How to solve hyperbolic equation with ratio?

Find out x in following equation in which 'y' and 'z' are known.

( y/z ) = (tanh x)/(tanh (x/z))

evaluate x from the following equation if 'y' and 'z' are known.

[Yazan]

Is this a Matlab-related question?

[ld ld]

yes it is related to it.

Even if it can be solved without matlab, it would be ok.

Answer 1

[David Goodmanson]

Hi L^2,

It never hurts to make a plot. Assume for the moment that z >1. Calling the right hand side of the equation f(x), then f(x) is a symmetric function with a peak around x=0, and 1 < f(x) < z. y/z has to fall within that range in order to have a real solution. There are similar results for z <1 and for negative z.

x = -40:.01:40;
z = 4;
y = 10;
f = @(x) tanh(x)./tanh(x/z)
g = @(x) f(x) - y/z;
x0 = fzero(g,[.001 20])    % root = 1.5252
figure(1);grid on
plot(x,f(x),x0,f(x0),'o',x,(y/z)*ones(size(x)))