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June 30th, 2007, 11:33 AM  #1 
Newbie 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 . . xxyy = 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 ). 

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