Derivation of Kepler's laws- differential equation question

  • Thread starter ddoctor
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  • #1
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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
 

Answers and Replies

  • #2
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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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