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We have BCs ##u(0,t) = u(a,t) = C.## Our standard method for finding u doesn't work here, since ##e^{-k(\frac{n\pi}a)^2t}sin(\frac{n\pi}a)## does not satisfy these BCs.

Make a change of variable from ##u## to ##v=u-C.## Show that ##v## satisfies the heat equation with BCs ##v=0## at ##x=0## and ##x=a.##

Write down the solution for ##v(x,t).##Deduce an expression for ##u(x,t)## in terms of constants ##c_1,c_2,\ldots,## and write down a formula for ##c_n.##

[Harder] Now suppose the BCs are ##u(0,t) = C##, ##u(a,t)=D## for constants ##C,D.## How could you solve the case?

My question: These are extensions to homework which I'd like try to attempt, but I don't know where to start with the change of variable