The study of first order algebraic differential equations produced fruitful results around the end of the last century. The classification of equations free of movable singularities was carried out successfully. The investigations were carried out in the complex plane and the main tool of investigation was "analytic continuation". Fuchs tried to clarify the algebraic aspect making use of "Puiseux series ", but his work was not developed fully at that time.

Read more at http://ebookee.org/First-order-algebraic-differential-equations-A-differential-algebraic-approach-by-Michihiko-Matsuda_1060123.html#DvwjDJQSdx7jIFbB.99