As my main goal is to find the path, or change in $\theta$ value as $k$ changes.
So I am trying to find the solution value for $\theta$.
The equation that I want to solve is :
given two gamma distributions $f_{1}(x), f_2(x)$, and $k$ is a constant,
$f_1(\theta)\left(2\int_{0}^{\theta}f_1(x)\mathrm dx -1\right) = kf_2(\theta)\left(2\int_{0}^{\theta}f_2(x)\mathrm dx -1\right)$
I thought about solving with Laplace transformation, but it will have convolution terms and get too complex. Do you have any thoughts or is this even feasible?
Thank you in advance!