Linear differential equations

Abstract

This chapter begins by proving the fundamental theorem on first-order ordinary differential equations in normal form, and by studying the differentiability of the solution. As a next step, it derives the canonical form of linear second-order ordinary differential equations with variable coefficients, which is very useful for qualitative investigations. Such equations are then extended to the complex domain, proving analyticity of the solution. The Sturm-Liouville problem is then studied with homogeneous boundary conditions in a closed interval, with emphasis on Green functions and the associated integral equations. Chapter 1 ends with a brief outline of the Heun equation. Chapter 2, after some basic examples of non-linear equations in the real domain, studies singular points in the complex domain for non-linear equations.