## Analytic solutions of a second-order functional differential equation with a state dependent delay.(English)Zbl 1017.34074

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 $\alpha^2 y''(\alpha z)y'(z)= \alpha y'(z)y''(z)+ (y'(z))^3[y(\alpha^2 z)- ay(\alpha z)].$
Reviewer: R.S.Dahiya (Ames)

### MSC:

 34K17 Transformation and reduction of functional-differential equations and systems, normal forms
Full Text:

### References:

 [1] E. Eder, The functional differential equation x’(t) = x(x(t)), J. Diff. Eq., 54(1984), 390–400. · Zbl 0497.34050 [2] E. Feckan, On certain type of functional differential equations, Math. Slovaca, 43(1993), 39–43. [3] K. Wang, On the equation x’(t) = f(x(x(t))), Funkcialaj Ekvacioj, 33(1990), 405–425. · Zbl 0714.34026 [4] S. Stanek, On global properties of solutions of functional differential equation x’(t) = x(x(t))+ x(t), Dynamic Sys. Appl, 4(1995), 263–278. · Zbl 0830.34064 [5] J. Si and S. S. Cheng, Analytic solutions of a functional differential equation with state dependent argument, Twiwanese Journal of Math., 1(4)(1997), 471–480. · Zbl 0892.30023 [6] V.R. Petahov, On a boundary value problem. Trudy Sem Teor Diff Uravnenii Otklon Argument, Univ. Druaby Narodov Patrisa Lumumby, 3(1965), 252–255. [7] J. Si, W. Li and S. S. Cheng, Analytic solutions of an iterative functional differential equation, Computers Math. Appl, 33(6)(1997), 47–51. · Zbl 0872.34042 [8] M. Kuczma, Functional equations in a single variable, Polish Scientific Publishers, Warszawa,1968. · Zbl 0196.16403
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.