Let $Y \sim \mathcal{N}(\mu,\sigma^2)$ and $F_Y(y)$ denote its cummulative distribution function.
Any hint to solve the equation below for unknown $x$, known constant $a,b,c$?
$$1-F_Y\left(a\sqrt{x}+bx\right) = cF_Y(bx)$$
I know that $0 < c, x <1$ and $a,b$ have values assuring the equation has solution (the solution can be found by using exhaustive search in [0,1] but I expect an analytical solution).
I got stuck because the CDF involves integral. If there were only one CDF, it should be ok with Q-function. But with two of them, I don't know where to start.