Question

Ordinary differential equation I'm having trouble solving

Answer 1

\[\frac{\sin x(y'\sin x)'}{y} + c=-k^2 \sin^2 x \]

k is an arbitrary real number, c is (I think only negative) but it could possibly be positive. I am fairly certain it's going to be only one sign, if that helps. I can show the PDE I came from, but I don't want to clutter this so much.

Answer 2

Real problem I have separated: \[\Psi(\theta, \phi)\] \(\phi\) is the azimuthal angle (goes \(2 \pi\) around equator) and \(\theta\) is the zenith angle, so it only goes \(\pi\). This will give us some boundary conditions I think. Here is the differential equation:
\[\frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial \Psi}{\partial \theta} \right)+ \frac{1}{\sin^2 \theta}\frac{\partial^2 \Psi}{\partial \phi^2} = -k^2 \Psi\]
\[\Psi(\theta, \phi) =\Theta(\theta) \Phi(\phi) \]
After plugging that in, the equation for \(\Phi\) is kind of trivial, just going to be an exponential, and I assume it will be a sine or cosine so this will make c a negative number to meet our boundary conditions. So here I replaced this for my equation:
\[y=\Theta, \ x = \theta \]
\[\frac{\sin x(y'\sin x)'}{y} -c^2=-k^2 \sin^2 x \]

And here I'm going to go ahead and claim that c can be any real number now since it will always give a negative number in that spot now, but if my reasoning is off change it to positive if you like.

Answer 3

... Unrelated to the math, but if you want motivation, I'm trying to find a generalization of this idea of aromaticity in organic chemistry similar to the Huckel rule that a cyclic compound with conjugated systems has 4n+2 electrons in what are essentially p-orbitals. However I would like to try to derive for particles on a sphere to see if I can try to match the energy levels and stability of buckyballs, which is similar but I am curious how close "particle on a sphere" gets to modelling this.

https://en.wikipedia.org/wiki/Particle_in_a_ring

Answer 4

Hmm, related to you unrelated post, I've done something similar to particle in a ring, as I think it maybe related to this one question I have done regarding a bead along a hoop which rotates, there is a nice way to find the hamiltonian to it and seeing if the energy is conserved or not...probably not the place for this discussion, just thought I'd share it lol if it has to do with it in anyway.

Answer 5

I think it was called particle in a ring..so yeah lol

Answer 6

Yeah sounds useful and interesting if you wanna explain it!

Answer 7

Ok since it's been some time since I've done this mechanics, ok so this is what I believe the problem went like it's a particle moving on a ring while it's rotating with angular velocity w so we can start of by finding the Lagrangian of this particle \[L = T-U = \frac{ 1 }{ 2 }(\dot x^2+ \dot y^2+ \dot z ^2)- mgz\] and we know the bead is constraint to the hoop...I don't know if this is related at all, still want me to continue?