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    solving system of differential equations and eigenvalues

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    (See attached file for full problem description)
    1) The slope field for the system
    dx/dt = 2x + 6y
    dy/dt = 2x - 2y
    is shown to the right
    a) determine the type of the equilibrium point at the origin.
    b) calculate all straight-line solution.

    2) show that a matrix of the form A =(a b; -b a) with b!=0 must have complex eigenvalues.

    3) Let A = (a b; c d), define the trace of A to be tr(A) = a +d. show that A has only one eignevlue if and only if
    (tr(A))^2 - 4det(A) = 0

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    Solution Summary

    The solution shows how to solve the system of differential equations. Furthermore, it characterizes the equilibrium at the origin.