I'm trying to find the position of the midpoint of the rear axis of a vehicle (Q), given the position of the midpoint of the front axis of the vehicle (P). I'm looking at the following paper: https://www.researchgate.net/publication/222417459_Mathematical_models_for_motion_of_the_rear_ends_of_vehicles
I understand how they derived the following equation on page 4:
$$\dot{B}=\left(\frac{\dot{P}\cdot B}{L^2}\right)B-\dot{P}$$
Where $B$ is the vector $\overrightarrow{PQ}$, $P$ is the position vector of point P at time $t$, and $L$ is the distance between P and Q, and is a constant.
My question is: how do you solve the equation to find $B$ in terms of $P$ and/or $L$?
I have considered making $e^{\int\left(\frac{\dot{P}\cdot B}{L^2}\right)B-\dot{P}}$ the integrating factor. However, I'm not sure this works, as $B$ is present in it. Furthermore, I'm unsure how to approach the multiplication of $B$ by a scalar, which is a function of the dot product of two vectors.