Sturm separation theorem

Remarks

The Wronskian $W = y_1 y_2' - y_1' y_2$ of two independent solutions never vanishes, so it has a constant sign on $I$. A common zero would make $W$ vanish there. If $y_2$ had no zero in $(x_1, x_2)$, the function $y_1 / y_2$ would be differentiable on $[x_1, x_2]$ with derivative $-W / y_2^2$ of constant sign. Yet it vanishes at $x_1$ and $x_2$, which contradicts Rolle's theorem. Applying the same argument with the roles exchanged shows that there is at most one zero. Example: for $y'' + y = 0$, the solutions $\sin x$ and $\cos x$ have zeros at $k\pi$ and at $\pi/2 + k\pi$, which alternate. Linear independence is needed: $y_1 = \sin x$ and $y_2 = 2 \sin x$ have the same zeros.

Going deeper

For a student

The theorem shows that oscillation is a property of the equation and not of the solution. If one nonzero solution has infinitely many zeros on an interval, then every nonzero solution does. It also shows that the numbers of zeros of two solutions on an interval differ by at most one. For equations $y'' + q y = 0$ it also follows from the Sturm comparison theorem with $q_1 = q_2$, and it is a basic tool in Sturm–Liouville theory.