My Math Forum  

Go Back   My Math Forum > College Math Forum > Differential Equations

Differential Equations Ordinary and Partial Differential Equations Math Forum

LinkBack Thread Tools Display Modes
June 30th, 2007, 11:33 AM   #1
Joined: Jun 2007
From: Milwaukee

Posts: 11
Thanks: 0

Study of a first order differential system

Consider the system

. .
x = - y, y = -x

(a) Sketch the vector field.

(b) Show that the trajectories of the system are hyperbolas of the form x^2 - y^2 = C.
(Hint: Show that the governing equations imply
. .
xx-yy = 0 and then integrate both sides.)

( c ) The origin is a saddle point; find equations for its stable and unstable manifolds.

(d) The system can be decoupled and solved as follows. Introduce new variables u and v, where u = x + y, v = x – y. Then rewrite the system in terms of u and v. Solve for u(t) and v(t), starting from an arbitrary initial condition (u0, v0).

(e) What are the equations for the stable and unstable manifolds in terms of u and v ?

(f) Finally, using the answer to (d), write the general solution for x(t) and y(t), starting
from an initial condition ( x0, y0 ).
Seng Peter Thao is offline  

  My Math Forum > College Math Forum > Differential Equations

differential, order, study, system

Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Study of the fixed points of a differential system Seng Peter Thao Differential Equations 4 August 25th, 2013 12:37 AM
System of nonlinear first-order differential equations Degnemose Differential Equations 0 March 20th, 2012 12:53 PM
First order differential system coupled oXis Differential Equations 0 October 29th, 2011 07:28 AM
Another system of differential equations silentwf Differential Equations 2 January 5th, 2010 12:08 AM
Differential system Seng Peter Thao Differential Equations 0 June 30th, 2007 11:58 AM

Copyright © 2019 My Math Forum. All rights reserved.