# The differential equation below models the temperature of a 95°C cup of coffee in a 21°C room, where it is known that the coffee cools at a rate of 1°C per minute when its temperature is 71°C. How to solve the differential equation?

Dec 2, 2017

See below.

#### Explanation:

This is a separable differential equation so it can be arranged as

$\frac{\mathrm{dy}}{y - 21} = - \frac{1}{50} \mathrm{dt}$

Now, integrating each side

${\log}_{e} \left\mid y - 21 \right\mid = - \frac{1}{50} t + {C}_{0}$ or

$y - 21 = {C}_{1} {e}^{- \frac{t}{50}}$

Now, at $t = 0$ the temperature is ${95}^{\circ}$ or

${95}^{\circ} - {21}^{\circ} = {C}_{1} {e}^{0} \Rightarrow {C}_{1} = {74}^{\circ}$ and finally

$y = {21}^{\circ} + {74}^{\circ} {e}^{- \frac{t}{50}}$