An introduction to linear ordinary differential equations using the impulsive response method and factorization
type: Item
instance of: scholarly article
title: An introduction to linear ordinary differential equations using the impulsive response method and factorization
zbMATH Open document ID: 1364.34002
DOI: 10.1007/978-3-319-49667-2
author: Roberto Camporesi
published in: PoliTO Springer Series
publication date: 2017-01-17T00:00:00Z
review text: This book develops the basic theory of \(n\)-th order linear differential equations of the form \[ y^{(n)}+a_1y^{(n-1)}+a_2y^{(n-2)}+\cdots+a_{n-1}y'+a_ny=f.\tag{1} \] In contrast to more traditional approaches (e.g., reduction to a first-order linear system, or the method of variation of parameters), the exposition is based on the method of factorization of linear differential operators. The book has two main parts; the first deals with the case when the coefficients \(a_1,\dots,a_n\) in (1) are constants and \(f\) is a continuous function. The basic idea is as follows: First, consider equation (1) together with the trivial initial conditions \[ y(x_0)=y'(x_0)=\cdots=y^{(n-1)}(x_0)=0.\tag{2} \] If \(\lambda_1,\dots,\lambda_n\) are the roots of the characteristic polynomial \[ \lambda^n+a_1\lambda^{n-1}+a_2\lambda^{n-2}+\cdots+a_{n-1}\lambda+a_n, \] then equation (1) can be rewritten as \[ (\frac{d}{dx}-\lambda_1)\cdots(\frac{d}{dx}-\lambda_n)y=f. \] By letting \(h=(\frac{d}{dx}-\lambda_n)y\), we get \[ (\frac{d}{dx}-\lambda_1)\cdots(\frac{d}{dx}-\lambda_{n-1})h=f,\quad h(x_0)=h'(x_0)=\cdots=h^{(n-2)}(x_0)=0, \] and \[ y'-\lambda_n y=h,\quad y(x_0)=0. \] Once we find \(h\), the solution of the first-order problem is \(y(x)=\int_{x_0}^x e^{\lambda_n(x-t)}h(t)dt\). Using the found \(h\), we apply the same procedure to the previous equation of order \(n-1\) and reduce it to a first-order equation and an equation of order \(n-2\). Proceeding further in the same way, one eventually finds that \[ y(x)=\int_{x_0}^x g_{\lambda_1,\ldots,\lambda_n}(x-s)f(s)ds,\tag{3} \] where \(g_{\lambda}(x)=e^{\lambda x}\) and \(g_{\lambda_1,\ldots,\lambda_k}=\int_{x_0}^x g_{\lambda_k}(x-t)g_{\lambda_1,\ldots,\lambda_{k-1}}(t)dt\) for \(k>1\). The function \(g=g_{\lambda_1,\ldots,\lambda_n}\) is called the impulse response (the reason becomes obvious in the context of distribution theory), and it is the unique solution of the homogeneous problem \[ y^{(n)}+a_1y^{(n-1)}+a_2y^{(n-2)}+\cdots+a…
reviewed by: Antonín Slavík
Mathematics Subject Classification ID: 34-01
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34A05
zbMATH DE Number: 6675952
zbMATH Keywords: linear differential equations
zbMATH Keywords: higher-order differential equations
zbMATH Keywords: factorization of differential operators
zbMATH Keywords: Mammana factorization
zbMATH Keywords: impulse response
zbMATH Keywords: method of undetermined coefficients
zbMATH Keywords: variation of parameters
zbMATH Keywords: Peano-Baker series
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-49667-2
OpenAlex ID: W2560283805
description: scientific article; zbMATH DE number 6675952