PDE problems with sources: nonhomogeneous solution methods

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We give an example of a heat equation that contains a source—a nonhomogeneity—and nonhomogeneous boundary conditions. The solution approach involves breaking the solution function u(x,t) into two pieces that you solve separately for.
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Do you happen to have a slightly harder version of this type of problem? Some of the problems I have to work on have An additional t term

catlandmath
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How would you solve nonhomogeneous heat equation with homogeneous boundary conditions?

jayjayf
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I really understand this solution but why we say the value of n =2k-1. Why not just k?

haceryerli
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There’s some wrong things, for example in the equation to find Bn u forgot the 1/L

gokuszz