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.
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