According to Stefan's law of radiation the absolute temperature T of a body cooling in a medium at constant absolute temperature T, is given by dT = k(T - Tm), where k is a constant. Stefan's law can be used over a greater temperature range than Newton's law of cooling. (a) Solve the differential equation. (b) Show that when T- Tm is small in comparison to T, then Newton's law of cooling approximates Stefan's law. [Hint: Think binomial series of the right-hand side of the DE dT = k(T4 - T4) dt - (- -([ ))'- = KTm + ])-:)- Using the binomial series, we expand the right side of the previous equation. (Enter the first three terms of the expansion.) dT = kT dt m dT When T-Tm is small compared to the third term in the expansion vx can be ignored, giving - z k¸(T – Tm), where k, = 4kT 3. m
According to Stefan's law of radiation the absolute temperature T of a body cooling in a medium at constant absolute temperature T, is given by dT = k(T - Tm), where k is a constant. Stefan's law can be used over a greater temperature range than Newton's law of cooling. (a) Solve the differential equation. (b) Show that when T- Tm is small in comparison to T, then Newton's law of cooling approximates Stefan's law. [Hint: Think binomial series of the right-hand side of the DE dT = k(T4 - T4) dt - (- -([ ))'- = KTm + ])-:)- Using the binomial series, we expand the right side of the previous equation. (Enter the first three terms of the expansion.) dT = kT dt m dT When T-Tm is small compared to the third term in the expansion vx can be ignored, giving - z k¸(T – Tm), where k, = 4kT 3. m
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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I need help solving a problem in my diffential equations class.
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![According to Stefan's law of radiation the absolute temperature Tof a body cooling in a medium at constant absolute temperature
Tm
is given by
dT
= k(T* - T),
'm
dt
where k is a constant. Stefan's law can be used over a greater temperature range than Newton's law of cooling.
(a) Solve the differential equation.
(b) Show that when T- Tm is small in comparison to T then Newton's law of cooling approximates Stefan's law. [Hint: Think binomial series of the right-hand side of the DE.]
dT
= k(T* - T)
dt
= kTm
1 +
Using the binomial series, we expand the right side of the previous equation. (Enter the first three terms of the expansion.)
dT
= kT
dt
m
When T- Tm is small compared to
the third term in the expansion
vx can be ignored, giving
- z k¸(T – Tm), where k, = 4kT3.
dt](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93e3d5a7-fb85-40b1-8aac-b02c08819a1b%2F8f3442ae-046b-4fda-8825-acb75082a98d%2Fx5lfzhg_processed.png&w=3840&q=75)
Transcribed Image Text:According to Stefan's law of radiation the absolute temperature Tof a body cooling in a medium at constant absolute temperature
Tm
is given by
dT
= k(T* - T),
'm
dt
where k is a constant. Stefan's law can be used over a greater temperature range than Newton's law of cooling.
(a) Solve the differential equation.
(b) Show that when T- Tm is small in comparison to T then Newton's law of cooling approximates Stefan's law. [Hint: Think binomial series of the right-hand side of the DE.]
dT
= k(T* - T)
dt
= kTm
1 +
Using the binomial series, we expand the right side of the previous equation. (Enter the first three terms of the expansion.)
dT
= kT
dt
m
When T- Tm is small compared to
the third term in the expansion
vx can be ignored, giving
- z k¸(T – Tm), where k, = 4kT3.
dt
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