Analytic solutions of a second-order functional differential equation with a state derivative dependent delay

This paper is concerned with a second-order functional differential equation of the form ${\displaystyle x\prime \prime \left(z\right)=x(az+bx\prime \left(z\right))}$ with the distinctive feature that the argument of the unknown function depends on the state derivative. An existence theorem is established for analytic solutions and systematic methods for deriving explicit solutions are also given.