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About this book

This book presents a method for solving linear ordinary differential equations based on the factorization of the differential operator. The approach for the case of constant coefficients is elementary, and only requires a basic knowledge of calculus and linear algebra. In particular, the book avoids the use of distribution theory, as well as the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and variation of parameters. The case of variable coefficients is addressed using Mammana’s result for the factorization of a real linear ordinary differential operator into a product of first-order (complex) factors, as well as a recent generalization of this result to the case of complex-valued coefficients.

Table of Contents


Chapter 1. Introduction

In view of their many applications in various fields of science and engineering, linear differential equations constitute an important chapter in the theory and application of ordinary differential equations.
Roberto Camporesi

Chapter 2. The Case of Constant Coefficients

Consider the linear first-order differential equation
$$\begin{aligned} y'+ay=f(x), \end{aligned}$$
where \(y'=\frac{dy}{dx}\), a is a real constant.
Roberto Camporesi

Chapter 3. The Case of Variable Coefficients

Most of the theory developed so far generalizes to linear differential equations with variable coefficients, namely the equations of the form
$$ Ly=y^{(n)} + a_1(x) \,y^{(n-1)}+a_2(x) \,y^{(n-2)}+\cdots +a_{n-1}(x)\,y'+a_n(x)\,y=f(x), $$
where \(a_1,a_2,\ldots ,a_n, f\) are complex-valued continuous functions in an interval \(I\subset \mathbb {R}\).
Roberto Camporesi


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