Induction factors BEM

I have a question concerning the calculation of the induction factors in BEM due to the effect of rotation:
In “Aerodynamics of Wind Turbines” of Martin Hansen (second edition) the starting point is equation 4.32:
x^2a’(1+a’)=a(1-a) (x local speed ratio)

In the “Wind Energy Handbook” of Burton et al (second edition) equation 3.62a states:
x^2a’= a(1-a)

Whereas the second equation leads to a = 1/3 for the optimal case , the Hansen equation leads to a solution dependent of x.

I do not see what is the difference between both approaches.

Can someone help me?