Rational Solutions of First-order Differential Equations

  title={Rational Solutions of First-order Differential Equations},
  author={A. Eremenko},
We prove that degrees of rational solutions of an algebraic differential equation F (dw/dz, w, z) = 0 are bounded. For given F an upper bound for degrees can be determined explicitly. This implies that one can find all rational solutions by solving algebraic equations. Consider the differential equation (1) F (w′, w, z) = 0 (w′ = dw/dz) where F is a polynomial in three variables. Theorem 1. For every F there exists a constant C = C(F ) such that the degree of every rational solution w of (1… CONTINUE READING