Answer
$$y = \frac{1}{2}\arcsin \left( {\frac{{x - 2}}{2}} \right) + \frac{1}{2}$$
Work Step by Step
$$\eqalign{
& \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {4x - {x^2}} }} \cr
& {\text{Separate the variables}} \cr
& dy = \frac{1}{{\sqrt {4x - {x^2}} }}dx \cr
& {\text{Complete the square}} \cr
& dy = \frac{1}{{\sqrt {4 - \left( {{x^2} - 4x + 4} \right)} }}dx \cr
& dy = \frac{1}{{\sqrt {4 - {{\left( {x - 2} \right)}^2}} }}dx \cr
& {\text{Integrate both sides}} \cr
& y = \frac{1}{2}\arcsin \left( {\frac{{x - 2}}{2}} \right) + C,{\text{ }}\left( {\bf{1}} \right) \cr
& {\text{Use the initial condition }}\left( {2,\frac{1}{2}} \right) \cr
& \frac{1}{2} = \frac{1}{2}\arcsin \left( {\frac{{2 - 2}}{2}} \right) + C \cr
& C = \frac{1}{2} \cr
& {\text{Substitute }}C{\text{ into }}\left( {\bf{1}} \right) \cr
& y = \frac{1}{2}\arcsin \left( {\frac{{x - 2}}{2}} \right) + \frac{1}{2} \cr
& \cr
& {\text{Graph}} \cr} $$
