify, do the algebra.

7. $\displaystyle 4y^3-4x-(2y^3-6x)=n(2xy^4-3x^2y-y^4x-4x^2y)$
$\displaystyle \frac{4y^3-4x-(2y^3-6x)}{(2xy^4-3x^2y-y^4x-4x^2y)}=n$
i cant get a number out of it