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d^2T/dr^2+1/r*dT/dr=0 (steady state) or

dT/dt=d^2T/dr^2+1/r*dT/dr (transient state)

The problem is simple: a ring with r1<r<r2, T(r1)=T1, T(r2)=T2.

I have searched the analytical solution for this kind of ODEs in polar coordinate systems, found a lot of solutions for more difficult problems

Can anybody help me with this one please?