Orthogonal trajectories for nonlinear equations

Abdelkader developed a method for solving the van der Pol equation numerically. He observed that the orthogonal trajectories of the van der Pol equation satisfy a Riccati equation, the solution to which can be expressed in terms of Bessel functions. In the present paper, all nonlinear equations whose orthogonal trajectories satisfy Riccati or generalized Riccati equations are determined. Further on, another representation of the trajectories of the van der Pol equation is derived. This representation appears to be simpler than Abdelkader's.