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June 30th, 2007, 11:33 AM   #1
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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 ).
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