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In the first part of the paper we characterize certain systems of first ordernonlinear differential equations whose space of solutions is an$\mathfrak{sl}_2(\mathbb{C})$-module. We prove that such systems, calledRamanujan systems of Rankin-Cohen type, have a special shape and are preciselythe ones whose solution space admits a Rankin-Cohen structure. In the secondpart of the paper we consider triangle groups $\Delta(n,m,\infty)$. By means ofmodular embeddings, we associate to every such group a number of systems of nonlinear ODEs whose solutions are algebraically independent twisted modularforms. In particular, all rational weight modular forms on $\Delta(n,m,\infty)$are generated by the solutions of one such system (which is of Rankin-Cohentype). As a corollary we find new relations for the Gauss hypergeometricfunction evaluated at functions on the upper half-plane. To demonstrate thepower of our approach in the non classical setting, we construct the space ofintegral weight twisted modular form on $\Delta(2,5,\infty)$ from solutions ofsystems of nonlinear ODEs.

Ramanujan systems of Rankin-Cohen type and hyperbolic triangles