Applications of Differential Equations

Temperature

The rate of change of temperature, T, of an object placed in a constant temperature medium, T_{m} (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 – T_{m})

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

T_{m}= 20C

dT/dt = k (T – T_{m})

What do you do next?

Plug T_{m}= 20C into the equation and separate the variables.

dT/dt = k (T – 20)

dT/(T – 20) = k dt

Now what?