General solution of a differential equation

Question

I'm a beginner in Mathematica and I'm learning about dealing with differential equations but I'm stuck in this problem

What I tried to do is I observed that I have the same differential operator acting on the function w(r), so I defined the operator

L = D[#, {r, 2}] + (1/r) D[#, r] - (m^2/r^2) &;

Then I tried to solve the equation L^2[w[r]] == q_0 using Solve but it did not work. I also tried to define the expression

L[f_] := D[f[r], {r, 2}] - (1/r) D[f[r], r] - (m^2/r^2) f[r]

then acting on it using the differential operator but it did not work tho.

Answer 1

I think you mis-coded something. Try the following:

L[funct_, r_] := 
 D[funct, {r, 2}] + (1/r) D[funct, r] - (m^2/r^2) funct &@funct 
DSolve[L[L[f[r], r], r] == 0, f[r], r]

to get your solution