{"paper":{"title":"The Ermakov-Pinney Equation: its varied origins and the effects of the introduction of symmetry-breaking functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Peter Gavin Lawrence Leach, Richard Michael Morris","submitted_at":"2015-10-30T07:40:07Z","abstract_excerpt":"The Ermakov-Pinney Equation, $$\\ddot{x}+\\omega^2 x=\\frac{h^2}{x^3},$$ has a varied provenance which we briefly delineate. We introduce time-dependent functions in place of the $\\omega^2$ and $h^2$. The former has no effect upon the algebra of the Lie point symmetries of the equation. The latter destroys the $sl(2,\\Re)$ symmetry and a single symmetry persists only when there is a specific relationship between the two time-dependent functions introduced. We calculate the form of the corresponding autonomous equation for these cases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.08992","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}