Applications of Differential Equations


The rate of change of temperature, T, of an object placed in a constant temperature medium, Tm (such as air, or a large body of water) is proportional to the temperature difference between the object and the medium.

Try writing that as a differential equation:

                dT/dt = k (T – Tm)

Now, let’s look at a speicifc problem:

An object which has a temperature of 100C is placed into air at 20C. Its temperature drops to 50C in 10 minutes. Express T(t).

Summarize the known information.

      t = 0    T = 100C

        t = 10     T = 50C

        Tm= 20C

         dT/dt = k (T – Tm)

    What do you do next?

    Plug Tm= 20C into the equation and separate the variables.

    dT/dt = k (T – 20)

    dT/(T – 20) = k dt

Now what?