Hi I have this question and the only hard part is completing it to give $u(x,t)$. 
The question is above and I've gone ahead to get as far as I can and then when coming to the end to get the coefficients I'm extremely stuck.
Right at the end I get this $$\sum_{n=1}^{\infty} A_n \sin(\pi n x) = x(x-1)$$ this is by just subbing the boundaries in.
Now usually, when I do these questions it's just in the form of the RHS being simple $sin x$ or $sin2x$ or something not TOO difficult. But i've never attempted a question where the RHS was just $x's$. So, I have no attempt of trying to share [as I'm not too sure what to do]
If someone could help me by walking me through this this would be really helpful -- thank you!
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Edit: The hard part is finding $A_n$ once this is found, it is simple to work out $u(x,t)$