As part of my research I get the following differential equation. I need to solve for $\mathcal{V}(\gamma )$. In fact the requirement is not to solve but to show that $\mathcal{V}(\gamma )$ is monotonic in ${a}_{j}$$\mathrm{\forall}j$, (which I hope it is) where ${a}_{j}$ are positive valued constants which do not depend on $\gamma $. If it can be shown without solving the differential equation that is sufficient. Please provide some suggestions.

$\frac{{\textstyle \gamma}}{{\textstyle \mathrm{log}\left(e\right)}}\frac{{\textstyle d}}{{\textstyle d\gamma}}\mathcal{V}\left(\gamma \right)=1-\eta \left(\gamma \right)$

$\eta \left(\gamma \right)=\frac{{\textstyle 1}}{{\textstyle 1+\gamma \sum _{j}\frac{{\textstyle {a}_{j}}}{{\textstyle 1-{a}_{j}(-\gamma \eta \left(\gamma \right))}}}}$

where $j=\{1,\dots ,n\}$

$\frac{{\textstyle \gamma}}{{\textstyle \mathrm{log}\left(e\right)}}\frac{{\textstyle d}}{{\textstyle d\gamma}}\mathcal{V}\left(\gamma \right)=1-\eta \left(\gamma \right)$

$\eta \left(\gamma \right)=\frac{{\textstyle 1}}{{\textstyle 1+\gamma \sum _{j}\frac{{\textstyle {a}_{j}}}{{\textstyle 1-{a}_{j}(-\gamma \eta \left(\gamma \right))}}}}$

where $j=\{1,\dots ,n\}$