2.2.2.2 ✓ Maple. Time used: 0.004 (sec). Leaf size: 84

ode:=diff(y(x),x)+((1-y(x)^2)/(-x^2+1))^(1/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \frac {\sqrt {\frac {-1+y^{2}}{x^{2}-1}}\, \sqrt {x^{2}-1}\, \ln \left (x +\sqrt {x^{2}-1}\right )}{\sqrt {y-1}\, \sqrt {y+1}}+\frac {\sqrt {-1+y^{2}}\, \ln \left (y+\sqrt {-1+y^{2}}\right )}{\sqrt {y-1}\, \sqrt {y+1}}+c_1 = 0 \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying homogeneous types: 
differential order: 1; looking for linear symmetries 
trying exact 
<- exact successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )+\sqrt {\frac {1-y \left (x \right )^{2}}{-x^{2}+1}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-\sqrt {\frac {1-y \left (x \right )^{2}}{-x^{2}+1}} \end {array} \]