I have two ways you could find a solution to that problem (actually 3 but the third is using a flow simulator :P)
Just assume that P2 and P3 are the same (or else you will find yourself in partial differential equation hell believe me been there and it isn't pretty)
1: Crane Flow of Fluids. Easy lazy, quick he gives you measured parameters for pressure loses and other things you just fit your numbers there http://www.waterlinefountains.com/wp-content/uploads/2017/09/Crane-410.pdf
Not the best method it has a lot of error. It is the first answer that comes up in google (though they do not provide you with the tables)
2: Bird transport phenomena:
Assume it is plug flow, define a three pipe system in which 1 is the feed and 2 and 3 are the exits, you calculate each exit with an equalized pressure (or so I remember). Just to be certain; I checked the book and problem 7Q4 can give you the answer, look up the solution and define n as 2.
So far I think this is the answer you want, you calculate everything by hand but it isn't a problem with an easy solution.
I recommend you checking if someone hasn't programmed it yet because you getting a short formula is a longshot the answer to 7Q4 is a differential equation.
3: Get a copy of EPAnet (free but works poorly), HYSYS (usually colleges with Chemical engineering majors have licences) or Olga, they do pressure drops though they use method 2 and 1 combined.
TLDR: To do it by hand is possible, has been done before and that is how most simulators work, though it is so hard to do that you use equivalent lengths instead.