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Power law trick DE

  1. Nov 28, 2007 #1
    1. The problem statement, all variables and given/known data


    I have no idea how to solve this differential equasion:(d^2y/ds^2)=L^2/(y^3)

    where L is constant. It looks like a inhomogenius DE but what should I do with y^3?
     
    Last edited by a moderator: Jul 15, 2014
  2. jcsd
  3. Nov 28, 2007 #2

    Dick

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    Try looking for a power law solution y=As^k.
     
  4. Nov 28, 2007 #3

    HallsofIvy

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    Dick, it that were [itex]y^2[/itex] on the right side that would work (it would be an "Euler-type" equation) but I don't think it works here. Since the independent variable, s, does not appear explicitely, I would try "quadrature".

    Let v= dy/dt so [itex]d^2y/dt^2= dv/dt= (dv/dy)(dy/dt)= v dv/dy[/itex]. The equation becomes v dv/dy= L/y3. vdv= Ly-3dy. Integrate that to get (1/2)v2= (-L/2)y-2+ C. Since v= dy/dt, that is
    [tex]\frac{dy}{dt}= \sqrt{C- Ly^{-2}}[/tex]
     
  5. Nov 28, 2007 #4

    Dick

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    Thanks, Halls. The power law trick does give you a particular solution proportional to s^(1/2), but that way you get a more general solution.
     
    Last edited: Nov 28, 2007
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