= Solution
Put $k_n=n\pi/L$ and $\lambda_n=\kappa k_n^2$. Direct <differentiation> of $T_n=A\sin(k_nx)e^{-\lambda_nt}$ gives
$$
\partial_tT_n=-\lambda_nT_n,\qquad \partial_x^2T_n=-k_n^2T_n,
$$
so $\partial_tT_n=\kappa\partial_x^2T_n$, the <heat equation>. At $x=0$, the sine factor vanishes; at $x=L$, it is $\sin(n\pi)=0$ for integer $n$. Thus the homogeneous <Dirichlet boundary conditions> hold. The nonzero, nonredundant spatial modes use $n=1,2,\ldots$; $n=0$ is the zero function and negative $n$ only change the amplitude sign.
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