Unformatted text preview:

MIT OpenCourseWarehttp://ocw.mit.edu 18.034 Honors Differential Equations Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.18.034 Recitation: May 14, 2009 1. Study the phase portraits of the systems � � � �� � x � 1 � x = y 1 −1 y and � � � �� � x � 0 −1 x = y � −1 y 2. Consider x� = y − x(x 2 + y 2), y� = −x − y(x 2 + y 2). (a) Find the critical point. (b) Determined the stability of the linear approximation at (0, 0). (c) Determined the stability of (0, 0). (d) Repeat for x� = y + x(x 2 + y 2), y� = −x − y(x 2 + y 2). 3. In the competitive system x� = x(k − ax − by), y� = y(m − cx − dy), k, m, a, b, c, d > 0 if the lines ax + by = k and cx + dy = m do not intersect in the first quadrangle x, y > 0 find the limit set. 4. If (x(t), y(t)) is a solution of the predator-prey equations x� = x(−k + by), y� = y(m − cx), k, m, b, c > 0 of period T > 0, show that 1 � T m 1 � T k x(t)dt = , y(t)dt = . T 0 c T 0 b 5. (a) Show that the differential equation x�� + (x 2 + 2(x�)2 − 1)x� + x = 0 has a nontrivial periodic solution. (b) Show that the system of differential equations x� = x + y 2 + x 3 , y� = −x + y + yx 2 has no nontrivial periodic solution.


View Full Document

MIT 18 034 - Lecture Notes

Documents in this Course
Exam 3

Exam 3

9 pages

Exam 1

Exam 1

6 pages

Load more
Download Lecture Notes
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Lecture Notes and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Lecture Notes 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?