On a structure of second order linear differential equations with periodic coefficients having the same discriminant. (English) Zbl 0584.34004

Let \(\Delta =\Delta(\lambda)\) be the discriminant of a differential equation \(y''=(q(t)+\lambda)y,\) \(q\in C^ 0({\mathbb{R}})\), \(q(t+\pi)=q(t)\) for \(t\in {\mathbb{R}}\), \(\lambda\in {\mathbb{R}}\). This paper presents all differential equations of the type \(y''=s(t,\lambda)y\), \(s\in C^ 0({\mathbb{R}}\times {\mathbb{R}})\), \(s(t+\pi,\lambda)=s(t,\lambda)\) for \((t,\lambda)\in {\mathbb{R}}\times {\mathbb{R}}\), whose discriminant is equal to \(\Delta(\lambda)\).


34A30 Linear ordinary differential equations and systems
Full Text: EuDML


[1] Arscott F. M.: Periodic Differential Equations. Pergamon Press, Oxford, 1964. · Zbl 0121.29903
[2] Borůvka O.: Linear Differential Transformations of the Second Order. The English Universities Press, London, 1971. · Zbl 0218.34005
[3] Борувка О.: Тєоруя глобалъных свойсмв овыкновєнных лунєйных дуффєрєнцуалъных уравнєнуй вморого порядка. Дифференциальные уравнения, No 8, t. XII, 1976, 1346-1383.
[4] Gregus M., Neuman F., Arscott F. M.: Three-point boundary problems in differential equations. J. London Math. Soc. (2), 3, 1971, 429-436. · Zbl 0226.34010 · doi:10.1112/jlms/s2-3.3.429
[5] Hartman P.: Ordinary Differential Equations. (In Russian) Moscow, 1970. · Zbl 0214.09101
[6] Якубович В. А., Старжинский В. А.: Лунєйныє дуффєрєнцуалъныє уравнєнуя с пєруодучєскум козффуцуєнмаму у ух пруложєнуя. Издательство ”Наука”, Москва 1970.
[7] Krbiľa J.: Vlastnosti fáz neoscilatorických rovnic y” = q(t)y definovaných pomocou hyperbolických polárnych súradnic. Sborník prací VŠD a VŰD, 19, 1969, 5-11.
[8] Krbiľa J.: Application von parabolischen Phasen der Differentialgleichung y” = q(t)y. Sborník prací VŠD a VÚD, IV ved. konf., 1. sekcia, 1973, 67-74.
[9] Krbiľa J.: Explicit solution of several Kummer’s nonlinear differential equations. Mat. Čas., 24 No. 4, 1974, 343-348.
[10] Magnus M., Winkler S.: Hill’s Equation. Interscience Publishers, New York, 1966. · Zbl 0158.09604
[11] Markus L., Moore R. A.: Oscillation and disconjugacy for linear differential equations with almost periodic coefficients. Acta Math., 96, 1956, 99-123. · Zbl 0071.08302 · doi:10.1007/BF02392359
[12] Neuman F., Staněk S.: On the structure of second-order periodic differential equations with given characteristic multipliers. Arch. Math. (Brno), 3, XIII, 1977, 149-158.
[13] Staněk S.: Phase and dispersion theory of the differential equation y” = q(t)y in connection with the generalized Floquet theory. Arch. Math. (Brno) 2, XIV, 1978, 109-122. · Zbl 0412.34025
[14] Staněk S.: A note on disconjugate linear differential equations of the second order with periodic coefficients. Acta Univ. Palackianae Olomucensis F. R. N., 61, 1979, 83-101.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.