Question

A sphere with radius 1m has temperature 15 degrees C. It lies inside a con-
centric sphere with radius 2m and temperature 25 degrees C: The temperature
T(r) at a distance r from the common centre of the spheres satisfies
the differential equation:

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

Set S = dT/dr . Then S satises a first-order differential equation. Solve
this to determine an expression for the temperature T(r) between the
spheres.

Answer 1

dS/dr + (2/r) S = 0
dS/dr = -2 S/r
dS/S = -2 dr/r

ln S = -2 ln r + C
ln S = -ln r^2 + C
ln(S r^2) = C
S r^2 = C

(dT/dr )r^2 = c
dT = c dr/r^2
T = - c /r + C
at r = 1, T = 15
at r = 2, T = 25
15 = -c + C
25 = -c/2 + C
------------
-10 = -c/2
c = 20
15 = -20 + C
C = 35
so T = -20/r + 35