Problem with PDE Ansatz

Question

Im currently reading "A stochastic differential game of capitalism" of Leong and Huang, and they try solving the PDE $$ -J_t^F(k) - \frac{1}{2} \sigma^2k^2 J_{kk}^F(k) = e^{-\rho t} \{f'(k)-\overline{x} - (\delta + n - \sigma^2)k \} $$ where, $\sigma, \delta, n$ are given constants and $f$ a given function. Also, $k$ and $\overline{x}$ are functions of time. They try the solution $J^F(k,t) = e^{-\rho t} B k^2$, where $B$ is a constant. My problem is, after solving for $B$, they arrive at $$ B = \frac{f'(k) - \overline{x} -(\delta +n - \sigma^2)k}{(p-\sigma^2)k},$$ so $B$ isn't a constant if $\overline{x}$ doesn't cancel the terms in the numerator. Furthermore, in Proposition 3.5, the condition for arriving at the above PDE imply $B = 1/(2k)$. So $B$ isn't a constant neither. Is there a mistake?