Evaluating the wave equation $u_{tt} = c^2 u_{xx}$ on $x \in (0,L)$ and $t \in (0,\infty)$ my lecture notes say:
Let us assume that the solution is of the form $u(x,t) = X(x)T(t)$ with two functions $X:(0,L) \to \mathbb{R}$ and $T:(0,\infty) \to \mathbb{R}$.
My question is where did they get this from? I understand how the solution to the wave equation can be split into $u(x,t) = f(x+ct) + g(x-ct)$ with functions $f,g:\mathbb{R} \to \mathbb{R}$ but I can't see where they got $X(x)T(t)$ (my printed lecture notes don't contain any intermediate steps or many proofs and I missed a fair few lecture due to illness). Also my lecture notes then say:
Then we obtain $X''(x) - \lambda X(x) = 0, \quad T''(t) - c^2 \lambda T(t) = 0.$
And I can't see how he got this either? I can get the result $X(x)T''(t) = c^2X''(x)T(t)$ when I differentiate $u(x,t) = X(x)T(t)$ with respect to $t$ and $x$ twice and then substitute back into the wave equation but I can't see where $\lambda$ comes from? It just appears with no explanation? Any help would be appreciated!