Chapter 9.3, Problem 42E

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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

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:

Explanation

**Given:**

Radius of the sphere whose temperature is

Radius of the sphere whose temperature is

The differential equation is

Here,

**Calculation:**

Consider the differential equation,

Separate the variables,

Integrate on both sides,

Taking exponentials on both sides,

Here, *C* is the constant.

Since

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