It is known that modular forms are solutions of differential equations. More precisely, let me cite the statement from the following question.

Differential Equations Satisfied by Modular Forms

Let $\Gamma$ be a discrete subgroup of $SL_{2}(\mathbb{R})$ commensurable with $SL_{2}(\mathbb{Z})$. For $f \in M_{k}(\Gamma)$ (the space of weight $k$ modular forms) and $t \in M_{0}(\Gamma)$ (the space of meromorphic weight 0 modular forms), if $f = \sum_{n \geq 0}b_{n}t^{n}$ near $t = 0$, then there is a linear order $k + 1$ differential equation satisfied by $g(x) = \sum_{n \geq 0} b_{n}x^{n}$, of the form $$P_{k + 1}(x)\frac{d^{k + 1}g}{dx^{k + 1}} + P_{k}(x)\frac{d^{k}g}{dx^{k}} + \cdots + P_{0}(x)g = 0 \tag{1}$$

I seem to remember this is a theorem of Zagier, but I cannot give a precise reference. Anyway, my question is:

Are there a similar result for quasi-modular forms? I.e., if $f$ is only a quasi-modular form, is the above statement valid? If not, could anyone explain why? or give an example?