I am trying to find a solution to the following ODE for $f:(0,1)\to \mathbb{R}$ $$\sqrt{1+(f')^2}\frac{\partial}{\partial x} \left(\frac{f'}{\sqrt{1+(f')^2}}\right)=c$$ for some constant $c>0$. Equivalently, $$\lambda f''-(f')^2-1=0$$ for $\lambda=1/c$.
How is this equation related to the minimal surface equation $$\mathrm{div}\left(\frac{\nabla F}{1+|\nabla F|^2}\right)=0$$ for a function $F:\Omega\to \mathbb{R}$ with $\Omega\subset \mathbb{R}^{n-1}$? Minimal surfaces have zero mean curvature. Does the above ODE describe constant mean curvature curves?