Viewing nonoscillatory second order linear differential equations from the angle of Riccati equations

TL;DR

This paper develops an existence theory for nonoscillatory second order linear differential equations by analyzing associated Riccati equations, providing explicit solution representations in terms of Riccati solutions.

Contribution

It introduces a novel approach linking Riccati equations to the existence and explicit construction of nonoscillatory solutions for second order differential equations.

Findings

Established existence of nonoscillatory solutions using Riccati equations.

Derived explicit exponential-integral representations of solutions.

Linked solutions of the differential equation to solutions of Riccati equations.

Abstract

We build an existence theory for nonoscillatory second order differential equations of the form (A) $(p(t)x')' = q(t)x, $ $p(t)$ and $q(t)$ being positive continuous functions on $[a,\infty)$, in which a crucial role is played by a pair of the Riccati differential equations (R1) $u' = q(t) - u^2/p(t)$, (R2) $ v' = 1/p(t) - q(t)v^2$, associated with (A). An essential part of the theory is the construction of a pair of linearly independent nonoscillatory solutions $x_1(t)$ and $x_2(t)$ of (A) enjoying explicit exponential-integral representations in terms of solutions $u_1(t)$ and $u_2(t)$ of (R1) or in terms of solutions $v_1(t)$ and $v_2(t)$ of (R2).