Solving $\sinh x = kx$ Can we solve the equation $\sinh x = kx$ for $x$ in terms of elementary functions? I've tried reexpressing the hyperbolic sine as exponentials and converting the equation into a quadratic in $e^x$, but this doesn't seem to make the problem any easier. I've considered expanding $\sinh x$ as a Taylor series, but this doesn't seem useful either.

As mentioned in the comments, the only solution for $k\le1$ is $x=0$ and for $k>1$ there are 3 solutions: $x=0,\pm x_\star$. Although there is no closed form in terms of special functions such as the Lambert W function known, it is not hard to numerically compute $x_\star$, the positive nonzero solution. For example, we have fixed-point iteration:
$$x_{n+1}=\ln(2kx_n+\exp(-x_n))$$
or Newton's method:
$$x_{n+1}=x_n-\frac{\sinh(x_n)-kx_n}{\cosh(x_n)-k}$$
or any other numerical method.
