
Contents
1 Second-order differentialequations in the phase plane
Get access-
Published:August 2007
Cite
Abstract
1.1. For the general equation x f (x, x), (see eqn (1.6)), equilibrium points lie on the x axis, and are given by all solutions of f (x, 0) 0, and the phase paths in the plane (x, y) (y x) are given by all solutions of the first-order equation Note that scales on the x and y axes are not always the same. Even though explicit equations for the phase paths can be found for problems (i) to (viii) below, it is often easier to compute and plot phase paths numerically from x f (x, x), if a suitable computer program is available. This is usually achieved by solving x y, y f (x, y) treated as simultaneous differential equations, so that (x(t), y(t)) are obtained parametrically in terms of t . The phase diagrams shown here have been computed using Mathematica.
Sign in
Personal account
- Sign in with email/username & password
- Get email alerts
- Save searches
- Purchase content
- Activate your purchase/trial code
- Add your ORCID iD
Purchase
Our books are available by subscription or purchase to libraries and institutions.
Purchasing informationMonth: | Total Views: |
---|---|
September 2024 | 2 |
March 2025 | 2 |
Get help with access
Institutional access
Access to content on Oxford Academic is often provided through institutional subscriptions and purchases. If you are a member of an institution with an active account, you may be able to access content in one of the following ways:
IP based access
Typically, access is provided across an institutional network to a range of IP addresses. This authentication occurs automatically, and it is not possible to sign out of an IP authenticated account.
Sign in through your institution
Choose this option to get remote access when outside your institution. Shibboleth/Open Athens technology is used to provide single sign-on between your institution’s website and Oxford Academic.
If your institution is not listed or you cannot sign in to your institution’s website, please contact your librarian or administrator.
Sign in with a library card
Enter your library card number to sign in. If you cannot sign in, please contact your librarian.
Society Members
Society member access to a journal is achieved in one of the following ways:
Sign in through society site
Many societies offer single sign-on between the society website and Oxford Academic. If you see ‘Sign in through society site’ in the sign in pane within a journal:
If you do not have a society account or have forgotten your username or password, please contact your society.
Sign in using a personal account
Some societies use Oxford Academic personal accounts to provide access to their members. See below.
Personal account
A personal account can be used to get email alerts, save searches, purchase content, and activate subscriptions.
Some societies use Oxford Academic personal accounts to provide access to their members.
Viewing your signed in accounts
Click the account icon in the top right to:
Signed in but can't access content
Oxford Academic is home to a wide variety of products. The institutional subscription may not cover the content that you are trying to access. If you believe you should have access to that content, please contact your librarian.
Institutional account management
For librarians and administrators, your personal account also provides access to institutional account management. Here you will find options to view and activate subscriptions, manage institutional settings and access options, access usage statistics, and more.