6.12 Solution to differential equation

You have already seen one way to solve the differential equation in the last frame, in MATH101 §3. But it is a separable equation, so we can also use the method outlined in 6.8 and 6.9. Dividing by $T-\tau$ and integrating, we obtain

and therefore $\log|T-\tau|=-\lambda t+c$. Taking exponentials, we obtain $T-\tau=\pm e^{-\lambda t+c}$, and therefore $T=\tau+Ae^{-\lambda t}$ where $A=\pm e^{c}$.

Note that as $t\rightarrow\infty$, the temperature tends to $\tau$. The greater the value of $\lambda$, the faster this convergence occurs. Given an initial condition $T(0)=T_{0}$, we obtain the value of $A={T_{0}-\tau.}$

NB. The Law of Cooling is only a good approximation which holds under certain conditions. Beware of physics lessons taught by mathematicians.