If I would ask for $\phi'(x) = f( \phi(x))$ and $\phi(0)=f(0)$, I would get that the inverse of $\phi$ is forced to be of the form:

$$\phi^{-1}(z) = \int_{0}^z \frac{1}{f(x)} d x.$$

Now it is natural to ask whether there is something similiar for the problem $\phi''(x) =f( \phi(x),\phi'(x))$?

Or a little bit mor restrictive: Given a (homogenous) polynomial $f_j(X_0, \dots X_n) \in \mathbb{C}[X_0, \dots, X_n]$, where $j=1, \dots, m$, such that the variety $V(f_1, \dots, f_m)$ is a non singular variety over $\mathbb{C}$.

Q1: Is it known in general, if there exists a function $\phi: \Omega \rightarrow \mathbb{C}$, such that $$f_j( \phi(s), \phi'(s), \dots, \phi^{(n)}(s))=0$$ or perhaps the easier problem with isolated variables $$\left( \phi^{(n+1)}(s) \right)^{\mathrm{deg} (f_j)}= f_j( \phi(s), \phi'(s), \dots, \phi^{(n)}(s)).$$ for all $s \in \Omega$? Here $\Omega$ is any open subset of $\mathbb{C}$. Are there global meromorphic solutions? What is the algebraic invariant, which counts the linear independent solutions here?

Q2: (see Denis Serre's answer): What is known in the case $m=1$? Or what if require all $f_j$'s to have the same degree?

Q3: Is there a general theory/algorithm, how to compute the solutions in special cases?

Q4: Are there some easy examples, which give an intuition, what we can expect to be true and what not?

I make this community wiki, since I do not know wheter my question is naive. I have little intuition about this stuff. Please comment if you do some changes. As a motivation, the Weierstrass $\wp $ function does the job for elliptic curves.