The title is only there to attract attention. In any case, I myself was not clever enough to conceive the following quite simple solution of the heat equation until today.
The following model differs from the previous one only in that it takes into account a position dependent thermal diffusivity. We assume that the diffusivity is inversely proportional to the density, which in the atmosphere decreases roughly exponentially with height. In other words:
D(x) = D*exp(x/H)
where H is the height scale.
Now we make the simple observation
heat transport per unit time = -D(x)*dT/dx
where T is temperature. If we let E denote the heat recieved from the sun, for a stationary solution we must have
-D(x)*dT/dx = constant = E
with the solution
T(x) = (EH/D)*exp(-x/H) + C
where C is a constant that has to be determined by imposing some boundary condition. The first thing to notice is that the model now predicts the existence of a stratosphere. It smothly approaches a zero decline in temperature with height. Of course, the temperaure decline in our atmosphere is not exponential but has more of a linear shape in the lower part, and this could possibly be explained by a "convective adjustment". Then there are other complications such as a temperature dependent density and things like that, but maybe Claes' computer simulations could bring more light on these issues some day.