Mathematica does not like hyperbolic functions

[blalien]

[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!

 

[George Jones]

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

$$\sinh x = n$$

$$\cosh x = nx.$$

Divide these equations and get

$$tanh x = \frac{1}{x}.$$

Solve this numerically for $x$ (use only the positive solution), and, to find $n$, plug this solution into the top equation.

 

[blalien]

Sweet, it worked. Thanks!