# Is there an alternative way to solve this equation? [closed]

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 $n$. – Baran 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
• @Alx would you explain more? – Baran Oct 11 '19 at 22:07
• 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

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 -> -1.9201894111730777583005519194488140695520.
]


0.*10^-20

With[{n = 1},
Coth[n x] Coth[\[Pi] x] - (5 - 2 x^2 + x^4)/(4 (-1 + x^2)) /.
x -> -1.5082193798592498307692559399054924215120.
]


0.*10^-19

Yup, it seems to be working.