Transcendental equation $2 x n\cot (2x)= x^2 - n^2$

Question

I have a transcendental equation and I have not a mathematical superiour formation (I'm an hydraulic engineer) necessary to solve it.

The equation is : $2 x n\cot (2x)= x^2 - n^2$ or (same equation) : $(n\cot (x)-x) (n \tan(x)+x) =0$.

Here $n$ is a constant (I have this $n$, for example $n = 0.5$). $x_p = $are the roots of equation above.

Answer 1

This is not a complete answer but this kind of transcendental equation are very hard to solve formally and they require very complex method. Anyway this is what I can tell you:

As you said you can rewrite your equation like this:

$$2xn\cot(2x)=x^2-n^2\Rightarrow(n\cot x-x)(n\tan x+x)=0$$

So you have to search from the solution of the two equations:

$$(1)\space\tan x=-\frac xn$$

$$(2)\space\cot x=\frac xn$$

Now taking reciprocals to $(2)$ you get:

$$(2)\space \tan x=\frac nx$$

And in this paper is shown how to solve this two equations: http://www4.ncsu.edu/~ces/pdfversions/52.pdf

Those are closed form solutions and may be not so useful for pratical purpouses so if you need numerical approximation I think that Newton's method is the best (like in the most kinds of situations).