A Brief Foray into the Regular Sturm Liouville Problem

for some λ ∈ C, x ∈ I = [a, b], and y ∈ C2(I). It was first introduced in a 1837 publication [7] by the eminent French mathematicians Joseph Liouville (1809 1882) and Jacques Charles François Sturm (1803 1855). At this point, our initial questions might be: why is this problem important? What can we say about the structure of this equation? How about the solutions? We will tackle a few basic results and hope to provide a little insight into each of these questions. First we note that the Sturm Liouville eqn. (1) can easily be transformed into a first order system, which will give us a better overview of its structure. We note that (1) ⇔ y′′(x) = (q(x)−λw(x))y(x)−p ′(x)y′(x) p(x) from which we see that the first order system has the form