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    Differential Equations Problem Set

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    Consider the differential equation

    dy/dx = -x/y.

    a) Sketch a direction field for this differential equation.
    b) Sketch solution curves of the equation passing through the points (0, 1), (1, 1) and (0, -2).
    c) State the regions of the xy-plane in which the conditions of the existence and uniqueness theorem are satisfied. Quote the theorem which you are using.
    d) Consider the autonomous equation

    dx/dt = -3/2 * third root of x.

    Show that the trajectory passing through a point x_0 > 0 reaches a fixed point in finite time. Explain why it is possible in this case to have more than one trajectory passing through a point of the phase space.

    Please refer to the attached document for the complete problem set.

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    Solution Preview

    (a) We wish to sketch a direction field for this differential equation.

    We do so by drawing the vector (y, -x) at each point (x, y). The result is shown ...

    Solution Summary

    This solution shows step-by-step calculations to solve various problems involving ordinary differential equations. Explanations are also included for understanding.