Is there an alternative way to solve this equation?

Question

Any suggestions to obtain $x$ in terms of $n$ in this equation. Again Mathematica gives: "This system cannot be solved with the methods available to Solve."

$\coth (\pi x) \coth (n x)-\frac{x^4-2 x^2+5}{4 \left(x^2-1\right)}=0$

Coth[n x] Coth[\[Pi] x] - (5 - 2 x^2 + x^4)/(4 (-1 + x^2)) == 0

Any comment to solve this equation is welcomed.

ContourPlot? Then you can use FindRoot with initial guess from that curve. — Alx, Oct 11 '19 at 14:30
I need to obtain an explicit expression for $x$ in terms of $n$. — , Oct 11 '19 at 15:41
Are you certain it's solvable? A lot of transcendental equations can't be solved analytically so you have to resort to numerical methods. — MassDefect, Oct 11 '19 at 16:26
@Baran, the idea is the same as in answer to another your question here: use ContourPlot to see all the picture and solve numerically then. I'm afraid there is no analythic expression of x(n), this can be done only numerically. — Alx, Oct 12 '19 at 00:24
I'm voting to close this question as off-topic because the OP is asking for an analytical solution for a problem where it does not exist. — m_goldberg, Oct 13 '19 at 17:30

score 1 · Accepted Answer · edited Nov 15 '19 at 10:37

It should be noted that the type of numeric search, based on ContourPlot, that others mention in the comments, has been automated by Wagon, in his book, Mathematica in Action. J.M. gives a version of Wagon's function in this answer.

Using his function, we get the following:

With[{n = 1},
 FindAllCrossings[Coth[n x] Coth[π x] - (5 - 2 x^2 + x^4)/(4 (-1 + x^2)), {x, -5, 5}, WorkingPrecision -> 20]
 ]

{-1.9201894111730777583, -1.5082193798592498308}

Let's check to make sure that the solution are ok:

With[{n = 1},
 Coth[n x] Coth[π x] - (5 - 2 x^2 + x^4)/(4 (-1 + x^2)) /. 
  x -> -1.92018941117307775830055191944881406955`20.
 ]

0.*10^-20

With[{n = 1},
 Coth[n x] Coth[\[Pi] x] - (5 - 2 x^2 + x^4)/(4 (-1 + x^2)) /. 
  x -> -1.50821937985924983076925593990549242151`20.
 ]

0.*10^-19

Yup, it seems to be working.

Is there an alternative way to solve this equation?

1 Answers1

Linked