I came across the following problem: solve
$(a^2+s^2)\sinh s + 2 b s = 2 s \cosh s$
for $s$, with $a$ and $b$ real parameters.
Is there anything known about the solution(s) of this problem?
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If $s$ is a solution then $-s$ is also a solution, so by symmetry you can already say that $0$ is a solution. The problem then reduces to finding a positive solution – Oussama Boussif May 12 '17 at 09:06
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Setting directly $s=0$ indeed shows that it is a solution. However there are cases where other solutions exists (I know this by numerical experimentation). – arovai May 12 '17 at 09:08
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I didn't say that other solutions do not exist, I just gave one root, we should investigate further to see wether there exists another one. – Oussama Boussif May 12 '17 at 09:21
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This equation looks interesting. I am afraid that only numerical methods could be used. Are they particular ranges for $a,b$ ? – Claude Leibovici May 12 '17 at 10:17
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In my problem I would take $|b|\leq 1$ but nothing else. – arovai May 12 '17 at 10:21