Differential Equation Examples
worksheet based on material provided by Dr. Eaton
1. Dr. Card is found dead in his office at 5:00pm one evening. The temperature of his
body was 80.0◦F. One hour later, at 6:00pm, the body has cooled to 75.0◦F. The room
is kept at a constant temperature of 70◦F. Assume Dr. Card had a normal temperature
of 98.6◦F at the time of death.
Let f(t) be the temperature of the body after thours.
(a) Since the rate a body cools is proportional to the difference in temperature be-
tween the body and the ambient temperature, we can write down a differential
equation. Recall “proportional to” means “is a constant ktimes”.
(b) The formula you wrote down above is known as Newton’s Law of Cooling and
happens to be a separable differential equation. Use the techniques we covered in
class today to solve the differential equation you wrote in part (a). Treat kas a
constant and leave the “+c” on the side with the t’s.
(c) Solve for f(t) as a function of t. Note, your function will involve the constants k
and c.
(d) Recall that you were given two data points in the above information. Let t= 0
correspond to the time the body was found and solve the initial value problem
and give us an explicit function f(t) that models the temperature of Dr. Card’s
body.
(e) When did the murder take place?
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