Second order equations and oscillations

Abstract

Differential equations of second order occur frequently in applied mathematics, particularly in applications coming from physics and engineering. The main reason for this is that most fundamental equations of physics, like Newton's second law of motion (2. 7), are second order differential equations. It is not clear why Nature, at a fundamental level, should obey second order differential equations, but that is what centuries of research have taught us. Since second order ODEs and even systems of second order ODEs are special cases of systems of first order ODEs, one might think that the study of second order ODEs is a simple application of the theory studied in Chapter 2. This is true as far as general existence and uniqueness questions are concerned, but there are a number of elementary techniques especially designed for solving second order ODEs which are more efficient than the general machinery developed for systems. In this chapter we briefly review these techniques and then look in detail at the application of second order ODEs in the study of oscillations. If there is one topic in physics that every mathematician should know about then this is it. Much understanding of a surprisingly wide range of physical phenomena can be gained from studying the equation governing a mass attached to a spring and acted on by an external force. Conversely, one can develop valuable intuition about second order differential equations from a thorough understanding of the physics of the oscillating spring.