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The non-homogeneous heat equation

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Concerning heat flow

I am confused about turning a non homogeneous equation (heat generation) into a homogeneous equation; could this process be explained in detail with an example....i unfortunately need this by noon on Thursday (EDT)

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The solution explains in details how to solve a general non-homogeneous heat equation with (mixed) homogeneous boundary conditions.

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Any homogenous heat equation with non-homogenous DirichletNeumann boundary conditions can be easily converted to a non-homogenous heat equation with homogeneous DirichletNeumann boundary conditions.
So we shall look at the general non-homogenous heat equation with Dirichlet BC.
In this case, the system looks like:
(1.1)
The associated homogenous system is
(1.2)
We know, using separation of variables, that the homogenous system gives rise to a set of orthogonal eigenfunctions
(1.3)
For brevity and generality sake we will just note that the eigenfunctions satisfy
(1.4)
Where is a constant.
Furthermore
(1.5)

For example, in the context of homogenous Dirichlet BC we see that and
And usually at this point we would find the expansion of the initial condition in terms of the eigenfunctions and equate coefficients.
However, in this case we have to look at the non-homogenous system.
We now assume that the non-homogenous solution is in the form:
(1.6)
Note that now the coefficient is yet-to-be defined function of t.
When substituted back into the non-homogenous system we obtain:

(1.7)
Since equation (1.7) can be written as:
(1.8)
Now, if we expand in ...

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