I need to find $x \in \mathbb{R}$ such that:
$a\Phi(x)x+b\Phi(x)+cx+d=\phi(x)$
Where $a,b,c,d$ are constants, $\phi(x)$ is the pdf of a standard normal distribution and $\Phi(x)$ the cdf of a standard normal distribution.
I will use eventually numerical methods to find a solution, but I was wondering if some analytical result is possible.
Thanks