Oscillation theory

Remarks

By the Sturm separation theorem, if one nonzero solution is oscillatory then every nonzero solution is oscillatory. So the equation is oscillatory exactly when all its nonzero solutions are. Example: $y'' + y = 0$ is oscillatory, since $y = \sin t$ vanishes at $t = k\pi$ for every integer $k$. Non-example: for $y'' = 0$ every nonzero solution $y = A t + B$ has at most one zero, so the equation is nonoscillatory. The Euler equation $y'' + \frac{c}{t^2} y = 0$ on $[1, \infty)$ is oscillatory exactly when $c > \frac{1}{4}$. A classical criterion says that the equation is oscillatory if $\int_{t_0}^\infty q(t) \, dt = +\infty$.