Pilot of a small aircraft flying VFR not in communication with ATC reported taking evasive action to avoid a passenger jet descending towards them. They reported the passenger jet appeared to take evasive action to avoid them too.

Narrative

I was flying VFR from ZZZ1 to ZZZ2 on a afternoon. I was at 4500 ft.in Class E airspace. I was not talking to any Air Traffic Controller. The airspace was pretty empty. I was monitoring 121.5 and squawking 1200. I was continuously monitoring my moving map page with traffic overlayed while flying. My aircraft does not have TCAS; only TAS. As I was flying; I noticed an aircraft on the screen rapidly approaching from my 10 o'clock; about 1000 ft. above me and descending. I received a traffic advisory: 'Traffic.' We were not exactly on a collision course. The relative vector indicator on my screen showed the aircraft would pass in front of me from left to right. I disabled the autopilot; turned the aircraft left; and descended immediately; to avoid getting too close. I felt this would be the safest way to avoid the other aircraft. We were not converging head-on; so I decided to turn left instead of right. If I turned right; I would have actually fallen even closer to the other aircraft's flight path. The other aircraft seemed to be a Boeing 737. That aircraft turned in the same direction that I did; but immediately started climbing. I estimate it climbed at least 500 ft. It was 1200 ft. above me and slightly to my right as it passed by me. I believe it was flying into ZZZ airport. My assumption would be that the B737 had a resolution advisory and decided to climb above me to avoid a loss of separation. Both airplanes landed safely at their respective destinations. While I was not required to be in contact with air traffic control while flying VFR in Class E airspace; in the future I will ask for flight following when flying near busy airspace so as to avoid situations like this.­