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This approach has two great problems: First the parameters $c$, $m$, $a$ seems to be easy to define when talking about a whole planet, but quite hard when we analyze only a small portion. Second, and more important, the equation seems to be unsolvable at least when Star 2 is visible. In general it's a Chini differential equation (see here). When both stars are not visible it becomes a Bernoulli equation and the solution is easy to find; when only Star 1 is visible the Chini invarant is constant ($C=0$ to be precise), so there is a precise solution to that case too. When Star 2 is visible, however, there seems to be no way to find a solution. So, after this wall of text, my question is:

This approach has two great problems: First the parameters $c$, $m$, $a$ seems to be easy to define when talking about a whole planet, but quite hard when we analyze only a small portion. Second, and more important, the equation seems to be unsolvable at least when Star 2 is visible. In general it's a Chini differential equation (see here). When both stars are not visible it becomes a Bernoulli equation and the solution is easy to find; when only Star 1 is visible the Chini invarant is constant ($C=0$ to be precise), so there is a precise solution to that case too. When Star 2 is visible, however, there seems to be no way to find a solution. So, after this wall of text, my question is:

Edit: I've done some computer simulations and it seems to work, but it needs some calibrations due to the not-so meaningful parameters and greenhouse effect, that acts only on the emitted thermal radiation. The model still doesn't take into account the heat redistribution due to air and water flows, but this is probably way too complicated to put in the model. A few plots can be seen in this gallery. Blue line indicates sunrise, red line sunset, green line is the instantaneous temperature, yellow line is the average temperature and the arrow field should give an idea of the slopes the differential equation produces, but it always seems less inclined than the green curve.

Edit: I've done some computer simulations and it seems to work, but it needs some calibrations due to the not-so meaningful parameters and greenhouse effect, that acts only on the emitted thermal radiation. The model still doesn't take into account the heat redistribution due to air and water flows, but this is probably way too complicated to put in the model. A few plots for Earth ( 1 star, $\tau = 1$ day) can be seen in this gallery. Blue line indicates sunrise, red line sunset, green line is the instantaneous temperature, yellow line is the average temperature and the arrow field should give an idea of the slopes the differential equation produces, but it always seems less inclined than the green curve.

Edit: I've done some computer simulations and it seems to work, but it needs some calibrations due to the not-so meaningful parameters and greenhouse effect, that acts only on the emitted thermal radiation. The model still doesn't take into account the heat redistribution due to air and water flows, but this is probably way too complicated to put in the model. A few plots can be seen in this gallery. Blue line indicates sunrise, red line sunset, green line is the instantaneous temperature, yellow line is the average temperature and the arrow field should give an idea of the slopes the differential equation produces, but it always seems less inclined than the green curve.

The plot is produced with a periodicity condition: the algorithm starts with a temperature of 300 K and cycles through a few days until the temperature at the beginning of the day matches the temperature at the end of the day. The seasonal variation is slow enough that this should be a good approximation.