Simplifying $ \frac{\sinh x}{xe^{\frac{1}{2}x^2}} = C \frac{\sinh y}{ye^{\frac{1}{2}y^2}} $

Question

I am searching for a simpler way of expressing the relationship between x and y given by the equation in the title where C is a known constant.

I recognize that the equation is of the form f(x)=Cf(y) where f is an even function that is increasing on (0,infinity). So when C = 1, we have |x|=|y|.

Any insight into the case when C is not 1 would be appreciated!