A Study of Certain Fourth Order Linear Differential Equations

The main goal is to study the pair of fourth order linear differential equations. (E) y - P(x)y = ° and (E+) y + P(x)y = ° Where P = P (x) is a positive, continuous function defined on [0, (0). These two equations have always been studied separately. The fact that either all the solutions of (E+) are oscillatory or all are non-oscillatory, while (E) always has a non-oscillatory solution, provides a demonstration of the apparent independence of the two equations in terms of oscillation. However, contrary to this evidence, it is shown that there is a relationship between (E) and (E+) that involves the oscillatory behavior of the solutions