# Newton's Law of Cooling ?

Nov 18, 2016

$T \left(t\right) = \frac{3}{k} \left(k t + {e}^{- k t} - 1\right)$

#### Explanation:

Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between the ambient temperature and its own temperature.

$\frac{d}{\mathrm{dt}} T \left(t\right) = k \left(3 t - T \left(t\right)\right)$ or

$\frac{d}{\mathrm{dt}} T \left(t\right) + k T \left(t\right) = 3 k t$

Here $k$ is the proportionality constant.

Solving this differential equation we have

$T \left(t\right) = \frac{3 \left(k t - 1\right)}{k} + {C}_{1} {e}^{- k t}$

Applying the initial conditions we have

$T \left(0\right) = 0$ so $- \frac{3}{k} + {C}_{1} = 0$ then

$T \left(t\right) = \frac{3}{k} \left(k t + {e}^{- k t} - 1\right)$