<s>[INST] Common integrating factor of two differential equations. The following differential equations have a common integrating factor.

$$(3y+4xy^2)dx+(4x+5x^2y)dy=0$$
$$(6y+x^2y^2)dx+(8x+x^3y)dy=0$$

I have to find this integrating factor.
I have arrived to the solution of this problem assuming that the integrating factor I was trying to find was of the form $x^ny^m$, and in this way, I have found that the common integrating factor is $x^2y^3$.
However, I was wondering if there might be a more direct way to find this integrating factor, without having to suppose that is has the form $x^ny^m$. Any ideas?
Thanks.
 [/INST] This may help
Let us re-write the first one as:
$$\frac{xdy}{ydx}=-\frac{3+6xy}{4+5xy}\implies\frac{xdy}{ydx}+1=-\frac{3+6xy}{4+5xy}+1. $$
$$\implies \frac{xdy+ydx}{ydx}=\frac{1-xy}{4+5xy}$$
$$\implies \frac{dv}{ydx}=\frac{1-v}{4+5v}, xy=v \implies \frac{dv}{dx}=\frac{v-v^2}{4+5v} \frac{1}{x}.$$
$$\implies \int \frac{4+5v}{v-v^2}dv=\int \frac{dx}{x}$$
$$\implies \ln \left( \frac{v^4}{(1-v)^9} \right)=\ln Cx \implies x^4y^4=Cx(1-xy)^9.$$
The second one can be re-written as:
$$\frac{xdy}{ydx}=-\frac{6+x^2y}{8+x^2y} \implies \frac{x^2dy}{2xydx}=-\frac{1}{2}~\frac{6+x^2y}{8+x^2y}. $$
$$\implies \frac{x^2dy}{2xydx}+1=1-\frac{1}{2}~\frac{6+x^2y}{8+x^2y}$$
$$\implies \frac{x^2dy+2xydy}{2xydx}=\frac{10+x^2y}{2(8+x^2y)}$$
Let $x^2y=u$, then 
$$\int \frac{2(8+u)du}{10u+u^2}=\int \frac{dx}{x}$$
$$\implies \ln u^8 +\ln (10+u)^2=\ln (Cx)^5$$
$$\implies u^8(10+u)^2=(Cx)^5, u=x^2y$$
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