Oscillatory Solutions of Differential Equation Show that the DE $$y'' +q(x)y = 0 \; (*)$$ is oscillatory if one of the following condition is satisfied: i) $q(x) \geq m^2 > 0$ eventually ii) $q(x) = 1 + \phi(x)$ where $\phi(x) \to 0 \; as \; x \to \infty$ iii) $q(x) \to \infty$ as $x \to \infty$ I know by Sturm's comparison theorem, if solution of (*) is oscillatory in $(a,b)$, then solution of $z'' + q_1(x)y = 0$ is oscillatory in $(a,b)$. But, I don't see the relationship in those conditions above. Can someone please walk me through it? Thank you.

Conditions ii) or iii) imply condition i). Under condition i), the angle $φ$ of $y=r\cos φ$, $v=r\sin φ$ for the first order system $y'=mv$, $v'=-q/m·y$ follows the differential equation $$ φ'=\frac{yv'-y'v}{y^2+v^2}=-\frac{mv^2+q/m·y^2}{y^2+v^2}=-m-\frac{(q-m^2)/m}{y^2+v^2} $$ Thus the angular velocity is $-m$ or larger (in negative direction).