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Homework Help: Solving ODE involving square of first derivative

  1. Aug 14, 2010 #1
    1. The problem statement, all variables and given/known data

    this is not from a math course, but from Gregory's classical mechanics book prob 2.10
    it's easy to obtain the desired ODE
    [tex]\dot{r}^{2}=\frac{u^{2}}{a^{2}}(\frac{U^{2}a^{2}}{a^{2}}-r^{2})[/tex]
    since it's non-linear, i have a difficult time to solve for r(t)
    u, U and a are some constants with unit speed, speed and length resp.
    2. Relevant equations



    3. The attempt at a solution
    all methods i know fail, I believe there is some trick that I am not aware of. great appreciate for any help:(
     
  2. jcsd
  3. Aug 14, 2010 #2

    hunt_mat

    User Avatar
    Homework Helper

    The equation you have can be written as:
    [tex]
    \dot{r}=\pm\frac{u}{a}\sqrt{U^{2}-r^{2}}
    [/tex]
    Dividing and integrating shows that:
    [tex]
    \int\frac{dr}{\sqrt{U^{2}-r^{2}}}=\pm\frac{u}{a}\int dt
    [/tex]
    The integral can be solved by the substitution:
    [tex]
    r=U\sin\alpha
    [/tex]
    I will leave you to slog through the algerbra.
     
  4. Aug 14, 2010 #3
    oh...thank you very much!!!
    I think i need to brush up my math skills....


    edit: in fact i type the ODE wrongly, but the method should be similar
     
    Last edited: Aug 14, 2010
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