2.6.9.4 ✓ Sympy. Time used: 0.339 (sec). Leaf size: 12

from sympy import * 
u = symbols("u") 
v = Function("v") 
ode = Eq(sqrt(Derivative(v(u), u) + 1) - exp(u)/2,0) 
ics = {} 
dsolve(ode,func=v(u),ics=ics)
 

\begin{gather*} \begin {aligned} v{\left (u \right )} = C_{1} - u + \frac {e^{2 u}}{8} \end {aligned} \end{gather*}

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=v(u)) 
 
('factorable', 'nth_algebraic', 'lie_group', 'nth_algebraic_Integral')