Does Lipschitz with Respect to x Imply Uniqueness for the Differential Equation

Of course, the answer to the question posed in the title is no, in general, but, surprisingly enough, yes in a significant situation to be detailed later. This paper aims to bring to the attention of the widest possible mathematical audience an elementary result, based on the theorem of differentiation of inverse functions, which extends the applicability of uniqueness theorems. Loosely speaking, it transforms every uniqueness theorem into an alternative version of it with the roles of the dependent and the independent variables interchanged in the assumptions. A remarkable example of such a theorem is the version of Lipschitz’s theorem alluded to in the first paragraph, but there are at least as many new possibilities as old uniqueness theorems.