"Homogeneous" second order differential equation: zeros separation principles.

In this paper we study the following problems: Problem 1. Let I ⊂ R be an open interval and F : R³ ✕ I → R be a continuous function with, F(0; 0; 0; x) = 0, for all x ε I.We consider the following differential equations F(y"; y'; y; x) = 0: (0.1) Let y ε C²(I) be a nontrivial solution of (0.1). In which conditions we have that: (1) the zeros of y and y' separate each other? (2) the zeros of y and y" separate each other? (3) the zeros of y' and y" separate each other? Problem 2. Let y₁; y₂ ε C²(I) be two linearly independent solutions of (0.1). In which conditions we have that: (1) the zeros of y₁ and y₂ separate each other? (2) the zeros of y'₁ and y'₂ separate each other? (3) the zeros of y"₁ and y"₂ separate each other? Problem 3. Let F;G : R³✕I→R be two continuous functions with F(0; 0; 0; x)=0, G(0; 0; 0; x) = 0, for all x ε I. We consider the following system of differential equations, F(y'; y; z; x) = 0; G(z'; y; z; x) = 0: (0.2) Let (y; z) ε C¹(I,R²) be a nontrivial solution of (0.2). In which conditions we have that: (1) the zeros of y and z separate each other? (2) the zeros of y' and z' separate each other? Problem 4. Let (y₁; z₁) and (y₂; z₂) be two linearly independent solutions of (0.2). In which conditions we have that: (1) the zeros of y₁ and y₂ separate each other? (2) the zeros of z₁ and z₂ separate each other? (3) the zeros of y'₁ and y'₂ separate each other? (4) the zeros of z'₁ and z'₂ separate each other? Some other problems are formulated. [ABSTRACT FROM AUTHOR]