Methods from Modelling with first-order equations

Build a first-order model by translating the verbal rate law:

• proportional change gives $P'=kP$;
• relaxation towards a target gives $T'=-k(T-A)$, with $k>0$ for decay toward the target;
• conservation gives $$\text{rate of change}=\text{inflow}-\text{outflow}.$$

Use initial or measured data to determine the constants, then interpret the resulting solution on the physical interval where the model is valid.