 Solution

For $k>0$, the harmonic oscillator equation gives $X=C\sin (kx)+D\cos (kx)$. The condition at zero requires $D=0$, and the condition at $L$ requires $C\sin (kL)=0$. A nonzero normal mode therefore hasA multiplicative normalization has been absorbed into the temporal amplitude. The $k=0$ candidate is affine and the two zero endpoint values force it to vanish. A positive separation constant gives hyperbolic functions, whose two zero endpoint values likewise give only the trivial solution. Negative $n$ adds no new eigenfunctions because it only reverses their signs.