A sphere with radius $1$ m has temperature $16$C$.$ It lies inside a concentric sphere with radius $2$ m and temperature $22$C$.$ The temperature

T$($r$)$

$($in $$C$)$ at a distance r $($in meters$)$ from the common center of the spheres satisfies the differential equation

d$2$T

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. $($Use T for

T$($r$).)$