Sturm comparison theorem

Remarks

The proof uses the function $W = y' z - y z'$, whose derivative is $(q_2 - q_1) y z$. If $z$ had no zero in $(a, b)$, one may take $y > 0$ and $z > 0$ on $(a, b)$. Integrating $W'$ over $[a, b]$ gives $y'(b) z(b) - y'(a) z(a) \ge 0$, while the signs $y'(a) > 0$, $y'(b) < 0$ and $z \ge 0$ force the left side to be $\le 0$. Analysing the case of equality gives $q_1 = q_2$ and $z(a) = z(b) = 0$, so $z$ is a multiple of $y$. Example: $y = \sin x$ solves $y'' + y = 0$ with zeros $0$ and $\pi$; every solution of $z'' + 4 z = 0$, such as $\cos 2x$, has a zero in $(0, \pi)$. The inequality $q_2 \ge q_1$ is needed: with $q_2 = 0$, the solution $z = 1$ of $z'' = 0$ has no zero at all. Mauro Picone extended the theorem to equations $(p_i y')' + q_i y = 0$ with $p_1 \ge p_2 > 0$ and $q_2 \ge q_1$; this is the Sturm–Picone comparison theorem.

Going deeper

For a student

The theorem lets one bound the zeros of solutions by comparison with equations with constant coefficients. If $m^2 \le q(x) \le M^2$ with $m, M > 0$, then the distance between consecutive zeros of a nonzero solution of $y'' + q y = 0$ lies between $\pi / M$ and $\pi / m$. It is the main tool of oscillation theory and it underlies the count of zeros of eigenfunctions in Sturm–Liouville theory. For equations of the form $y'' + q y = 0$, the case $q_1 = q_2$ gives the Sturm separation theorem.