Solution of the boundary value problem to obtain the temperature distribution in the rod.

(3.7)

This equation is non-homogeneous. Let us make it homogeneous by the change of variable

(3.10)

Click here for Math Link 2, a discussion of second order linear DE's, homogeneous and non-homogeneous.

Then equation (3.7) becomes

(3.11)

for which the boundary conditions are

(3.12)
 

The general solution to equation (3.11) is                     (see Boyce and Di Prima)

(3.13)

Inserting the boundary condition at z = 0, we have

(3.14)

The boundary condition at z = L requires

 

Solving for c1, we have

(3.15)

Using equations (3.14), (3.15) and (3.10) we can write (3.13) in the form

(3.16)

Solving for T = T(z), we have the temperature distribution in the rod.

Explore this distribution in the next discover section.


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