LECTURE 3 - THE CLASSIFICATION OF LINEAR EQUATIONS OF THE SECOND ORDER

Abstract

This chapter focuses on the classification of linear equations of the second order. The chapter describes the properties of linear equations of the second order in more detail by discussing certain properties of their coefficients. It examines how the coefficients of the linear equations of the second order transform under an arbitrary change of the independent variables, or what comes to the same thing, under any geometrical transformation of the space of the variables X1, X2, …, Xn.The coefficients of the transformation can always be chosen so that a quadratic form is reduced to a sum of squares. If the equation has constant coefficients, then under a linear change of variables, it will transform into an equation with coefficients constant again. The chapter also highlights that the character of a second-order equation is completely determined by the number r of positive coefficients and the number s of negative coefficients of the second-order derivatives after such a transformation.