3 Mathematica. Time used: 0.417 (sec). Leaf size: 23

ode=D[y[x],{x,2}]==m/k*Sqrt[1+D[y[x],x]^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]

\begin{align*} y(x)&\to \frac {k \cosh \left (\frac {m x}{k}+c_1\right )}{m}+c_2 \end{align*}