Nonlinear Heat Loss Model

Background:

In the derivation for the heat-loss model with no internal heat source (so the furnace is broken), the outside temperature was assumed to be constant. This is clearly not realistic for many applications.

1. Does the model derivation remain valid for this more general case? Is the appropriate heat-loss model the obvious one:

   

2. When is constant, it isn't hard to solve the model and show that

   

   (This was done in lecture.) Is it still true for the more general model that

   

3. When is constant and , it isn't hard to solve the model and show that for all t. Does this remain true for the more general model? (Can it happen that the temperature inside the house would actually fall belowthe outside temperature when the furnace breaks down?)
4. Do some numerical examples (with MATLAB) to illustrate your conclusions for the last two parts.