Derivation of Kepler's laws- differential equation question

Click For 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
Messages
9
Reaction score
0
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
 
Physics news on Phys.org
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
 

Thread 'Reference frames, center of rotation, etc'

Topic about reference frames, center of rotation, postion of origin etc Comoving ref. frame is frame that is attached to moving object, does that mean, in that frame translation and rotation of object is zero, because origin and axes(x,y,z) are fixed to object? Is it same if you place origin of frame at object center of mass or at object tail? What type of comoving frame exist? What is lab frame? If we talk about center of rotation do we always need to specified from what frame we observe?
View full post »

Similar threads

Graduate Doubt in a step while deriving Bertrand theorem
  • · Replies 2 ·
Replies
2
Views
849
High School Sign conventions in torque and non-uniform circular motion
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Mathematical disagreement with inverse square law?
  • · Replies 33 ·
2
Replies
33
Views
3K
Undergrad What is the derivation of the radial acceleration in Kepler's 1st Law?
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Find period of circular-orbiting source based on max observed freq
  • · Replies 1 ·
Replies
1
Views
1K
Differential Equation of a Pendulum
  • · Replies 32 ·
2
Replies
32
Views
2K
Undergrad Missing exponent in "Theoretical Minimum"?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Alternative Approach to Solving Foucault's Pendulum Behavior
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad How Does Temperature Affect Wire Tension at Constant Length?
  • · Replies 4 ·
Replies
4
Views
2K
Deriving ODEs for straight lines in polar coordinates for a given Lagrangian
  • · Replies 8 ·
Replies
8
Views
2K