Identify the integrating factor for $x > 0$ for the first order linear equation $xw'+ w = 0$
I'm really confused how to identify the integrating factor for $xw'+ w = 0$ because it appears to be solvable by separation of variables, without the need to find $\mu(t)$.
$$x\frac{dw}{dx}+w=0$$ $$x\frac{dw}{dx}=-w$$ $$\frac{x}{dx}=\frac{-w}{dw}$$ $$\frac{dx}{x}=\frac{dw}{-w}$$
When integrating:
$$\int\frac{1}{x}\,dx=\int\frac{1}{-w}\,dw$$ $$\ln(x)=-\ln(w)+\ln(a)$$ $$\ln(x)=\ln\left(\frac{a}{w}\right)$$
Again, I don't see how an integrating factor would come into play with this separable, first order homogenous linear equation. What would it be?
Any help would be greatly appreciated!