Finding the Root of a Transcendental Equation with Sinh

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SUMMARY

The discussion centers on solving the transcendental equation 2a sinh(25/a) = 51 for the variable a. The user attempted to manipulate the equation into a form involving exponentials but encountered difficulties due to the multiplicative factor of a. The consensus is that this equation cannot be solved algebraically and requires numerical methods or the Lambert W function for a solution.

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Shukie
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I have the following equation:

[tex]2a \sinh(\frac{25}{a}) = 51[/tex]

How do I solve this for a?

I tried changing it to:

[tex]a(e^{\frac{25}{a}} - e^{\frac{-25}{a}}) = 51[/tex], but that didn't get me any further. Anyone?
 
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Well, if it weren't for that overall multiplicative factor of [itex]a[/itex] on the LHS, you could multiply both sides by e^(25/a) to get a quadratic equation in terms of e^(25/a). But, that [itex]a[/itex] ruins everything, leaving you with a transcendental equation that I'm afraid you'll have to solve numerically (or perhaps in terms of the lambert W function), I believe.
 

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