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Derivation of Kepler's laws- differential equation question

  1. Sep 21, 2005 #1
    Hi group,

    Could someone 'remind me' why the equation u" + u = km/L^2 has the solution of the form u(theta) = km/L^2 + C cos(theta - theta(o)).
    Any references would be appreciated.
    Thanks

    Dave
     
  2. jcsd
  3. Sep 21, 2005 #2
    a linear differential equation with non zero additional term has a soulution which is a sum of a general solution of this equation with zero additional term and a particular solution of this equation. To be more clear (I assume that theta is independent variable) the equation

    [tex]u^{''}+u=0[/tex]
    has solution
    [tex]u_0=C\times cos(\theta-\theta_0)[/tex]

    on the other hand, your equation has a particular solution

    [tex]u_p=km/L^2=const[/tex]
    so the general solution is

    [tex]u=u_0+u_p[/tex]
     
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