Dsolve to find vector potential component's

Question

I want to find the vector potential component's. I expand the Poisson’s equation in cylindrical coordinates which yields the three equations as in the picture. The equation along rho direction equal zero. The symbols e, hpar, m ,l, k, and N are constants. I try to use Mathematica to find them by solve the three related equations together but the results doesn't make sense. Any help!!

My code is

Assuming[\[Rho] >= 0 && Element[l, Integers] && Element[e, Integers] && 
  Element[\[HBar], Integers] && Element[B, Integers] && 
  Element[m, Integers] && Element[k, Integers], 
   DSolve[{(1/\[Rho])*
      D[\[Rho]*D[X[\[Rho]], \[Rho]], \[Rho]] - (l^2/\[Rho]^2)*
      X[\[Rho]] + 
     k^2*X[\[Rho]] + (1/\[Rho]^2)*X[\[Rho]] - ((2*Il)/\[Rho]^2)*
      Y[\[Rho]] == 0, 
       ((1/\[Rho])*
        D[\[Rho]*D[Y[\[Rho]], \[Rho]], \[Rho]] + (1/\[Rho]^2)*l^2*
        Y[\[Rho]] + k^2*Y[\[Rho]] - 
       Y[\[Rho]]/\[Rho]^2) + (2/\[Rho]^2)*
      IlX[\[Rho]] == (eB\[HBar]l*BesselJ[l, k\[Rho]]^2)/
     m\[Rho]}, {X[\[Rho]], Y[\[Rho]]}, \[Rho]]]
&&
Assuming[\[Rho] >= 0 && Element[l, Integers] && Element[e, Integers] && 
  Element[\[HBar], Integers] && Element[B, Integers] && 
  Element[m, Integers] && Element[k, Integers], 
   DSolve[(1/\[Rho])*
     D[\[Rho]*D[A[\[Rho]], \[Rho]], \[Rho]] - (l^2/\[Rho]^2)*
     A[\[Rho]] - k^2*A[\[Rho]] == (eB\[HBar]k*BesselJ[l, k\[Rho]]^2)/
    m, A[\[Rho]], \[Rho]]]

Answer 1

Two fundamental advices for physicists:

• Mention the name of the equation you are copying from literature, in this case "Helmholtz vector equation for the radial part of a cylindrical field" arising from the "Schrödinger equation".

• write physical constants by there names mass, charge, plancks etc-

Advice for Mathematica beginners: Don't use implicit multiplication. It makes error detection nearly impossible expecially if narrow fonts come into use as here.

Because of personal interest I nevertheless decoded your first equation

     Assuming[\[Rho] >= 0 && {l, e, \[HBar], B, m, k} \[Element]  Integers,
 With[{Dr = (1/\[Rho]*D[\[Rho]*D[#, \[Rho]], \[Rho]] &)},
  DSolve[{Dr[X[\[Rho]]] - (l^2 - 1)/\[Rho]^2* X[\[Rho]] + 
      k^2* X[\[Rho]] - (2 Il)/\[Rho] * Y[\[Rho]] == 0,
    Dr[Y[\[Rho]]] + (l^2 - 1)/\[Rho]^2 *Y[\[Rho]] + 
      k^2* Y[\[Rho]] + (2 Il)/\[Rho] * X[\[Rho]] ==  (e* B * \[HBar] )/
      m * 1/\[Rho] * BesselJ[l, k \[Rho]] }, {X[\[Rho]], 
    Y[\[Rho]]}, \[Rho]] ]]


Out= $Aborted

Of course Mathematica is unable to solve this inhomogeneous system of coupled equations. But the system needs infinite time to find out. Its a kind of the

Wikipedia Halting problem

The solutions can be found in Moon/Spencer A Field Theory Handbook

There exist free download adresses in the internet, at least via a student account. A secondary source is

PDF Handbook of Differntial equations