# help again (temperature between two spheres)

#### jellybean

##### New member
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

##### Elite Member
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.
$$\displaystyle \text{I solved it with an integrating factor: }\,I\:=\:r^2$$

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

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

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

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

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

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

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

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

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