3.5 Extracting DSolve solutions

one way

Clear[y, x] 
DSolve[{y'[x]^2 == 1 - y[x]^2, y[0] == 0}, y[x], x] 
 
{{y[x] -> -Sin[x]}, {y[x] -> Sin[x]}}
 

And now

\verb|y[x] /. %| 
 
   {-Sin[x],Sin[x]}