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 ″ ( x ) + h ( x ) u ( x ) = 0 , with differentiable h ( x ) , locally admits two linearly independent solutions which exist on an open interval around any x 0 ∈ R : u top ( x ) = exp ⁡ [ ∫ x 0 x d ξ Φ ( ξ ) Φ ′ ( ξ ) − [ h ( ξ ) − Φ 2 ( ξ ) ] 2 + [ Φ ′ ( ξ ) ] 2 h ( ξ ) − Φ 2 ( ξ ) ] , u bot ( x ) = exp ⁡ [ ∫ x 0 x d ξ Φ ( ξ ) Φ ′ ( ξ ) + [ h ( ξ ) − Φ 2 ( ξ ) ] 2 + [ Φ ′ ( ξ ) ] 2 h ( ξ ) − Φ 2 ( ξ ) ] , where Φ ( x ) is any geodesic curve in a two dimensional hyperbolic geometry of a Riemannian manifold M h , which is non-vertical at x 0 . I define M h to be an upper half plane { ( x , Φ ) ∈ R 2 | Φ > 0 } , with points in which Φ 2 = h ( x ) being removed, equipped with metric g h = [ ( h ( x ) − Φ 2 ) 2 d x 2 + d Φ 2 ] / Φ 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 M h and Poincaré upper half plane H is induced by any pair of linearly independent solutions of u ″ ( x ) + h ( x ) u ( x ) = 0 . If this pair is selected to be u top ( x ) and u bot ( x ) , the associate geodesic curve Φ ( x ) is mapped to a vertical geodesic curve on H . With these results, supported by complementary remarks and examples, I establish a fundamental link between linear second order ordinary differential equations and two dimensional hyperbolic geometry, adding to textbook knowledge about these profound subareas of mathematics.