According to Newton's law of cooling, the temperature of a body changes at a rate proportional to the difference between the temperature of the body and the temperature of the surrounding medium. Thus, if Tm is the temperature of the medium and T = T(t) is the temperature of the body at time t, then T'= -k(T – Tm) where k is a positive constant. In this exercise, you will verify that T=Tm+ (To-Tm)e -kt is a solution to the differential equation. Here To denotes the initial temperature T(0) of the body. (a) First substitute the above expression for T into the left side of the differential equation. In other words, compute T'. Use an underscore "_" to write the subscripts on To and Tm T'= (b) Next substitute the above expression for T into the right side of the differential equation. In other words, compute -k(T - Tm). -k(T - Tm) (c) Are your answers in parts (a) and (b) equal? No Yes
According to Newton's law of cooling, the temperature of a body changes at a rate proportional to the difference between the temperature of the body and the temperature of the surrounding medium. Thus, if Tm is the temperature of the medium and T = T(t) is the temperature of the body at time t, then T'= -k(T – Tm) where k is a positive constant. In this exercise, you will verify that T=Tm+ (To-Tm)e -kt is a solution to the differential equation. Here To denotes the initial temperature T(0) of the body. (a) First substitute the above expression for T into the left side of the differential equation. In other words, compute T'. Use an underscore "_" to write the subscripts on To and Tm T'= (b) Next substitute the above expression for T into the right side of the differential equation. In other words, compute -k(T - Tm). -k(T - Tm) (c) Are your answers in parts (a) and (b) equal? No Yes
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:According to Newton's law of cooling, the
temperature of a body changes at a rate proportional
to the difference between the temperature of the
body and the temperature of the surrounding
medium. Thus, if Tm is the temperature of the
medium and T = T(t) is the temperature of the
body at time t, then
T'= -k(T - Tm)
where k is a positive constant.
In this exercise, you will verify that
T = Tm+ (To-Tm)e -kt is a solution to the
differential equation. Here To denotes the initial
temperature T(0) of the body.
(a) First substitute the above expression for T into
the left side of the differential equation. In other
words, compute T'.
Use an underscore "_" to write the subscripts on To
and Im
T'=
(b) Next substitute the above expression for T into
the right side of the differential equation. In other
words, compute -k(T – Tm).
-
-k(T - Tm)
(c) Are your answers in parts (a) and (b) equal?
No
Yes
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