  We prove a fundamental theorem for tropical partial differential equations
analogue of the fundamental theorem of tropical geometry in this context. We
extend results from Aroca et al., Falkensteiner et al. and from Fink and
Toghani, which work only in the case of trivial valuation as introduced by
Grigoriev, to differential equations with power series coefficients over any
valued field. To do so, a crucial ingredient is the framework for tropical
partial differential equations introduced by Giansiracusa and Mereta. Using
this framework we also add a fourth statement to the fundamental theorem,
seeing the tropicalization as the set of evaluations of points of the
differential Berkovich analytification on the generators of a differential
algebra for a given presentation. Lastly, as a corollary of the fundamental
theorem, we have that the radius of convergence of solutions of an ordinary
differential equation over a nontrivially valued field can be computed
tropically.
