In this paper we prove that Ramanujan's differential equations for the Eisenstein series P, Q, and R are invariant under a simple one-parameter stretching group of transformations. Using this, we show that the three differential equations may be reduced to a first order Riccati differential equation, the solution of which may be represented in terms of hypergeometric functions. The resulting formulas allow for the derivation of parametric representations of P, Q, and R, analogous to representations in Ramanujan's second notebook. In contrast, in the classical approach, one first needs to derive the fundamental formula connecting theta functions with elliptic integrals. This theorem is not needed in the present approach.
Hill, J. M., Berndt, B. C., & Huber, T. (2007). Solving Ramanujan’s differential equations for Eisenstein series via a first order Riccati equation. Acta Arithmetica, 128, 281–294. https://doi.org/10.4064/aa128-3-6