Sturm’s theorems

Learning objectives

Deduce properties of sine and cosine using only the fact that they are solutions of the differential equation $y^{\prime\prime} + y = 0$
Find qualitative information about solutions of the differential equation $y^{\prime\prime} + q(x)y=0$

In class

Definitions of sine and cosine
Prove Sturm’s theorems in a series of exercises, key

On your own

Qualitative analysis of Van der Pol’s equation