## Analytic solutions of a second-order iterative functional differential equation.(English)Zbl 0983.34056

Authors’ abstract: This paper is concerned with the second-order iterative functional-differential equation $x''(x^{[r]}(z))=c_{0}z+c_{1}x(z)+\dots +c_{m}x^{[m]}(z),$ where $$r$$ and $$m$$ are nonnegative integers, $$x^{[0]}(z)=z,$$ $$x^{[1]}(z)=x(z),$$ $$x^{[2]}(z)=x(x(z)),$$ etc., are the iterates of the function $$x(z).$$ By constructing a convergent power series solution $$y(z)$$ to a companion equation of the form $\alpha^{2}y''(\alpha^{r+1}z)y'(\alpha^{r}z)=\alpha y'(\alpha^{r+1}z)y''(\alpha^{r} z)+[y'(\alpha^{r} z)]^{3}\left[\sum_{i=0}^{m}c_{i}y(\alpha^{i}z)\right],$ analytic solutions of the form $$y(\alpha y^{-1}(z))$$ to the original differential equation are obtained.

### MSC:

 34K07 Theoretical approximation of solutions to functional-differential equations 34K25 Asymptotic theory of functional-differential equations 34K05 General theory of functional-differential equations

### Keywords:

functional-differential equation; analytic solution
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### References:

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