Heat Equation without Fourier Series

The problem:

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Two rods of length and , which have heat diffusion constants and , respectively, are welded together.

The left end of the left rod (length ) is maintained at temperature , the right end of the right rod is kept at temperature . The temperature and heat flux are continuous across the weld.

I need to find the equilibrium temperature distribution across the welded rod of length .

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I started setting up the data:

Since nothing was given for and , I assume they're

If we call the solution , it seems that

with and

We found in class that the general solution for the heat equation on a rod of infinite length with is

However, this doesn't seem to help because we have finite length and boundary conditions. And here I get lost (before I really even begin!) because there is just a ridiculous amount of information and I'm not really sure what to do with it.

Help is appreciated!