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The motion of a simple pendulum with a linear damping is governed by the equation u''+vu'+\omega ^{2}\sin u=0\,. where v>0\,.

In the x-y\, phase plane for which x=u\, and \omega y=u'\,, find all the critical points, classify them, and sketch the orbits near each critical point. Sketch the phase plane.

Notice: I missed a negative sign in the scanned homework: {\frac  {y'}{\omega }}={\frac  {-vy}{\omega }}-\omega ^{2}\sin x\,

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