A Question on a Wikipedia article on Bertrand's theorem

[Kashmir]

Wikipedia while deriving Bertrands theorem writes after some steps :

(...For the orbits to be closed, $β$ must be a rational number. What's more, it must be the same rational number for all radii, since $β$ cannot change continuously.Using the definition of $J$ along with equation (1) we get$J^{\prime}\left(u_{0}\right)=\frac{2}{u_{0}}\left[\frac{m}{L^{2} u_{0}^{2}} f\left(\frac{1}{u_{0}}\right)\right]-\left[\frac{m}{L^{2} u_{0}^{2}} f\left(\frac{1}{u_{0}}\right)\right] \frac{1}{f\left(\frac{1}{u_{0}}\right)} \frac{d}{d u_{0}} f\left(\frac{1}{u_{0}}\right)=-2+\frac{u_{0}}{f\left(\frac{1}{v_{0}}\right)} \frac{d}{d u_{0}} f\left(\frac{1}{u_{0}}\right)=1-\beta^{2}$where $J^{\prime}\left(u_{0}\right)=\left.\frac{d J}{d u}\right|_{u_{0}}$Since this must hold for any value of $u_0$ then${\displaystyle {\frac {df}{dr}}=(\beta ^{2}-3){\frac {f}{r}}}$ and hence${\displaystyle f(r)=-{\frac {k}{r^{3-\beta ^{2}}}}.}$and then we find that${\displaystyle J(u)={\frac {mk}{L^{2}}}u^{1-\beta ^{2}}.}$

However by a similar argument we can say that since $β$ is a constant then we can directly solve
$J^{\prime}(u)=1-\beta^{2}$ and find that $J=\left(1-β^{2}\right) u$) which is wrong.

What went wrong?

Link: https://en.m.wikipedia.org/wiki/Bertrand's_theorem#math_2

 

[LightHero]

The mistake in the derivation is that the author mistakenly assumed that the rational number β is the same for all radii. This is incorrect, as Bertrand's theorem states that for each different radius there is a different rational number β associated with it. Thus, the equation J'(u) = 1 - β2 does not hold for any value of u, since the value of β varies depending on u. Therefore, the equation J(u) = (1 - β2)u is incorrect.