New bounds for the solutions of second order linear differential equations

This paper presents a new method to construct explicit bounds for the solutions of second order linear differential equations of the type [p(x)y′(x)]′+q(x)y(x)=0[p(x)y′(x)]′+q(x)y(x)=0,p(x),q(x)>0p(x),q(x)>0, x>x0x>x0, which complements the method described in recent papers of the authors. The method lies in the use of Beesack’s version of Opial’s inequality. Some results on asymptotics of the solution for the case p(x)=1p(x)=1 are also included.