MATH 4240 Final: Math 4240 2008 Spring Final Exam
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Rewrite as a 2 2 rst-order system. Describe the phase portrait of this system. (4) two point masses in the plane move according to the gravitational attrac- tion between them (i. e. , inverse-square law; don"t worry about the correct physical constants of proportionality). X yf (x2 + y2), where f (0) = 0 and f > 0 otherwise. Write down the linearization of this system at the origin. With respect to the linearized system is the origin asymptotically stable? (6) let h be a smooth function of two variables. Show that h(x, y) is conserved (not changing with time) under the ow of the system x = h/ y, y = Use it to show that the system x = y + x xr2, y = y x yr2 has a cycle about the origin. (8) prove: let a be a 2 2 constant matrix.
