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
$\displaystyle \int{\frac{du}{\sqrt{1-u^2}}} = \int{\frac{dx}{x+1}}$
The left side is a standard integral and is equal to $\displaystyle arcsin(u) +A $
wikipedia link (and don't forget to put y-1 wherever u appears)
For the right side recall that $\displaystyle \int{\frac{f'(x)}{f(x)}} = ln|f(x)| + B$
Where A and B are constants and so C = B-A which is also a constant (your constant of integration)