The analytic solution of the differential equation of the steady-state one-dimensional temperature distribution in a plate with linearly tempeature-dependent thermal conductivity and internal heat generation is presented. It is shown that the solution of this differential equation with a numerical method and its analytic solution with the assumption that the thermal conductivity of the material of the plate is constant while it depends on the temperature of the plate linearly are trivial for some conditions, which may be important for practical applications. Simple criteria are given to avoid these trivial solutions.