Abstract: Professors sometimes have to teach courses outside their area of expertise. For example, I’ve taught a course in perturbation methods several times, even though I don't know much about the subject. In this talk, I'll discuss a singularly perturbed nonlinear boundary-value problem that has surprised me with something new, every time I teach it. The surprises include a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the solutions' initial conditions that can be calculated by elementary means.