Almost all real linear second order ordinary differential equations are solved by geodesic curves in two dimensional Riemannian hyperbolic geometry

I show that a real linear second order ordinary differential equation $u^{\mathrm{\prime }\mathrm{\prime }}\left(x\right)+h\left(x\right)u\left(x\right)=0$, with differentiable $h(x)$, locally admits two linearly independent solutions which exist on an open interval around any $x_0\in \mathbb{R}$: $$u_{\mathtt{t}\mathtt{o}\mathtt{p}}(x)=\exp \left[{\int }_{x_0}^x\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}d\xi \mathrm{\Phi }\left(\xi \right)\frac{{\mathrm{\Phi }}^{\mathrm{\prime }}\left(\xi \right)-\sqrt{{[h\left(\xi \right)-{\mathrm{\Phi }}^2\left(\xi \right)]}^2+{\left[{\mathrm{\Phi }}^{\mathrm{\prime }}\left(\xi \right)\right]}^2}}{h\left(\xi \right)-{\mathrm{\Phi }}^2\left(\xi \right)}\right],$$ $$u_{\mathtt{b}\mathtt{o}\mathtt{t}}(x)=\exp \left[{\int }_{x_0}^x\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}d\xi \mathrm{\Phi }\left(\xi \right)\frac{{\mathrm{\Phi }}^{\mathrm{\prime }}\left(\xi \right)+\sqrt{{[h\left(\xi \right)-{\mathrm{\Phi }}^2\left(\xi \right)]}^2+{\left[{\mathrm{\Phi }}^{\mathrm{\prime }}\left(\xi \right)\right]}^2}}{h\left(\xi \right)-{\mathrm{\Phi }}^2\left(\xi \right)}\right],$$ where $\mathrm{\Phi }(x)$ 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 $\{(x,\Phi )\in {\mathbb{R}}^2\mathrm{\mid }\Phi >0\}$, with points in which ${\Phi }^2=h(x)$ being removed, equipped with metric $g_h=[{(h(x)-{\Phi }^2)}^{2}d{x}^2+d{\Phi }^2]\mathrm{/}{\Phi }^2$. 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 $u^{\mathrm{\prime }\mathrm{\prime }}\text{\hspace{0.17em}⁣}\left(x\right)+h\left(x\right)u\left(x\right)=0$. If this pair is selected to be $u_{\mathtt{t}\mathtt{o}\mathtt{p}}(x)$ and $u_{\mathtt{b}\mathtt{o}\mathtt{t}}(x)$, the associate geodesic curve $\mathrm{\Phi }(x)$ 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.