# As part of my research I get the following differential equation. I need to solve for <mrow class

As part of my research I get the following differential equation. I need to solve for $\mathcal{V}\left(\gamma \right)$. In fact the requirement is not to solve but to show that $\mathcal{V}\left(\gamma \right)$ 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.
$\frac{\gamma }{\mathrm{log}\left(e\right)}\frac{d}{d\gamma }\mathcal{V}\left(\gamma \right)=1-\eta \left(\gamma \right)$
$\eta \left(\gamma \right)=\frac{1}{1+\gamma \sum _{j}\frac{{a}_{j}}{1-{a}_{j}\left(-\gamma \eta \left(\gamma \right)\right)}}$
where $j=\left\{1,\dots ,n\right\}$
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Tristin Case
It seems that ${\eta }_{a}\left(\gamma \right)$ solves the equation ${F}_{\gamma }\left({\eta }_{a}\left(\gamma \right),a\right)+n=1$ where
${F}_{\gamma }\left(h,a\right)=h-\sum _{i}\frac{1}{1+\gamma h{a}_{i}}.$
Each function is increasing. Each function is increasing. Hence $a↦{\eta }_{a}\left(\gamma \right)$ is decreasing, for each $\gamma$. This proves that $a↦\frac{\mathrm{d}}{\mathrm{d}\gamma }{V}_{a}\left(\gamma \right)$ is increasing.
Thus, assuming that there exists some ${\gamma }_{0}$ such that ${V}_{a}\left({\gamma }_{0}\right)$ does not depend on a, one sees that $a↦{V}_{a}\left(\gamma \right)$ is increasing if $\gamma >{\gamma }_{0}$ and decreasing if $\gamma <{\gamma }_{0}$.
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