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The one-dimensional unsteady state heat conduction equation in a hollow cylinder with a constant heat source $q$ is

$$ \frac{\partial T}{\partial t}=\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial T}{\partial r})+q$$

If $A$ and $B$ are arbitrary constants, then the steady solution to the above equation is

- $T(r) = -\frac{{qr^2}}{2} + \frac{A}{r} + B$
- $T(r) = -\frac{{qr^2}}{4} + A\ln r + B$
- $T(r) = A\ln r + B$
- $T(r) = \frac{{qr^2}}{4} + A\ln r + B$