Motion in a Central Gravitational Force

[Macykc2]

Homework Statement

Discuss the motion of a particle in a central inverse-square-law force field for a superimposed force whose magnitude is inversely proportional to the cube of the distance from the particle to the force center, that is:

F(r) = -k/r² - λ/r³ and k,λ>0​

Show that the motion is described by a precessing ellipse.

Homework Equations

The one given in the question

The Attempt at a Solution

I honestly don't know where to begin, we derived an equation in class that we could find the force law if a particular known orbit r=r(θ), and I was thinking of using it:

(d²/dθ²)(1/r) + (1/r) = -μr²F(r)/l²​

but I don't know what to do with the left hand side, but again it's just a guess as to what to start with.
Sorry for not using the math commands, they didn't want to work for some reason.

 

[Simon Bridge]

You need to solve the differential equation.

For equations, you can type out LaTeX markup directly.

You may want to back up a bit and make sure you understand the motivation for deriving the equation you used in the first place... see if the same approach still applies here.