Analytic solutions of a second-order functional differential equation with a state dependent delay (Q5943578)

The authors consider a second-order functional-differential equation of the form \(x''(z)= x(az+ bx(z))\). They obtain an analytical solution to this differential equation in the form \((y(\alpha y^{-1}(z))- az)/b\) by constructing a convergent power series solution to an auxiliary equation of the form \[ \alpha^2 y''(\alpha z)y'(z)= \alpha y'(z)y''(z)+ (y'(z))^3[y(\alpha^2 z)- ay(\alpha z)]. \]