The exact analytic solutions of a nonlinear differential iterative equation

This paper is concerned with a second-order nonlinear iterated differential equation of the form c 0 x ″ ( z ) + c 1 x ′ ( z ) + c 2 x ( z ) = x ( p ( z ) + b x ( z ) ) + h ( z ) . By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. We discuss not only the general case | β | ≠ 1 , but also the critical case | β | = 1 , especially when β is a root of unity. Furthermore, the exact and explicit analytic solution of the original equation is investigated for the first time. Such equations are important in both applications and the theory of iterations.