But I found 1770 and bought them. There is only one question that worries me. Will the 1770 work on an EXDOS v.2.0 board?
Yes, it will work. The only important difference is the step rates, with the 1770 6,12,20,30ms is possible, with 1772 6,12,2,3.
Only some drives can work with fastest 2ms (for example Teac FD55F or GF), but near all of them work with 3ms, with faster and quieter operation than the 6ms.
So the original EXDOS ROM modified for using 3ms for 1772 (variable 73 set to 3). If you only have a 1770 then you should use the rom which uses 6ms setting (variable 73 set to 0).