We study Newton’s Law of Cooling as an application of a first order separable differential equation.

Newton’s Law of Cooling

Newton’s law of cooling states that if an object with temperature at time is in a medium with temperature , the rate of change of at time is proportional to ; thus, satisfies a differential equation of the form

Here , since the temperature of the object must decrease if , or increase if . We’ll call the temperature decay constant of the medium.

For simplicity, in this section we’ll assume that the medium is maintained at a constant temperature . This is another example of building a simple mathematical model for a physical phenomenon. Like most mathematical models it has its limitations. For example, it’s reasonable to assume that the temperature of a room remains approximately constant if the cooling object is a cup of coffee, but perhaps not if it’s a huge cauldron of molten metal.

To solve (eq:4.2.1), we rewrite it as Since is a solution of the complementary equation, the solutions of this equation are of the form , where , so . Hence, so If , setting here yields , so

Note that decays exponentially, with decay constant .

Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)