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Property $A$ of the $ (n+1)^{th}$ Order Differential Equation $\left[ \frac
1 {r_1(t)} \ \left( x^{(n)}(t) + p(t) x(t) \right)\ \right]' = f(t, x (t),
\cdots , x^{(n)}(t))$

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*Monika Kovacova*

**Address.** Dept. of Math., Faculty of Mechanical Engineering, Slovak
Technical University, Namestie Slobody 17,

812 31 Bratislava, Slovak Republic
**E-mail:** kovacova_v@sjf.stuba.sk

**Abstract.** The aim of this contribution is to study properties
of solutions of the $n +1 ^{th}$-order differential equation of the form
\begin{equation} \left[ \frac 1 {r_1(t)} \ \left( x^{(n)}(t) + p(t) x(t)
\right)\ \right]' = f(t, x (t), \cdots , x^{(n)}(t))\,.\label{moni1} \end{equation}
where $ n\ge 2 $ is a natural number. A new approach using ``submersivity''
of a solution of an equation is presented, by means of it a sufficient
condition for the property A is proved. This approach can be also used
to prove necessary condition for the property A.

**AMSclassification.** 34C10, 34C15

**Keywords.** Property $A$, oscillatory solutions