Dear Crossley Mjojo,
I understand that you have a problem at hand.
The problem is whether to find the variable values or prediction parameters.
Here x is the location and T is the temperature at this location x. These two are variables as they vary with locations.
The parameters here are r and s as they define the complicated equation suggested by you for the application you are trying to understand. They are fixed by the domain of (x, T).
Therefore there are two problems at hand.
(1) To find the prediction equation (r , s and the equation)
(2) Using the prediction, finding the T values at various S.
No realtime applications in my opinion involve more than a second order differential equation.
Since you want to find the s value, either you need as many conditions as there are variables unknown or the field data (x, T).
Once the formation is known, then Maple will do it for you instantly. Maple needs what you want and how do you want?
Hope these opinions shared here contribute to a little more clarity in understanding.