3 Mathematica. Time used: 0.015 (sec). Leaf size: 20

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

\begin{align*} v(u)&\to -u+\frac {e^{2 u}}{8}+c_1 \end{align*}