Sturm-Liouville for a solid cylinder problem

Question

I am trying to find the solution of the problem with boundary conditions

$$ \frac{d}{dR}\left(r \frac{dR}{dr}\right)+(\beta^2r-(v^2/r)) $$ in $0\le r\le b$

with the condition $k\dfrac{dR}{dr}+hR=0$.

I am trying to solve this like a simple Sturm-Liouville problem, but I had a lot of trouble with this case, every prove that i made was inspired by a 2D Sturm-Liouville problem.

Answer 1

I will answer to what I think you mean.

ode = D[r*D[R[r], r], r] + (β^2*r - v^2/r) == 0
(*β^2 r + r R''[r] + R'[r] - v^2/r == 0*)

DSolve[ode, R[r], r] // Flatten
(*{R[r] -> C[1] Log[r] + C[2] - (β^2 r^2)/4 + 1/2 v^2 Log[r]^2}*)

R[r_] = R[r] /. % /. {C[1] -> c1, C[2] -> c2}
c1 Log[r] + c2 - (β^2 r^2)/4 + 1/2 v^2 Log[r]^2

Applying the condition at r = b:

(k R'[r] + h R /. r -> b) == 0
(*k (-(1/2) (b β^2) + c1/b + (v^2 Log[b])/b) + h R == 0*)

Solve[%, c1] // Flatten // Simplify
(*{c1 -> 1/2 b (b β^2 - (2 h R)/k) - v^2 Log[b]}*)

c1 = c1 /. %

R[r]
(*Log[r] (1/2 b (b β^2 - (2 h R)/k) - 
v^2 Log[b]) + c2 - (β^2 r^2)/4 + 1/2 v^2 Log[r]^2*)

You need another condition to solve for c2, so this solution is correct within a constant

Check the solution

ode // Simplify
(*True*)

(k R'[r] + h R /. r -> b) == 0 // Simplify
(*True*)