Eq. (49) solves Eq. (44): $\dot F_k + F_k^2 + k^2 = 0$

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 every real $k$ and every time $t$ with $\cos(kt)\ne0$, the mode coefficient $F_k(t)=-k\tan(kt)$ of Eq. (49) satisfies the mode form of Eq. (44),

Eq. (44), $\dot F_{yz}+\int dx\,F_{yx}F_{xz}-\partial_z^2\delta(y-z)=0$, is the Riccati equation for the kernel of the classical Hamilton–Jacobi functional $S^c[\phi,t]=\tfrac12\iint\phi_yF_{yz}(t)\phi_z$; it is independent of $g$ and therefore also holds in the interacting solution of Sec. IV B.

Formalization Note The time derivative is taken with deriv; the hypothesis $\cos(kt)\ne0$ excludes the poles of $\tan$.