# A Differential Equation for Heat Transfer According to Newton's Law of Cooling

A Differential Equation for Heat Transfer According to Newton's Law of Cooling

Let be the temperature of a building (with neither heat nor air conditioning running) at time and let be the temperature of the surrounding air. Newton's law of cooling states that

u(t)

t

T(t)

du

dt

where is a constant independent of time.

k

In this example, is plotted in red; is the initial temperature of the air surrounding the building. The value of depends on a number of factors including the insulation of the building. In this example, you can modify the proportionality constant and mean temperature.

T(t)=T(0)+15sin(2πt)

T(0)

k

Five specific solutions around the initial temperature are plotted as solid blue curves through the dashed blue lines that represent the direction fields.