Lazarus Fuchs' transformation for solving rational first-order differential equations (Q1340601)

L. Fuchs' method of solving a rational first-order differential equation is re-formulated and is neatly explained. For a rational curve \(F(X, Y)= 0\) over the field \(E\) of meromorphic functions on a domain of the complex plane parametrized by \(z\), consider the differential equation \(F(w, dw/dz)= 0\). By the help of a rational parametrization \(X= \phi(T)\), \(Y= \psi(T)\), where \(\phi\) and \(\psi\) are rational functions over the algebraic closure \(\overline E\) of \(E\), the differential equation above can be transformed into a first-order differential equation of degree 1 of the following form: \(H_ 0(t) dt/dz+ H_ 1(t)= 0\), where \(H_ 0(T)\) and \(H_ 1(T)\) are polynomials in \(T\) over \(\overline E\). Many examples are given, for which the actual procedure of transformations and of solving the resulting equations for \(t\) are shown.