I get stuck at the integration step
use the method of separation of variables to find the solution of the differential equation:
dy/dx=sqrt(2y-y^2)/x+1 , y(0)=1
Danny pointed it out:
u = y-1 so du = dy
Therefore your equation becomes
The left side is a standard integral and is equal to
wikipedia link (and don't forget to put y-1 wherever u appears)
For the right side recall that
Where A and B are constants and so C = B-A which is also a constant (your constant of integration)