How to solve hyperbolic function ratio equation

find x from the following equation

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

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

Answer 1

[Wan Ji]

you can solve it with symbolic expression

syms z x y
eq = y/z - tanh(x)/tanh(x/2);
sol = solve(eq, x, 'ReturnConditions',true);
x = sol.x
% conds = sol.conditions % see conditions 

x equals to

x =
 
 log((z + (-y*(y - 2*z))^(1/2))/(y - z)) + pi*k*2i
 log((z - (-y*(y - 2*z))^(1/2))/(y - z)) + pi*k*2i
 

where k is an integer.

[Wan Ji]

Given y=1, z=2, k=0, evalute x as

x =[ log((z + (-y*(y - 2*z))^(1/2))/(y - z)); ...
    log((z - (-y*(y - 2*z))^(1/2))/(y - z))];
x = subs(x, [y,z],[1,2])

Then

x0 =
 
 log(- 3^(1/2) - 2)
   log(3^(1/2) - 2)