Equivalence of differential equations of order one

The notion of strict equivalence for order one differential equations of the form f(y′,y,z)=0f(y′,y,z)=0 with coefficients in a finite extension K   of C(z)C(z) is introduced. The equation gives rise to a curve X over K and a derivation D   on its function field K(X)K(X). Procedures are described for testing strict equivalence, strict equivalence to an autonomous equation, computing algebraic solutions and verifying the Painlevé property. These procedures use known algorithms for isomorphisms of curves over an algebraically closed field of characteristic zero, the Risch algorithm and computation of algebraic solutions. The most involved cases concern curves X of genus 0 or 1. This paper complements work of M. Matsuda and of G. Muntingh & M. van der Put.