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These are 5 questions that I need help with. I have access to the solutions but some parts are missing and I can't seem to follow them. I would be very thankful if someone is able to help.

I know that for the first one we need to use the method that involves an equilibrium solution. For the second one, we rely on the Fourier transform (I get this for the most part). For the fourth and fifth, we use the method of separation of variables. While the general approach isn't impossibly hard, I seem to hit roadblocks where there are gaps in the solution. Thanks!