Using Rolle's to find relationships between solutions:

If y1 and y2 are solutions of y' + P(x)y = 0, then between consecutive zeros of y1, there is at least one zero of y2 (Sturm's separation theorem for first-order case).

Proof idea: Consider phi = $\frac{y1}{y2}$ (assuming y2 != 0). phi' = $\frac{y1'*y2 - y1*y2'}{y2}$^2. Using the ODE: y1' = -Py1 and y2' = -Py2. Then phi' = (-Py1y2 - y1(-Py2))/$y2^2$ = 0. This is trivial — a deeper result needs second-order ODEs.

More useful application: If f is differentiable and f(a) = f(b) = 0, and f satisfies f'(x) = g(x)f(x), then either f is identically zero on [a,b] or g(c) = 0 for some c in (a,b). Apply Rolle's: f'(c) = 0 = g(c)f(c). If f(c) != 0, then g(c) = 0.