Answer to Problem 56

Use the above result to rewrite the equation (6.2):



Equation 82



Equation 83

Using equation (6.1) to substitute for , we obtain the differential equation as follows.

Since



Equation 84

then



Equation 85



Equation 86

Let



Equation 87

Then



Equation 88

NOTE: Equation (6.3), together with the boundary temperatures and at and , respectively, define the boundary value problem

BVP

Equation 89

in which the constants , , , and are given and the constant is to be determined along with the solution . The BVP above is solved in Problems 6.2, 6.3, and 6.4.

Generally, two boundary conditions arise in conjunction with a second order ordinary differential equation (ODE). But equation (6.3) is a first order ODE which usually has a single associated boundary condition to fix the arbitrary constant of integration.

The additional boundary condition is needed to determine the value of .