Derivation of Kepler's laws- differential equation question

AI Thread Summary
The equation u" + u = km/L^2 has a solution of the form u(theta) = km/L^2 + C cos(theta - theta(o)) due to the principle of superposition in linear differential equations. The general solution consists of the homogeneous part, u_0 = C cos(theta - theta_0), and a particular solution, u_p = km/L^2. The presence of the non-zero term km/L^2 allows for the addition of the constant solution to the general solution. This approach combines the effects of both the homogeneous and particular solutions. Understanding this concept is crucial for solving linear differential equations effectively.
ddoctor
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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
 
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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

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
 
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