I am working on a $2D$ steady state heat equation (Laplacian). I did Separation of Variables and am evaluating $3$ cases ($k>0, k<0, k=0$). The problem is a cylinder with height 1 and radius 1, where the temperature is 1 on the top, and 0 on the curved wall and the bottom. Thus $T(r,0)=0$, $ T(r,1)=1$, $T(1,z)=0$.
The problem is axisymmetric so no theta terms are included.
I started with: $${∂^2T\over∂r^2} +{1\over r}{∂T\over∂r} + {∂^2T\over∂z^2}=0$$
Tried Separation of Variables as follows: $T(r,z) = R(r)Z(z)$
Which resulted in these two ODEs:
$${1\over R}{∂^2T\over∂r^2} +{1\over r}{∂T\over∂r} = k$$ $$-{1\over Z}{∂^2Z\over∂z^2} = k$$
I am struggling to understand which case is appropriate for this problem. I believe $k=0$ results in a singularity at $r=0$ due to a natural logarithm term, so that case does not seem correct. I am having a difficult time with $k>0$ and $k<0$ so any guidance would be appreciated.
I think, but am not certain, that $k>0$ leads to an exponential solution for Z (not sure about R).
I think, but am not certain, that $k<0$ leads to an oscillatory solution for Z, and a Bessel function for R.
