I have a question on this
We denote ${\cal\widetilde{M}_{\ast}}(\Gamma_1)={\cal\widetilde{M}_{\ast}}(\text{SL}_2(\mathbb{Z}))=\mathbb{C}[E_2,E_4,E_6]$ with the Eisenstein series $E_i$. If $f,f',f'',f'''$ are algebraically dependent, I can find $\lambda_i\in\mathbb{C}$ with $$\lambda_1 f+\lambda_2f'+\lambda_3f''+\lambda_4f'''=0.$$ So this is a linear ODE and not a non-linear ODE like it is said in the proposition.
Where is my mistake?
To say that $f, f', f'', f'''$ are algebraically dependent over $\mathbb{C}$ means that there is some nonzero polynomial $p \in \mathbb{C}[x_1, x_2, x_3, x_4]$ such that $p(f, f', f'', f''') = 0$. (For more on this, see here.) If, for instance, $p = x_1 x_2 + x_3 x_4$, then the corresponding differential equation is $$ 0 = p(f, f', f'', f''') = f f' + f'' f''' $$ which is nonlinear.
You seemed to assume that $f, f', f'', f'''$ were linearly dependent.