# Sturm-Liouville for a solid cylinder problem

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.

• Are you trying to solve this in the software Mathematica & Wolfram language ? Aug 18, 2020 at 15:26
• I assume you mean d/dr instead of d/dR in that equation. Otherwise, it is nowhere near a Sturm Liouville equation. Aug 18, 2020 at 17:49

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 Log[r] + C - (β^2 r^2)/4 + 1/2 v^2 Log[r]^2}*)

R[r_] = R[r] /. % /. {C -> c1, C -> 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*)