   Chapter 9.3, Problem 42E

Chapter
Section
Textbook Problem

# A sphere with radius 1 m has temperature 15°C. It lies inside a concentric sphere with radius 2 m and temperature 25°C. The temperature T(r) at a distance r from the common center of the spheres satisfies the differential equation d 2 T d r 2   +   2 r d T d r   =   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.

To determine

To find: The function T(r) by solving the differential equation: d2Tdr2+2rdTdr .

Explanation

Given:

Radius of the sphere whose temperature is 15C = 1m.

Radius of the sphere whose temperature is 25C = 2m.

The differential equation is d2Tdr2+2rdTdr=0 .

Here, S=dTdr be first order differential equation.

Calculation:

Consider the differential equation, dSdr+2rS=0 .

Separate the variables,

dSdr=2rSdS2S=drr

Integrate on both sides,

dS2S=drr12(lnS)=lnr+C(lnS)=2lnr+C

Taking exponentials on both sides,

S=e2lnr+C=elnr2.eC=r2C

Here, C is the constant.

Since S=dTdr , dT is obtained

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