{"paper":{"title":"On the integrability of the Abel and of the extended Li\\'{e}nard equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","nlin.SI"],"primary_cat":"math.CA","authors_text":"Man Kwong Mak, Tiberiu Harko","submitted_at":"2019-08-10T11:00:22Z","abstract_excerpt":"We present some exact integrability cases of the extended Li\\'{e}nard equation $y^{\\prime \\prime }+f\\left( y\\right) \\left(y^{\\prime }\\right)^{n}+k\\left( y\\right) \\left(y^{\\prime }\\right)^{m}+g\\left(y\\right) y^{\\prime }+h\\left( y\\right) =0$, with $n>0$ and $m>0$ arbitrary constants, while $f(y)$, $k(y)$, $g(y)$, and $h(y)$ are arbitrary functions. The solutions are obtained by transforming the equation Li\\'{e}nard equation to an equivalent first kind first order Abel type equation given by $\\frac{dv}{dy} =f\\left( y\\right) v^{3-n}+k\\left( y\\right) v^{3-m}+g\\left( y\\right) v^{2}+h\\left( y\\right) "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03730","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.03730/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}