Derivation of Kepler's laws- differential equation question

[ddoctor]

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

 

[shyboy]

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

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

on the other hand, your equation has a particular solution

$$u_p=km/L^2=const$$
so the general solution is

$$u=u_0+u_p$$