On splitting solutions of a certain \(n\)-th order differential equation. (English) Zbl 0611.34023

Vlček, Vladimir

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. 82, Math. 24, 71-80 (1985).

The article investigates a linear homogeneous nth order differential equation \(y^{(n)}(t)+\sum^{n}_{i=1}a_ i(t)y^{(i-1)}(t)=0,\) where \(a_ i(t)=a_ i[q(t),q'(t),...,q^{(n-2)}(t)];\) the basis of the space of all solutions \(y(t)=\sum^{n}_{i=1}C_ iu^{n-i}(t)v^{i- 1}(t),\) \(C_ i\in {\mathbb{R}}\), \(\sum^{n}_{i=1}C^ 2_ i>0\) \((i=1,2,...,n;n>1)\) constitutes an ordered n-tuple of functions \([u^{n- 1}(t),u^{n-2}(t)v(t),...,u(t)v^{n-2}(t),v^{n-1}(t)]\). Thereby [u(t),v(t)] is an oscillatory basis of all solutions \(y(t)=C_ 1u(t)+C_ 2v(t)\) of the second order differential equation \(y''(t)+q(t)v(t)=0,\) \(q(t)\in {\mathbb{C}}^{(n-2)}(-\infty,+\infty)\); \(n>1.\)
It is shown that an arbitrary (nontrivial) solution of the differential equation under investigation vanishing together with the function u(t) in the \(k\in \{1,2,...,n-1\}\)-fold zero \(t_ 0\in (-\infty,+\infty)\) may be written (up to a suitable multiplicative constant \(C\in {\mathbb{R}})\) as \(y(t)=u^ k(t)\prod^{n-k-1}_{j=1}[c_{j1}u(t)+c_{j2}v(t)],\) where \(c_{j1},c_{j2}\in {\mathbb{R}}\), \(c_{j2}\neq 0\), \(c_{j1}c_{m2}- c_{j2}c_{m1}\neq 0\); j,m\(\in \{1,2,...,n-k-1\}\), \(j\neq m\) or where \(j=m\) for any j [possibly \((c_{j1},c_{j2})\), \((c_{m1},c_{m2})\) constitute two complex conjugate pairs of coefficients in the corresponding factorization, f.e. \(Rec_{j1}Imc_{j2}- Imc_{j1}Rec_{j2}\neq 0\) for \(j=1,2]\); where by \(C_{n-k}=\prod^{n- k-1}_{j=1}c_{j2}\neq 0\).

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A30	Linear ordinary differential equations and systems

Keywords:

linear homogeneous nth order differential equation; oscillatory basis

References:

[1]	Vlček V.: Conjugate points of solutions of an iterated differential equation of the N-th order. Acta UP Olom., F. R. N., Tom 76, 1983. · Zbl 0545.34025
[2]	Borůvka O.: Lineare Differentialtransformationen 2. Ordnung. VEB Deutscher Verlag der Wissenschaften, Berlin, 1967. · Zbl 0153.11201