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We find their corresponding invariants which reduce them to first order nonlinear ordinary differential equations. Using particular solutions of such EF equations, the two classes are set in the autonomous nonlinear oscillator form d^2\\nu/dt^2 +ad \\nu/dt +b(\\nu-\\nu^n)=0, where the coefficients a,b depend only on \\lambda, n. For both classes, we write closed-form"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1604.04807","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-04-16T22:53:52Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"82745846738ff5bbeea9bc5e9d3414d50fa90415173f6cd383e9564b35d28bcb","abstract_canon_sha256":"acbad1763d7259cf0ead51152e33f302e576f3c61f78931611e78ff87fd2a0c2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:40.013889Z","signature_b64":"8COuJj6HrD91t9d/+3zyywgF4au7o6to5PkqJQOZiMIP2aSBXxBBKnqCoeidbayMuRq77EW8psgBoVuwsTBqDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e7b8c854e4b3a5a17c239b55ad8f8451797229a00cf0e4827288da522f38e54","last_reissued_at":"2026-05-17T23:51:40.013245Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:40.013245Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two integrable classes of Emden-Fowler equations with applications in astrophysics and cosmology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"H.C. Rosu, S.C. Mancas","submitted_at":"2016-04-16T22:53:52Z","abstract_excerpt":"We show that some Emden-Fowler (EF) equations encountered in astrophysics and cosmology belong to two EF integrable classes of the type d^2z/d{\\chi}^2=A \\chi^{-\\lambda-2}z^n for \\lambda=(n-1)/2 (class one), and \\lambda=n+1 (class two). We find their corresponding invariants which reduce them to first order nonlinear ordinary differential equations. Using particular solutions of such EF equations, the two classes are set in the autonomous nonlinear oscillator form d^2\\nu/dt^2 +ad \\nu/dt +b(\\nu-\\nu^n)=0, where the coefficients a,b depend only on \\lambda, n. 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