Particle board desk top reinforcement - do i need it? how to do it?
I would make a frame underneath about 2” in from each edge screwed and glued. 2*1 would be good and that will give some support to the legs, possibly some bracing for the legs - a horizontal bar that could also be a footrest across the back or diagonal bracing like an X.

Edit: the desk I am at has a frame of 20mm * 65mm and the same 38mm thickness top that you mention but it is not particle board (particle board is weaker than some glued timber block table tops) so take note of Isherwood's comment about 2*3"...