Applications of Differential Equations
Temperature
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?
