{"paper":{"title":"Algebraic and qualitative remarks about the family $yy'= (\\alpha x^{m+k-1} + \\beta x^{m-k-1})y + \\gamma x^{2m-2k-1}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.DS","authors_text":"Alberto Reyes-Linero, Jorge Rodr\\'iguez-Contreras, Primitivo B. Acosta-Hum\\'anez","submitted_at":"2018-07-10T09:43:16Z","abstract_excerpt":"The aim of this paper is the analysis, from algebraic point of view and singularities studies, of the 5-parametric family of differential equations\n  \\begin{equation*}\\label{folpz}\n  yy'=(\\alpha x^{m+k-1}+\\beta x^{m-k-1})y+\\gamma x^{2m-2k-1}, \\quad y'=\\frac{dy}{dx}\n  \\end{equation*}\n  where $a,b,c\\in \\mathbb{C}$, $m,k\\in \\mathbb{Z}$ and\n  $$\\alpha=a(2m+k) \\quad \\beta=b(2m-k), \\quad \\gamma=-(a^2mx^{4k}+cx^{2k}+b^2m).$$ This family is very important because include Van Der Pol equation. Moreover, this family seems to appear as exercise in the celebrated book of Polyanin and Zaitsev. Unfortunatel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.03551","kind":"arxiv","version":3},"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"}