Ordinary Differential Equations

Carl M. Bender and Steven A. Orszag
Additional contact information
Carl M. Bender: Washington University, Department of Physics
Steven A. Orszag: Yale University, Department of Mathematics

Chapter Chapter One in Advanced Mathematical Methods for Scientists and Engineers I, 1999, pp 3-35 from Springer

Abstract: Abstract An nth-order differential equation has the form $$ {y^{\left( n \right)}}\left( x \right) = F\left[ {x,y\left( x \right),y'\left( x \right), \ldots ,{y^{\left( {n - 1} \right)}}\left( x \right)} \right], $$ where y (k)= d k y/dx k . Equation (1.1.1) is a linear differential equation if F is a linear function of y and its derivatives (the explicit x dependence of F is still arbitrary). If (1.1.1) is linear, then the general solution y(x) depends on n independent parameters called constants of integration; all solutions of a linear differential equation may be obtained by proper choice of these constants. If (1.1.1) is a nonlinear differential equation, then it also has a general solution which contains n constants of integration. However, there sometimes exist special additional solutions of nonlinear differential equations that cannot be obtained from the general solution for any choice of the integration constants. We omit a rigorous discussion of these fundamental properties of differential equations but illustrate them in the next three examples.

Keywords: General Solution; Riccati Equation; Nonlinear Differential Equation; Linear Differential Equation; Bernoulli Equation (search for similar items in EconPapers)
Date: 1999
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4757-3069-2_1

Ordering information: This item can be ordered from
http://www.springer.com/9781475730692

DOI: 10.1007/978-1-4757-3069-2_1

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().