I am reading the paper Large financial crashes and I noticed that it uses the renormalization group (RG) formalism. Here are some excerpts from the paper:
It continues by writing that
we notice that the solution (3) of the RG equation (2) together with (1) and the linear approximation $\phi(x)=\lambda x$ can be rewritten as $$ \frac{d F(x)}{d \log x}=\alpha F(x), $$
Anyway, I tried to obtain this result by calculating explicitly the derivative of $F(x)$ with respect $\log x$ but I came across a different outcome: $$ \frac{d F(x)}{d \log x}=x\frac{d F(x)}{d x}=\alpha F(x)+C (\omega - \lambda x) \sin\left(\lambda x - \omega \log x\right)$$ Could someone explain to me where I'm going wrong please?