2.2.2.3 ✓ Mathematica. Time used: 0.45 (sec). Leaf size: 39

ode=D[y[x],x]+Sqrt[ (1-y[x]^2)/(1-x^2)]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\cosh \left (2 \text {arctanh}\left (\frac {1}{\sqrt {\frac {x-1}{x+1}}}\right )-c_1\right )\\ y(x)&\to -1\\ y(x)&\to 1 \end{align*}