I am trying to solve the following long equation for $\mu$:
$$\left((s_1-n)\mu + s_2\right) e^{\left(-\frac{1}{2\sigma^2}\right)(s_3-2\mu s_2+(n-s_1)\mu^2)}\left(e^{-\frac{(b-\mu)^2}{2\sigma^2}}-e^{-\frac{(a-\mu)^2}{2\sigma^2}}\right)^{-1}=\Omega $$
where we have $s_1\le n$, $\sigma > 0$ ,$(b-\mu)^2< (a-\mu)^2$, and $\Omega\neq 0$. All quantities are real numbers except $n$, which is a natural number, and all quantities except $\mu$ are given.
My question: Is it possible to solve/approximate this equation for $\mu$, and if yes, can you show me how this can be done?
For $\Omega=0$, the task is much easier, and I get $\mu=\frac{s_2}{n-s_1}$. For the harder case $\Omega\neq 0$, I wanted to transform the equation into the form $f(\mu,n,a,b,s_1,s_2,s_3)e^{f(\mu,n,a,b,s_1,s_2,s_3)}=\Omega$ for some "simple" function $f$ and then use the Lampert-$W$ function, but I failed.