Chapter 5: Applications of Integration
Section 5.6: Differential Equations
Example 5.6.9
Obtain the general solution of the differential equation u.=k u−us, where ut represents the temperature of a body in thermal contact with its surroundings at a fixed temperature us. This equation, sometimes called Newton's law of cooling, simply states that the rate of change of temperature of the body is proportional to the difference in temperature between the body and its surroundings.
Solution
Mathematical Solution
The differential equation is separable, and can be solved with the same approach taken in Example 5.6.7. After the variables are separated, elementary integration lead to the logarithm of an absolute value on the left. Exponentiation of both sides removes the logarithm. The exponential of the arbitrary constant of integration is denoted by A^. Removing the absolute value on the left might require introducing a minus sign. If so, let this sign change be provided by the arbitrary constant, now called A.
duu−us
=k dt
ln(u−us)
=k t+λ
u−us
=ek t+λ
=ek t⋅eλ
=A^ ek t
=A ek t
u
=us+A ek t
Maple Solution
Control-drag the differential equation. Convert us to the Atomic Identifier u__s.
Context Panel: Solve DE≻ut
u.=k u−u__s
ⅆⅆtut=kut−u__s
→solve DE
ut=u__s+_C1ⅇkt
Maple writes _C1 for the arbitrary constant, the lead underscore indicating that Maple introduced this symbol into the calculation.
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