= Solution
For each of the four standard pairs, <integration by parts> gives
$$
\langle W,KW\rangle=A\int_0^L|W''|^2\,dx\geq0.
$$
Consequently the <eigenvalues> are nonnegative. For a positive <eigenvalue> $\mu_n=A k_n^4$, the <eigenvalue equation> is $W_n''''=k_n^4W_n$. Its four characteristic roots are $\pm k_n,\pm i k_n$, so
$$
\boxed{W_n(x)=C_1\cos(k_nx)+C_2\sin(k_nx)+C_3\cosh(k_nx)+C_4\sinh(k_nx).}
$$
Here $C_j$ are coefficients, avoiding a collision with the <filament bending modulus> $A$. The regular finite-interval <self-adjoint operator> has <compact resolvent>; applying the <spectral theorem for compact self-adjoint operators> to a shifted inverse supplies a complete <orthonormal basis> of <eigenfunctions>.
For <clamped boundary conditions> at zero, a <clamped--clamped bending mode> can be written
$$
W=C(\cosh kx-\cos kx)+D(\sinh kx-\sin kx).
$$
At $L$, writing $\beta=kL$, the two remaining <boundary conditions> are
$$
\begin{pmatrix}\cosh\beta-\cos\beta&\sinh\beta-\sin\beta\\
\sinh\beta+\sin\beta&\cosh\beta-\cos\beta\end{pmatrix}
\binom CD=0.
$$
The <determinant> is $2(1-\cos\beta\cosh\beta)$. Hence \b[the positive <wavenumbers> obey]
$$
\boxed{\cos\beta_n\cosh\beta_n=1,\qquad k_n=\beta_n/L.}
$$
Equivalently, intersect $\cos\beta$ with $\operatorname{sech}\beta$. Numerical root bracketing gives
$$
\beta_1\simeq4.730041,\quad\beta_2\simeq7.853205,\quad\beta_3\simeq10.995608,\quad\beta_4\simeq14.137165,\quad\beta_5\simeq17.278760.
$$
The entire sequence has the useful large-$n$ description
$$
\boxed{\beta_n=(n+\tfrac12)\pi+2(-1)^{n+1}e^{-(n+1/2)\pi}+O(e^{-2(n+1/2)\pi}),\qquad n=1,2,\ldots.}
$$
Indeed, put $\beta=(n+\tfrac12)\pi+\delta$ in $\cos\beta=\operatorname{sech}\beta$ and use $\cos\beta=(-1)^{n+1}\delta+O(\delta^3)$ and $\operatorname{sech}\beta=2e^{-\beta}+O(e^{-3\beta})$.
The apparent root $\beta=0$ in the <determinant> equation is spurious for the clamped problem: at zero <eigenvalue>, $W$ is a cubic polynomial, and its four clamped conditions force $W=0$. For other endpoint choices, <zero-energy filament modes> must be treated separately from the trigonometric formula. The free-free <kernel> consists of affine functions, the torqued-torqued <kernel> consists of constants, and the hinged-hinged <kernel> is trivial. Including those <kernels> is necessary for a complete <eigenfunction expansion>.
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