CHAPTER II - LINEAR DIFFERENTIAL EQUATIONS. SUPPLEMENTARY REMARKS ON THE THEORY OF DIFFERENTIAL EQUATIONS

Abstract

This chapter discusses linear homogeneous and nonhomogeneous equations of any order. The solutions of the linear homogeneous equation of second order can be obtained by multiplying existing solutions by arbitrary constants and adding. The general solution of a nonhomogeneous linear equation of the second order is equal to the sum of the general solution of the corresponding homogeneous equation and any solution of the nonhomogeneous equation. However, the solutions of a linear equation of a higher order than the first with variable coefficients are not generally expressible in terms of elementary functions, and the integration of such an equation does not, in general, reduce to a quadrature. The most useful method is to represent the required solution as a power series. This device is particularly applicable to linear differential equations. The chapter discusses the second-order equation and also discusses the existence and uniqueness theorem for differential equations.