How can I solve for $x\,$? $:\;\;x-x_r=(x-x_1)e^{\large -(x-x_1)^2}$

Question

I want to find $x$ for given values of $x_r$ and $x_1\,$ (domain $\mathbb{R}$):

$$x-x_r=(x-x_1)e^{\large -(x-x_1)^2}$$

Thanks

Answer 1

You can rewrite the equation as ${{x-x_r}\over{x-x_1}}=e^{-(x-x_1)^2}$ and graph each side. The left is a hyperbola with a vertical asymptote at $x=x_1$ and a horizontal one at $y=1$. The right side is a bell shaped curve with max at $(x_1,1)$. Now decide based on $x_r>x_1$ or $x_r<x_1$ to see where the unique root is going to be. Next you my try to get some asymptotics of the answer. It all depends.