Eq. (54) solves Eq. (47): $\frac{d}{dt}(\beta_kK_k)+\beta_kK_kF_k=0$

Throughout, kernels of Sec. IV A are written in the real orthonormal eigenbasis $f^{(k)}$ of $-\partial_x^2$ (eigenvalue $k^2$): a kernel $A_{xy}=\sum_k A_k f^{(k)}_x f^{(k)}_y$ is represented by its mode coefficient $A_k$, so $\int dx\,A_{yx}B_{xz}\mapsto A_kB_k$, $\partial_z^2\delta(y-z)\mapsto -k^2$ and $\int dx\,A_{xx}\mapsto\sum_k A_k$.

For all real $w_k,\tau_k,k$ and every time $t$ with $\cos(kt)\ne0$, the coefficients $\beta_k=w_k\cos(kt)$ (Eq. (54)), $K_k=\tau_k/\cos^2(kt)$ (Eq. (51)) and $F_k=-k\tan(kt)$ (Eq. (49)) satisfy the mode form of Eq. (47),

Eq. (47) fixes the centre $\beta_x(t)$ of the classical Gaussian ensemble $P^c$; the solution oscillates like the mean position of a classical oscillator.