## Discussion on the analytic solutions of the second-order iterated differential equation.(English)Zbl 1131.34048

The paper deals with the second order differential equation with state-dependent argument $c_0 x''(z) + c_1 x'(z) + c_2 x(z) = x(az + bx(z)) + h(z), z \in \mathbb{C}, \eqno(1)$ where $$h$$ is a given complex function, $$a,b$$ and $$c_i, i=0,1,2,$$ are complex numbers. Constructing analytic solutions for some auxiliary equation, the authors prove the existence of analytic solution for (1) in a neighborhood of the initial conditions $$x(0)=0, x'(0) = \alpha \neq 0.$$

### MSC:

 34K05 General theory of functional-differential equations 34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
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