Lecture 11: Regular perturbation methods for ODEs

This lecture introduces the simplest perturbation methods for analyzing ordinary differential equations (ODEs). These methods go by the name of "regular perturbation theory" (as opposed to "singular perturbation theory," which is much more subtle and rich, and which will occupy us for several lectures to come). The basics of regular perturbation theory are illustrated here with a problem about a toy rocket ship that is launched straight up from the surface of the Earth. How high does the toy rocket go, and how long does it take to fall back to the ground? The problem is easy if we assume the acceleration due to gravity is constant, as in a first-year physics course. But the problem gets more interesting if we take account of the fact that the strength of gravity weakens slightly as the rocket moves farther from the center of the Earth. What corrections does this small perturbation make on the solution? This is what regular perturbation theory can tell us. Prof. Strogatz also introduces the important concept of "uniform validity" of an asymptotic expansion, and gives an simple example of an approximation that is asymptotic but not uniformly valid. These ideas set the stage for boundary layer methods, the next topic in the course.