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I need to solve a problem in which I find an equation like:

$$C=e^{a*x}(A*\cosh(b*x)+B*\sinh(b*x))$$

I would like to express $x$ in function of $C,a,b,A,B$.

However I am starting to wonder if it is simply possible to find analytic solution to this... Is it a kind of non solvable analytically transcendent equation ?

StarBucK
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1 Answers1

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Hint:

put $$A= M\sinh(b\,c)\quad B=M\cosh(b\,c)$$ which invert to $$B^2-A^2=M^2 \quad \tanh(bc)=a/B$$

Then use the fact that $\sinh(x+y)=\sinh(x)\cos(y)+\sinh(y) \cosh(x)$

G Cab
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  • Thank you for your answer. If I follow you method I would end up with $\frac{C}{M}=e^{ax}sinh(b(c+x))$ but then I don't see either how this can be solved. If I expand I could have on the right handside a sum involving two exponentials which I don't see how it can be solved analytically but maybe I'm confused – StarBucK Oct 25 '19 at 23:41
  • You are right, and I don't see how further you can go analytically . – G Cab Oct 26 '19 at 00:37
  • See related Math Overflow solution after substituting $b(c+x)\to x$ – Тyma Gaidash Nov 11 '23 at 15:38