3 Mathematica. Time used: 0.277 (sec). Leaf size: 44

ode=Sqrt[1-u^2]*D[v[u],u]==2*u*Sqrt[1-v[u]^2]; 
ic={}; 
DSolve[{ode,ic},v[u],u,IncludeSingularSolutions->True]

\begin{align*} v(u)&\to -\sin \left (2 \sqrt {1-u^2}-c_1\right )\\ v(u)&\to -1\\ v(u)&\to 1\\ v(u)&\to \text {Interval}[\{-1,1\}] \end{align*}