The general d.e. (for constant area, and long wire) that you are looking for looks as follows:
(A/rho) *V'' - (A/rho^2)*V'*T'* (drho/dt) = 0 The current constancy equation.
V'=dV/dx
rho=rho0(1+alpha (T-T0)), alpha can be suitably obtained from Handbooks.
This d.e. can be readily solved.
Then...
