Find $x$ such that \begin{equation} x\tanh(x\sqrt{2\alpha})=\frac{2}{\sqrt{2\alpha}} \end{equation}
2026-03-28 20:07:52.1774728472
How to find $x$ such that $f(x)$ takes a prescribed value
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Your equation is equivalent to $$x\sqrt{2\alpha}\tanh(x\sqrt{2\alpha})=2$$ so your question boils down to solving $$t\tanh(t)=2.$$ This I believe cannot be solved analytically, but you approximately have $t \approx \pm 2.07 \approx 2$, whence $x \approx \pm 2.07/\sqrt{2\alpha} \approx \sqrt{2/\alpha}$.