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Unformatted text preview: EA 4 Workshop questions  week of 11/17/08 1. Consider the system x = x 2 + y 2 , (1) y = x 2 + y 2 . (2) (a) Determine all critical points for system (1). Answer the only critical point is x = 0 and y = 0 . (b) How does this critical point differ from those discussed in class? Answer there are no linear terms. (c) Determine an equation for the trajectories of this system. Answer from (1) we get dy dx = 1 , so that all trajectories are lines with slope 1 in the phase plane. (d) Consider the solution with the initial condition x (0) = 1 , y (0) = 2 . Determine the trajectory of the solution and the limits of x ( t ) and y ( t ) as t → t * , where t * is the upper limit of times for which the solution exists (will be finite). Answer the limit is ∞ for both x and y . The trajectory is simply the line y + 2 x + 1 = 1 . There are no critical points on this trajectory. The only thing that can happen is that x and y increase indefinitely. (e) Consider the solution with the initial condition x (0) = y (0) = 1 . Determine the limits of x ( t ) and y ( t ) as t → ∞ . Answer Facilitators this question is designed to test student’s understanding of trajectories and critical points and is the key aspect of this problem. The initial point is on the line y = x . This line has slope 1 and so students might expect this line to be the trajectory of the solution. However, this is not true . The reason is that the trajectory cannot include the critical point. The critical point is a solution (a point not a curve in phase space). Thus, no other trajectory, even if it forms a continuous curve withcurve in phase space)....
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This note was uploaded on 09/08/2010 for the course GEN_ENG 205 taught by Professor Taflove during the Spring '08 term at Northwestern.
 Spring '08
 TAFLOVE

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