Chapter 9.3, Problem 42E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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 15âˆ˜C = 1m.

Radius of the sphere whose temperature is 25âˆ˜C = 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=âˆ’2rSdS2S=âˆ’drr

Integrate on both sides,

âˆ«dS2S=âˆ’âˆ«drr12(lnS)=âˆ’lnr+C(lnS)=âˆ’2lnr+C

Taking exponentials on both sides,

S=eâˆ’2lnr+C=elnrâˆ’2.eC=râˆ’2C

Here, C is the constant.

Since S=dTdr , dT is obtained

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started