We are in $\mathbb{R}$ and $x\geq 0$.
I have an equation: $(1-\alpha)x^{\frac{1}{1-\alpha}}-W(y)x=g(y)W(y)$
Where $\alpha \in (0,1)$, y is a parameter $0<y<1$, W and g are constants in $x$ and $W$ is increasing in y and g is decreasing in y. g is such that the equation has two solutions $x_1$ and $x_2$, in particular $g(y)<0$. Finally $W>0$.
I would like to know what is the methodology to look at the evolution of $x_2$-$x_1$ when y increases. This is not straigthforward as both the RHS and the LHS of the equation then decrease. Is comparing the derivative, holding x constant enough?
Thank you for your help
All the best T.