Other fundamental systems of solutions of the differential equation (D2 – 2αD + α2 + β2)ny = 0, β ≠ 0
-
Jozef Fecenko
Abstract
This paper generalizes the results of the first part of the paper [Fecenko, J.—Diekema, E.: On the linear (in)dependence of sequences of derivatives of the functions xn sin x and xn cos x, http://arxiv.org/abs/2305.11184]. The main goal is to prove that the sequence of functions f(x), Df(x), …, D2n–1 f(x), where f(x) is xn−1eαx sin βx or xn−1 eαx cos βx for β ≠ 0, forms a fundamental system of solutions of the differential equation (D2 – 2αD + α2 + β2)ny = 0. Or more generally, that the sequence of functions Dkf(x), Dk+1f(x), …, D2n+k–1 f(x), k ∈ ℕ also forms a fundamental system of solutions of the aforementioned differential equation. The problem is solved using a suitable transformation of the Wronskian matrix, by solving a nonlinear differential equation and then calculating the determinant of the modified Wronskian matrix by applying Laplace’s generalized expansion.
Communicated by Michal Fečkan
References
[1] Collective of Authors: Applied Mathematics I, II, SNTL - Publishing House of Technical Literature, Prague, 1978 (in Czech).Search in Google Scholar
[2] Fecenko, J.: Matrix differential operator method of finding a particular solution to a nonhomogeneous linear ordinary differential equation with constant coefficients, https://arxiv.org/abs/2101.02037.Search in Google Scholar
[3] Fecenko, J.—Diekema, E.: On the linear (in)dependence of sequences of derivatives of the functions xn sin x and xn cos x, http://arxiv.org/abs/2305.11184.Search in Google Scholar
[4] Gatto, L.—Scherbak, I.: On Generalized Wronskians, https://www.impan.pl/pragacz/gatto2.pdf.Search in Google Scholar
[5] Spiegel, M. R.: Schaum’s Outline of Theory and Problems of Advanced Mathematics for Engineers and Scientists, McGraw-Hill, 2002.Search in Google Scholar
[6] Weisstein, E. W. “Linearly Dependent Functions”. From MathWorld–A Wolfram Web Resource, https://mathworld.wolfram.com/LinearlyDependentFunctions.html.Search in Google Scholar
© 2024 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- On monoids of endomorphisms of a cycle graph
- The influence of separating cycles in drawings of K5 ∖ e in the join product with paths and cycles
- Completely hereditarily atomic OMLS
- New notes on the equation d(n) = d(φ(n)) and the inequality d(n) > d(φ(n))
- On the monogenity and Galois group of certain classes of polynomials
- Other fundamental systems of solutions of the differential equation (D2 – 2αD + α2 + β2)ny = 0, β ≠ 0
- Characterization of continuous additive set-valued maps “modulo K” on finite dimensional linear spaces
- Hermite–Hadamard inequalities for Riemann–Liouville fractional integrals
- Global Mild Solutions For Hilfer Fractional Neutral Evolution Equation
- The discrete fractional Karamata theorem and its applications
- Complex Generalized Stancu-Schurer Operators
- New oscillatory criteria for third-order differential equations with mixed argument
- Cauchy problem for a loaded hyperbolic equation with the Bessel operator
- Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p0)-condition
- On the rate of convergence for the q-Durrmeyer polynomials in complex domains
- Some fixed point theorems in generalized parametric metric spaces and applications to ordinary differential equations
- On the relation of Kannan contraction and Banach contraction
- Extropy and statistical features of dual generalized order statistics’ concomitants arising from the Sarmanov family
- Residual Inaccuracy Extropy and its properties
- Archimedean ℓ-groups with strong unit: cozero-sets and coincidence of types of ideals
Articles in the same Issue
- On monoids of endomorphisms of a cycle graph
- The influence of separating cycles in drawings of K5 ∖ e in the join product with paths and cycles
- Completely hereditarily atomic OMLS
- New notes on the equation d(n) = d(φ(n)) and the inequality d(n) > d(φ(n))
- On the monogenity and Galois group of certain classes of polynomials
- Other fundamental systems of solutions of the differential equation (D2 – 2αD + α2 + β2)ny = 0, β ≠ 0
- Characterization of continuous additive set-valued maps “modulo K” on finite dimensional linear spaces
- Hermite–Hadamard inequalities for Riemann–Liouville fractional integrals
- Global Mild Solutions For Hilfer Fractional Neutral Evolution Equation
- The discrete fractional Karamata theorem and its applications
- Complex Generalized Stancu-Schurer Operators
- New oscillatory criteria for third-order differential equations with mixed argument
- Cauchy problem for a loaded hyperbolic equation with the Bessel operator
- Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p0)-condition
- On the rate of convergence for the q-Durrmeyer polynomials in complex domains
- Some fixed point theorems in generalized parametric metric spaces and applications to ordinary differential equations
- On the relation of Kannan contraction and Banach contraction
- Extropy and statistical features of dual generalized order statistics’ concomitants arising from the Sarmanov family
- Residual Inaccuracy Extropy and its properties
- Archimedean ℓ-groups with strong unit: cozero-sets and coincidence of types of ideals
