On the solution to the equation $\dfrac{1}{2} f'' = f^3 - f$

Question

I am trying to find some textbooks which present on the derivation of the equation $\dfrac{1}{2} f''= f^3 - f$ with boundary conditions $f(0) = f(\infty) = 0$ and its solution ($f(x) = \tanh x$).

Thank you very much for your suggestions.

Answer 1

Multiplying by $f'$ we have

$$ \frac 14((f')^2)'-\frac 14 (f^4)' + \frac 12(f^2)'=0 $$

or integrating

$$ (f')^2=f^4-2f^2+C_0 = (f^2-1)^2+C_1 $$

with solution

$$ f(x) = \frac 12C_2^{-1} e^{\pm x}\left(1-C_1 +C_2^2e^{\mp 2x}\right) $$

etc.