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

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
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### References:

  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  Borůvka O.: Lineare Differentialtransformationen 2. Ordnung. VEB Deutscher Verlag der Wissenschaften, Berlin, 1967. · Zbl 0153.11201
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