help again (temperature between two spheres)

jellybean

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Joined
Feb 9, 2006
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A sphere with radius 1 m has temperature 15 degrees C. It lies inside a concentric sphere with radius 2 m and temperature 25 degrees C.

The temperature T(r) at a distance r from the common center of the spheres satisfies the differential equation:

. . .(d^2T)/(dr^2) + (2/r)(dT/dr)=0

If we let S=dT/dr, then S satisfies a first-order differential equation. Solve it to find an expression for the temperature T(r) between the spheres.
 

soroban

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Jan 28, 2005
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5,587
Hello, jellybean!

A sphere with radius 1 m has temperature 15 degrees C.
It lies inside a concentric sphere with radius 2 m and temperature 25 degrees C.

The temperature \(\displaystyle T(r)\) at a distance \(\displaystyle r\) from the common center of the spheres

satisfies the differential equation: \(\displaystyle \L\,\frac{d^2T}{dr^2}\,+\,\frac{2}{r}\cdot\frac{dT}{dr}\;=\;0\)

Find an expression for the temperature \(\displaystyle T(r)\) between the spheres.
\(\displaystyle \text{I solved it with an integrating factor: }\,I\:=\:r^2\)

\(\displaystyle \text{We have: }\L\,r^2\left(\frac{d^2T}{dr^2}\right)\,+\,2r\left(\frac{dT}{dr}\right)\;=\;0\)

\(\displaystyle \text{Then: }\L\,\frac{d}{dr}\left(r^2\cdot\frac{dT}{dr}\right)\;= \;0\)

\(\displaystyle \text{Integrate: }\L\,r^2\left(\frac{dT}{dr}\right)\;=\;C_1\;\;\Rightarrow\;\;dT\;=\;C_2\cdot r^{-1}\,dr\)

\(\displaystyle \text{Integrate: }\L\,T(r)\;=\;-C_1\cdot r^{-1}\,+\,C_2\)


\(\displaystyle \text{When }r\,=\,1,\:T\,=\,15:\L\;\;15\;=\;-C_1\,+\,C_2\)

\(\displaystyle \text{When }r\,=\,2,\:T\,=\,25:\L\;\;25\;=\;-\frac{C_2}{2}\,+\,C_2\)

\(\displaystyle \text{We have a system of equations: }\:\L\begin{Bmatrix}-C_1\,+\,C_2\:=\:15 \\ -C_1\,+\,2C_2\;=\;50\end{Bmatrix}\)

\(\displaystyle \;\;\text{from which we get: }\L\,C_1\,=\,20,\;C_2\,=\,35\)


\(\displaystyle \text{Therefore: }\L\:T(r)\;=\;-\frac{20}{r}\,+\,35\)
 
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