I began by reminding the class of the form of general solutions to the ODE y" + omega^2 y = 0: the pair of solutions normalized at 0 is cos(omega x), (1/omega) sin(omega x): so the initial value problem at x = 0 has solution y(0) cos(omega x) + y'(0) (1/omega) sin(omega x). Omega is the "circular frequency" or "angular speed"; it is measured in radians per unit time; the true frequency is given by omega/2 pi.
Sometimes "initial conditions" come in a different form. For example we may want solutions such that y(0) = 0 and y(l) = 0 for some nonzero l. The first condition forces y(x) = b sin(omega x), which can only be zero at x = l if omega l = n pi for some integer n (or b = 0 - you always have the trivial solution). This discrete condition relating omega and l is the origin of quantum numbers, among other things.
As an example I developed the spinning rope, as in the book (except that instead of appealing to trigonometry I linearized directly by approximating y'^2 by 0). The result is
y" + (rho omega^2 / T) y = 0
where rho is the linear density, T is the tension, and omega is the circular frequency of the rotation. y(0) = 0 forces the solutions b sin (omega_0 x) where
omega_0 = omega sqrt{rho/T}.
y(l) = 0 forces either b = 0 or
omega = (pi / l) sqrt{T/rho} n , n = 1, 2, ...
It's not a surprise that a spinning rope assumes a sinusoidal shape, or that there is a necessary relationship between T, rho, and l. The interesting consequence of this model is that it gives us a quantitative relationship. I illustrated with a spinning clothesline.
As a second example I discussed the equilibrium analysis of a stack of pancakes, as in EP pp 279 ff. The solution is a rescaled multiple of the "Airy cosine" as described in the handout on Airy functions. I pointed out the similarity of theta" + lambda x theta = 0 to y" + omega^2 y = 0, and that this leads us to expect oscillatory solutoins whose frequency increases with x. This is what happens. Equilibrium soltions correspond to zeros of this function, and the handout describes a way of estimating the location of those zeros. I drew a few cases.