Almost all real linear second order ordinary differential equations are solved by geodesic curves in two dimensional Riemannian hyperbolic geometry
Łukasz Rudnicki
TL;DR
The paper links real linear second-order ODEs of the form $u''(x)+h(x)u(x)=0$ to geodesic curves in a two-dimensional hyperbolic geometry by introducing the Linear-$2^{\mathrm{nd}}$-order-ODE upper half-plane $\mathbb{M}_h$ with metric $g_h$ of constant curvature $-1$. A central result shows that every solution pair of the ODE can be encoded by a geodesic $\Phi(x)$ on $\mathbb{M}_h$, with two independent ODE solutions $u_{\text{top}}$ and $u_{\text{bot}}$ constructed from $\Phi$ via exponential integrals, and that a local diffeomorphism from $\mathbb{M}_h$ to the Poincaré upper half-plane $\mathbb{H}$ is induced by these solutions. The framework provides explicit geodesic equations, a Riccati representation for associated quantities, and a universal geometric reduction that holds locally for any differentiable $h$, as illustrated by canonical examples ($h=0$, $h=-\omega^2$, and $h=+\omega^2$). This establishes a deep link between linear ODE theory and hyperbolic geometry, offering geometric intuition and new tools for analyzing second-order linear equations. Potential extensions include relaxing regularity, addressing singularities, and exploring the complex-domain counterpart explored in subsequent work.
Abstract
I show that a real linear second order ordinary differential equation $u''\left(x\right)+h\left(x\right)u\left(x\right)=0$, with differentiable $h(x)$, locally admits two linearly independent solutions which exist on an open interval around any $x_0\in\mathbb{R}$: \[ u_\mathtt{top}(x)=\exp\left[\int_{x_0}^{x}\!\!dξ\,Φ\left(ξ\right)\frac{Φ'\left(ξ\right)-\sqrt{\left[h\left(ξ\right)-Φ^{2}\left(ξ\right)\right]^{2}+\left[Φ'\left(ξ\right)\right]^{2}}}{h\left(ξ\right)-Φ^{2}\left(ξ\right)}\right], \] \[ u_\mathtt{bot}(x)=\exp\left[\int_{x_0}^{x}\!\!dξ\,Φ\left(ξ\right)\frac{Φ'\left(ξ\right)+\sqrt{\left[h\left(ξ\right)-Φ^{2}\left(ξ\right)\right]^{2}+\left[Φ'\left(ξ\right)\right]^{2}}}{h\left(ξ\right)-Φ^{2}\left(ξ\right)}\right], \] where $Φ(x)$ is any geodesic curve in a two dimensional hyperbolic geometry of a Riemannian manifold $\mathbb{M}_h$, which is non-vertical at $x_0$. I define $\mathbb{M}_h$ to be an upper half plane $\{\left(x,\varPhi\right)\in\mathbb{R}^2\,|\,\varPhi>0\}$, with points in which $\varPhi^2=h(x)$ being removed, equipped with metric $g_h=\left[\left(h(x)-\varPhi^2\right)^2dx^2+d\varPhi^2\right]/\varPhi^2$. A non-trivial character of the presented result stems from the fact that $g_h$ is solely defined in terms of the function $h(x)$. I also show that a local diffeomorphism between $\mathbb{M}_h$ and Poincaré upper half plane $\mathbb{H}$ is induced by any pair of linearly independent solutions of $u''\!\left(x\right)+h\left(x\right)u\left(x\right)=0$. If this pair is selected to be $u_\mathtt{top}(x)$ and $u_\mathtt{bot}(x)$, the associate geodesic curve $Φ(x)$ is mapped to a vertical geodesic curve on $\mathbb{H}$. Thus, I establish a link between linear second order ordinary differential equations and two dimensional hyperbolic geometry.