= Solution
For distinct nonzero recurrence roots, the <determinant> formula gives
$$
\boxed{D_N=0\quad\Longleftrightarrow\quad\left(\frac{\beta_1}{\beta_2}\right)^{N+1}=1,\qquad\beta_1\ne\beta_2.}
$$
The ratio one is excluded: at that repeated root, part (iii) gives $(N+1)(\pm k)^N\ne0$. Thus the naive extra root corresponding to equal recurrence roots is spurious.
To count the physical solutions without ambiguity from exchanging $\beta_1$ and $\beta_2$, write $\beta_{1,2}=ke^{\pm iq}$ and $d=2k\cos q$ with $0<q<\pi$. Then
$$
D_N=k^N\frac{\sin[(N+1)q]}{\sin q}.
$$
Its $N$ distinct zeros occur at $q=j\pi/(N+1)$, $j=1,\ldots,N$. They give \b[exactly $N$ distinct positive normal frequencies], or equivalently $N$ distinct roots in $\omega^2$:
$$
\omega_j=2\sqrt{\frac km}\sin\frac{j\pi}{2(N+1)}.
$$
If signed $\omega$ is counted, the <characteristic equation> has $2N$ roots $\pm\omega_j$. The sine quotient is the <Chebyshev polynomial of the second kind> representation of the <fixed-end spring-chain determinant and its repeated-root limit>.
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