DSolve[ (Sin[x1]^2 Sin[
     x2] ((-1 + l) (3 + l) q1[z] + 3 z Derivative[1][q1][z] -
      z^2 (q1^[Prime][Prime])[z]))/z^5 + (
   Sin[x1]^2 Sin[
     x2] ((1 - l) (3 + l) z^2 q1[z] - z^3 Derivative[1][q1][z] +
      z^4 (q1^[Prime][Prime])[z]))/(L^2 z^5) == 0, q1[z], z]

{{q1[z] ->
   I^(1 - l) L^(-1 + l) z^(1 - l)
      C[1] Hypergeometric2F1[-(l/2) -
       1/2 Sqrt[-2 + 2 l + l^2], -(l/2) +
       1/2 Sqrt[-2 + 2 l + l^2], -l, z^2/L^2] +
    I^(3 + l) L^(-3 - l) z^(3 + l)
      C[2] Hypergeometric2F1[1 + l/2 - 1/2 Sqrt[-2 + 2 l + l^2],
      1 + l/2 + 1/2 Sqrt[-2 + 2 l + l^2], 2 + l, z^2/L^2]}}

q1[z_] :=
 I^(1 - l) L^(-1 + l) z^(1 - l)
    Hypergeometric2F1[-(l/2) - 1/2 Sqrt[-2 + 2 l + l^2], -(l/2) +
     1/2 Sqrt[-2 + 2 l + l^2], -l, z^2/L^2] +
  I^(3 + l) L^(-3 - l) z^(3 + l)
     Hypergeometric2F1[1 + l/2 - 1/2 Sqrt[-2 + 2 l + l^2],
    1 + l/2 + 1/2 Sqrt[-2 + 2 l + l^2], 2 + l, z^2/L^2]
(Sin[x1]^2 Sin[
      x2] ((-1 + l) (3 + l) q1[z] + 3 z Derivative[1][q1][z] -
       z^2 (q1^[Prime][Prime])[z]))/z^5 + (
    Sin[x1]^2 Sin[
      x2] ((1 - l) (3 + l) z^2 q1[z] - z^3 Derivative[1][q1][z] +
       z^4 (q1^[Prime][Prime])[z]))/(L^2 z^5) == 0 //
  Factor // Simplify

(1/(l (2 + l)))
 I^-l L^(-1 - l)
   z^(-1 - l) (2 l (-2 + l + l^2) L^(6 + 2 l)
      Hypergeometric2F1[1/2 (-l - Sqrt[-2 + 2 l + l^2]),
      1/2 (-l + Sqrt[-2 + 2 l + l^2]), -l, z^2/L^2] -
    2 (-2 + l + l^2) L^(
     4 + 2 l) (z^2 + l (L^2 - z^2)) Hypergeometric2F1[
      1/2 (2 - l - Sqrt[-2 + 2 l + l^2]),
      1/2 (2 - l + Sqrt[-2 + 2 l + l^2]), 1 - l, z^2/L^2] +
    z^2 (3 (-2 + l + l^2) L^(
        2 + 2 l) (L^2 - z^2) Hypergeometric2F1[
         1/2 (4 - l - Sqrt[-2 + 2 l + l^2]),
         1/2 (4 - l + Sqrt[-2 + 2 l + l^2]), 2 - l, z^2/L^2] +
       I^(2 l) l (3 + l) z^(
        2 l) (2 (2 + l) L^4 Hypergeometric2F1[
            1/2 (2 + l - Sqrt[-2 + 2 l + l^2]),
            1/2 (2 + l + Sqrt[-2 + 2 l + l^2]), 2 + l, z^2/L^2] +
          2 L^2 (-(2 + l) L^2 + (3 + l) z^2) Hypergeometric2F1[
            1/2 (4 + l - Sqrt[-2 + 2 l + l^2]),
            1/2 (4 + l + Sqrt[-2 + 2 l + l^2]), 3 + l, z^2/L^2] +
          3 z^2 (-L^2 + z^2) Hypergeometric2F1[
            1/2 (6 + l - Sqrt[-2 + 2 l + l^2]),
            1/2 (6 + l + Sqrt[-2 + 2 l + l^2]), 4 + l, z^2/
            L^2]))) Sin[x1] Sin[x2] == 0