separation constant

The single constant that a separated PDE's x-dependent side and t-dependent side must both equal, since one depends only on x and the other only on t.

The separation constant is the common value (often written $-\lambda$ ) that both sides of a separated equation such as $\dfrac{X''}{X}=\dfrac1k\dfrac{T'}{T}$ must share -- a function of $x$ alone that equals a function of $t$ alone for every $x$ and $t$ can only be a constant. Its allowed values are then pinned down by requiring the resulting $x$ -equation, together with the boundary conditions, to have a nontrivial solution -- exactly the Sturm-Liouville eigenvalue problem of ch.04.