So my book says that to solve a PDE by separation of variables, we check the three cases where λ, the separation constant, is equal to 0, -a^2, and a^2. But in this particular problem, instead of substituting λ=0, λ = a^2, λ= -a^2, they substitute the entire coefficient of X, (λ-1)/k =0, (λ-1)/k =a^2, and (λ-1)/k =-a^2. I've linked a picture of what it says in the solution manual. I don't understand why in this particular problem they substituted the entire coefficient of X rather than just λ.
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