A singular Jacobian indicates that the initial guess causes the solution to diverge. The BVP4C function finds the solution by solving a system of nonlinear algebraic equations. Nonlinear solvers are only as effective as the initial guess they start with, so changing your starting guess may help. Also, BVP4C is responsible for providing a guess for the solution from one iteration to another.
A detailed explanation and possible solutions for this error are given in the link below:
Also, based on the picture of the equation you are solving, I believe the rod_BV function should be written as follows:
function dydx = rod_BV(r,y)
dydx = zeros(2,1);
dydx(1) = y(2)
dydx(2) = -(q/r)-((1/r)*y(2))
This is because T = y(1), T' = y(2), which implies, T'' = -(q/r)-(1/r)*y(2).
Hope this helps!