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.