Solve DE: m, h, E, Z, t, k, a (1-4)

[OneByBane]

I am currently trying to solve this differential equation:

r²/F(r) d²F(r)/dr² + 2mr²/h²(E + Zt²/kr) - a² = 0

Wher m, h, E, Z, t and k are other variables and 'a' can have values 1, 2, 3, 4... (Whole numbers)

I have come across this while solving a problem in physics and have no clue if this even has a solution.
Any help will be appreciated greatly.

 

[Strum]

If your equation is given by ( please please please learn to tex )
$$ \frac{r^2}{f(r)} \frac{d^2 f(r)}{dr^2} + \frac{2mr^2}{h^2}\left(E + \frac{zt^2}{kr}\right) -a^2 = 0 $$
Then you can rewrite to
$$ \left( \frac{d^2}{dr^2} + \frac{2mE}{h^2} + \frac{2mzt^2}{h^2 kr} - \frac{a^2}{r^2} \right) f(r) = 0 $$
Which is pretty close to the Whittaker equation ( https://en.wikipedia.org/wiki/Whittaker_function ).

 

[Ssnow]

This is a second order differential equation, you can rewrite it as:

$\frac{d^2}{dr^2}F(r)+ \left[\frac{2m}{h^2}\left(E-\frac{h^2a^2}{2mr^2} + \frac{Zt^2}{kr}\right)\right]F(r) =0$

calling $a(r)=\frac{2m}{h^2}\left(E-\frac{h^2a^2}{2mr^2} + \frac{Zt^2}{kr}\right)$ we have that

$\frac{d^2}{dr^2}F(r)+ a(r)\cdot F(r) =0$

to solve this DE you must find a particular solution in order to find the general ...

 

[the_wolfman]

The solutions to this equation are called Coulomb wave functions. You can also write the solution in terms of confluent hypergeometric functions.