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Homework Help: Linearizing a differential equation

  1. Jan 4, 2012 #1


    User Avatar

    This question popped up while trying to solve problem 7.41 from Taylor's Classical Mechanics book.

    Basically we have the differential equation

    [tex](1+4k^2\rho^2)\ddot{\rho} = \rho \omega^2 - 4k^2\rho{\dot{\rho}}^2-2gk\rho[/tex]

    and we are looking if the equilibrium position [tex]\rho = 0[/tex] is stable.

    After "linearizing" this you are supposed to end up with

    [tex]\ddot{\rho} = (\omega^2 - 2gk)\rho[/tex]

    but I really don't understand the correct way to arrive at this. Particularly, how are you supposed to justify dropping the [tex]4k^2\rho{\dot{\rho}}^2[/tex] term?
  2. jcsd
  3. Jan 4, 2012 #2


    Staff: Mentor

    My guess is that at the equilibrium position, ρ ≈ 0, and [itex]\dot{\rho}[/itex] is either zero or close to it, which would make [itex]\dot{\rho}^2[/itex] negligible.
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