On solutions of functional-differential equations f′(x) = a(x)f(g(x)) + b(x)f(x) + c(x) in the large

We consider solutions of functional-differential equations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$f'(x) = a(x)f(g(x)) + b(x)f(x) + c(x)$$ \end{document} in both real and complex variables. We characterize entire solutions g when f is a meromorphic function in the complex plane and a ≠ 0, b, c are constants or polynomials. We also examine questions of existence and uniqueness of the solutions in the real variable for initial value problems and provide theorems that are valid “in the large”.