= Solution
For a nonzero <separable solution> $u=XT$, <separation of variables> gives
$$
\frac{T'}{DT}=\frac{X''}{X}=-\kappa,
\qquad
\boxed{X''+\kappa X=0,\qquad X(0)=X(L)=0.}
$$
These <Dirichlet boundary conditions> follow because $T$ is not identically zero. If $\kappa=0$, the <affine function> $X=A+Bx$ obeys both conditions only when $A=B=0$. If $\kappa=-\rho^2<0$, $X=A\cosh(\rho x)+B\sinh(\rho x)$ also has only the zero solution under both conditions.
For $\kappa=\omega^2>0$, $X=A\cos(\omega x)+B\sin(\omega x)$. The first <boundary condition> forces $A=0$, and the second forces $\omega L=n\pi$ for a nonzero solution. Hence all nonzero spatial modes are
$$
\boxed{\kappa_n=\left(\frac{n\pi}{L}\right)^2,\qquad
X_n(x)=B_n\sin\frac{n\pi x}{L},\quad n=1,2,\ldots,\quad B_n\ne0.}
$$
The identically zero <function> is also a solution. Allowing $B_n=0$ includes it in the displayed family. Here $L>0$ is the interval length and $n$ labels the <eigenvalues> of the operator $-d^2/dx^2$ with these <boundary conditions>.
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