Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - \frac{1}{\psi(x)} \right),$$ with initial conditions $x(0) = 0$ and $\dot x(0) = 1$. I would like to find an explicit choice of $\psi(x)$ which satisfies $$\lim_{x\to\infty} \psi(x) = \infty,$$ and for which the solution $x(t)$ is strictly increasing and asymptotically linear (i.e. $x(t) = O(t)~$). If the ODE could be solved explicitly, that would be even better.

I have used Mathematica to experiment with $\psi(x) = x+1$. Based on the numerical evidence, this example seem to work, but I don't know how to rigorously analyze the solution.

Can you come up with a function $\psi(x)$ for which the above holds? If not, could you point me toward a reference which discusses how to rigorously show the properties above in the absence of an explicit solution?

**Edit:** This is the equation governing the $x$-position of geodesics in the plane $\mathbb R^2$ under the conformal Riemannian metric $g_{ij}(x,y) = \sqrt{\psi(x)} \delta_{ij}(x).$