During an experiment of compressed air and water flowing in a 1/2 in. diameter gas pipe with a small hole(about 1/10 in. diameter) on its wall, an interesting self-excited oscillation took place by accident. Air and water spurted out of the hole alternately.
The oscillation is represented alternately two distinct linear differential equations which are of the third order and have constant coefficients.
This paper presents a simple method of analysing these mutually switching differential equations of the third order by the trajectories on the phase-plane.