Bibliography request for differential equation involving Dedekind Eta Function

Question

I would like to find the reference or bibliographic source of the fact that the following differential equation:

$$36(y')^2-24y''y+y''' = 0$$

is satisfied by $y(z) = \frac{\eta'(z)}{\eta(z)}$ ($\eta(z)$ denotes Dedekind eta function).

I have found this ODE in Wolfram Functions. After a while searching around the internet, I have not found any reference.

Thank you.

Answer 1

I found one reference and there must be others. The arXiv article Integrable systems and modular forms of level 2 by Ablowitz, Chakravarty and Hanh on page 2 give an ODE $$ y''' = 2yy'' - 3y'^2, \tag{1.2} $$ and note that it is related to Darboux-Halphen and Chazy. They remark that

Indeed, a particular solution of $(1.2)$ is given by $$ y(z) :=\pi i E_2(z),$$

Now we have $\,\eta'(z)/\eta(z) = \frac{2\pi i}{24} E_2(z)\,$ which leads to the ODE you give in your question. Now $E_2(z)$ is a quasi-modular form and is closely related to the modular forms $E_4(z)$ and $E_6(z)$.