Need help to solve one differential equation

[TimJ]

Does anybody know how to solve this differential equation:

$$\frac{d^2 \theta}{dt^2}-A\sin(\theta)- \frac{B^2 \cos(\theta)}{ \sin^3(\theta) }=0$$

This is an equation that i get from the brachistochronic motion on the surface of a sphere.

 

[tiny-tim]

Welcome to PF!

Hi TimJ! Welcome to PF!

(have a theta: θ and an omega: ω )

$$\frac{d^2 \theta}{dt^2}-A\sin(\theta)- \frac{B^2 \cos(\theta)}{ \sin^3(\theta) }=0$$

Try writing d²θ/dt² = dω/dt = dω/dθ dθ/dt = ω dω/dθ = 1/2 d(ω²)dθ,

so d(ω²)/dθ = 2Asinθ + 2Bcotθcosec²θ

 

[TimJ]

Thank you for your help.