the Sturm comparison theorem
/ STURM /
If two runners start together and one always runs faster, the faster one laps the track more often. There is a version of this for oscillating solutions of differential equations: if one equation 'pushes back toward zero' more strongly than another, its solutions wiggle across the axis more frequently. The Sturm comparison theorem makes this intuition precise — it lets you compare the zeros of solutions of two related equations without solving either one.
Consider two equations in self-adjoint form, (p y')' + q_1 y = 0 and (p z')' + q_2 z = 0 on the same interval, with the same p > 0, where the second has a larger restoring coefficient: q_2(x) >= q_1(x) everywhere (and strictly larger somewhere). The theorem says that the more strongly restored solution z oscillates at least as fast: between any two consecutive zeros of y, the solution z must have a zero of its own. Bigger q means tighter spacing of zeros. The proof is a short, clever computation: form the combination p(y z' minus z y'), differentiate it using both equations, and the difference q_2 minus q_1 forces a sign that cannot persist unless z crosses zero in the gap.
This is the foundational tool for understanding oscillation. It explains, qualitatively and without any formula for the solutions, why a stiffer spring vibrates faster, why solutions of y'' + q(x) y = 0 oscillate where q is large and grow monotonically where q is negative, and why bigger eigenvalues come with more wiggly eigenfunctions. It is the engine behind the Sturm oscillation theorem and a recurring device throughout the qualitative theory of second-order equations and special functions.
Compare y'' + y = 0 (q_1 = 1) with z'' + 4 z = 0 (q_2 = 4). Their solutions are sin(x), with zeros spaced pi apart, and sin(2x), with zeros spaced pi/2 apart. Since q_2 = 4 > 1 = q_1, the comparison theorem promises z has a zero between every pair of zeros of y — and indeed sin(2x) crosses zero twice as often.
A larger restoring coefficient q squeezes the zeros closer together — the solution oscillates faster, with no need to solve either equation.
The two equations must share the same p(x) for the standard comparison to apply directly; if their p's differ you first recast them or use a more general version. And the conclusion is about where zeros lie, not about amplitude — a faster-oscillating solution is not necessarily larger.