  The notions of equivalence and strict equivalence for order one differential
equations are introduced. The more explicit notion of strict equivalence is
applied to examples and questions concerning autonomous equations and equations
having the Painleve property. The order one equation determines an algebraic
curve. If this curve has genus zero or one, then it is difficult to verify
strict equivalence. However, for higher genus strict equivalence can be tested
by an algorithm sketched in the text. For autonomous equations, testing strict
equivalence and the existence of algebraic solutions are shown to be
algorithmic.
