Lecture Notes in Differential Equations - Bruce E. Shapiro

49
This is identical to
with
Therefore the voltage is given by
dV
dt = V ∞ − V
τ
(6.30)
V ∞ = V batt (6.31)
τ = RC (6.32)
V = V batt + (V 0 − V batt )e −t/RC (6.33)
where V 0 is the voltage at t = 0. Regardless of the initial voltage, the
voltage always tends towards the battery voltage. The current through the
circuit is
i = C dV
dt = V 0 − V batt
e −t/RC (6.34)
R
so the current decays exponentially.
Example 6.3. Newton’s Law of Heating (or Cooling) says that the rate
of change of the temperature T of an object (e.g., a potato) is proportional
to the difference in temperatures between the object and its environment
(e.g., an oven), i.e.,
dT
dt = k(T oven − T ) (6.35)
Suppose that the oven is set to 350 F and a potato at room temperature
(70 F) is put in the oven at t = 0, with a baking thermometer inserted into
the potato. After three minutes you observe that the temperature of the
potato is 150. How long will it take the potato to reach a temperature of
350?
The initial value problem we have to solve is
Dividing and integrating,
From the initial condition
}
T ′ = k(400 − T )
T (0) = 70
∫
∫
dT
400 − T =
(6.36)
kdt (6.37)
− ln(400 − T ) = kt + C (6.38)
C = − ln(400 − 70) = − ln 330 (6.39)