Local analytic solution of a second-order functional differential equation with a state derivative dependent delay

This paper is concerned with a second-order functional differential equation x″(z)=1x(az+bx′(z)) with a state derivative dependent delay. By reducing the equation to another functional differential equation with proportional delay, an existence theorem is established for analytic solutions of the original equation, and systematic methods for deriving explicit solutions are also given. We not only prove the convergence of the formal solution under the Diophantine condition (i.e. eigenvalues is “far from” unit roots), but also make progresses without the Diophantine condition (i.e. the convergence is equivalent to the well-known “small divisor problems”).