Mathematica does not like hyperbolic functions

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Main Question or Discussion Point

[SOLVED] Mathematica does not like hyperbolic functions

So, consider the equation cosh(x)=n*x

For a given n, the equation has 0, 1, or 2 possible values of x. If n is below the critical value, the equation has no solutions. If n is above the critical value, the equation has two solutions. And if n is exactly the critical value, the equation has one solution. My goal is to use Mathematica to show that the critical value is approximately 1.51.

Theoretically, the line Length[Solve[Cosh[x] == n*x, x]] should give the number of solutions for a given n. Then I can make a table of n's and find the point where the number of solutions goes from 0 to 2.

Unfortunately, I keep getting the error:
Solve::tdep: The equations appear to involve the variables to be solved for in an essentially non-algebraic way.

NSolve has the exact same problem. FindRoot always gives exactly one solution, whether there are zero or two solutions to the equation. Is there a way to make Mathematica more cooperative, or another way to go about this problem? Since the TI-89 can handle this problem (but is too slow to be useful), it seems like Mathematica should be able to as well.

Thanks for your help!
 

Answers and Replies

George Jones
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Since [itex]\cosh x[/itex] and [itex]nx[/itex] have one intersection point, their derivatives are equal. Along with the original equation, this gives two equations for the two unknowns [itex]n[/itex] and [itex]x[/itex] (at intersection).

[tex]\sinh x = n[/tex]

[tex]\cosh x = nx.[/tex]

Divide these equations and get

[tex]tanh x = \frac{1}{x}.[/tex]

Solve this numerically for [itex]x[/itex] (use only the positive solution), and, to find [itex]n[/itex], plug this solution into the top equation.
 
Last edited:
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Sweet, it worked. Thanks!
 

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