Incorporate axis symmetric case

[Dustinsfl]

With this code, I can't look at the $\mathcal{J}_0$ or $\mathcal{Y}_0$. How can I altered to to pick up those orders?

ClearAll["Global'*"];
inits = Table[
   FindRoot[
    BesselJ[n, x]*BesselY[n, 2*x] - BesselJ[n, 2*x]*BesselY[n, x] == 
     0, {x, n + 3}], {n, 1, 5}];
g1 = x /. inits

zeros = Table[
   FindRoot[
    BesselJ[n, x]*BesselY[n, 2*x] - BesselJ[n, 2*x]*BesselY[n, x] == 
     0, {x, g1[[n]] + (m - 1) Pi}], {m, 1, 5}, {n, 1, 5}];
g = x /. zeros;
g // TableForm

x = r Cos[t]; y = r Sin[t];
m = 2; n = 2;
ParametricPlot3D[{x, 
  y, (BesselJ[n, g[[m, n]]*r]*BesselY[n, 2*g[[m, n]]*r] - 
     BesselJ[n, 2*g[[m, n]]*r]*BesselY[n, g[[m, n]]*r]) Cos[n t]}, {r,
   0, 1}, {t, 0, 2 Pi}]

 

[NateD]

To get the $\mathcal{J}_0$ and $\mathcal{Y}_0$ you need to alter the code as follows: ClearAll["Global'*"];inits = Table[ FindRoot[ BesselJ[n, x]*BesselY[n, 2*x] - BesselJ[n, 2*x]*BesselY[n, x] == 0, {x, n + 3}], {n, 0, 5}];g1 = x /. initszeros = Table[ FindRoot[ BesselJ[n, x]*BesselY[n, 2*x] - BesselJ[n, 2*x]*BesselY[n, x] == 0, {x, g1[[n]] + (m - 1) Pi}], {m, 1, 5}, {n, 0, 5}];g = x /. zeros;g // TableFormx = r Cos[t]; y = r Sin[t];m = 2; n = 2;ParametricPlot3D[{x, y, (BesselJ[n, g[[m, n]]*r]*BesselY[n, 2*g[[m, n]]*r] - BesselJ[n, 2*g[[m, n]]*r]*BesselY[n, g[[m, n]]*r]) Cos[n t]}, {r, 0, 1}, {t, 0, 2 Pi}] The key change is changing the line that reads:inits = Table[ FindRoot[ BesselJ[n, x]*BesselY[n, 2*x] - BesselJ[n, 2*x]*BesselY[n, x] == 0, {x, n + 3}], {n, 1, 5}]; to this: inits = Table[ FindRoot[ BesselJ[n, x]*BesselY[n, 2*x] - BesselJ[n,