I am given this equation
$$(4y-10x)dx+(4x-6x^2y^{-1})dy=0$$
where I must find an integrating factor to turn this into an exact differential. The integrating factor is supposed to be in the form $\mu=x^ny^m$.
I have found $M_{y}=4$ and $N_{x}=(4-12xy^{-1})$. It is here where I get stuck. How do I go about finding the integrating factor in the form $\mu=x^ny^m$, and what would the $n$ and $m$ end up being? Any help would be greatly appreciated!
Best Answer
In comments you find $x^3y^3$ as integrating factor of your equation. It is nice approach you found and here is another. With $M_{y}=4$ and $N_{x}=(4-12x\dfrac{1}{y})$ we have $$p(z)=\dfrac{M_y-N_x}{Ny-Mx}=\dfrac{\frac{12x}{y}}{4x^2}=\dfrac{3}{xy}=\dfrac{3}{z}$$ means your integrating factor is of the form $\mu(z)=\mu(xy)$. So $$I=e^{\int p(z)dz}=z^3=(xy)^3$$ and finally the answer is $$\color{blue}{x^4y^4-2x^5y^3=C}$$