I am trying to follow a book example where I need to solve the following ODE on a circular domain: $$\frac{1}{r}\frac{d}{dr}\left(r\frac{dT}{dr} \right) = -A$$ subject to two Neumann boundary conditions: boundedness at $r = 0$ and $$\frac{dT(R)}{dr} = B,$$ where $A$ and $B$ are known constants.
The solution of the differential equation is: $$T(r) = -Ar^2 + c_1 \ln r + c_2,$$ where the boundedness condition forces $c_1 = 0$. So how can I use the other condition to find $c_2$ when it's going to disappear when I take the derivative? The text presents it as if it's intuitively obvious but I don't see how it's possible. Am I misunderstanding something? I don't see how I have any free parameters in applying the other Neumann condition.
The value of $c_2$ cannot be determined from the information given. If the answer to the problem would be: $$ T(r) = -Ar^2 + c_2, $$ where $c_2$ is an arbitrary real constant.
That was assuming that all solutions were expected. If the goal is to find just one, then take a convenient value for $c_2$, such as $0$.