We construct a general framework for tropical differential equations based on idempotent semirings and an idempotent version of differential algebra. Over a differential ring equipped with a valuation enhanced with additional differential information, we define tropicalization of differential equations and tropicalization of their solution sets. This framework includes rings of interest in the theory of p-adic differential equations: rings of convergent power series over a non-archimedean normed field. The tropicalization records the valuation of the coefficients. This gives a significant refinement of Grigoriev's framework for tropical differential equations. Time permitting, we then prove a differential analogue of Payne's inverse limit theorem: the limit of all tropicalizations of a system of differential equations is isomorphic to a differential variant of the Berkovich analytification.