Normal forms of functional-differential equations of Beardon type and Briot–Bouquet type equations in the complex domain

Abstract

We study local analytic solutions of the functional-differential equation of the form ${h(\psi(z)) = b(z) h(z) h^\prime(z) + d(z)h(z)^{2}}$ which are called Beardon type functional-differential equations. All functions involved are supposed to be holomorphic in a neighbourhood of zero. Special cases are the equations f(kz) =  k f(z) f′(z) where k is a complex number, ${k \neq 0}$, and ${f(\varphi(z)) = a(z) f(z) f'(z)}$ with given ${\varphi}$ and a. The class of these equations is invariant under transformations ${h \to \alpha h, \alpha(z) \neq 0}$ for all z in a neighbourhood of zero, of the unknown function and ${z \to T(z)}$ of the argument z. In particular, we are interested to know under which conditions a Beardon type functional-differential equation can be transformed to the simplified (normal form) ${h(kz) = k h(z) h'(z) + c(z) h(z)^2}$ where ${k \in \mathbb {C} \backslash\left\{0\right\}}$. We solve this normal form by another transfomation to a so-called Briot–Bouquet type functional-differential equation.