Applied Mathematical Methods

33

Second Order Linear Homogeneous ODE’s

Second order ordinary differential equations occupy a very important place in applied mathematics, because a large majority of physical systems can be modelled as or reduced to such ODE’s. The theory of linear second order ODE’s is well-developed and a wealth of fertile ideas and methods are available in the area. Before embarking upon a complete study of the general (non-homogeneous) case in the next chapter, we first establish the theory and methods of the homogeneous linear second order ODE’s in the present chapter.

Introduction

A general second order ordinary differential equation can be expressed in the form

f (x, y, y′, y″) = 0,           (33.1)

which can be nonlinear in general. Analytical treatment, ...