{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:46:28Z","timestamp":1760240788273,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2019,9,17]],"date-time":"2019-09-17T00:00:00Z","timestamp":1568678400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Noether symmetries and first integrals of a class of two-dimensional systems of second order ordinary differential equations (ODEs) are investigated using real and complex methods. We show that first integrals of systems of two second order ODEs derived by the complex Noether approach cannot be obtained by the real methods. Furthermore, it is proved that a complex method can be extended to larger systems and higher order.<\/jats:p>","DOI":"10.3390\/sym11091180","type":"journal-article","created":{"date-parts":[[2019,9,17]],"date-time":"2019-09-17T10:42:58Z","timestamp":1568716978000},"page":"1180","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Comparison of Noether Symmetries and First Integrals of Two-Dimensional Systems of Second Order Ordinary Differential Equations by Real and Complex Methods"],"prefix":"10.3390","volume":"11","author":[{"given":"Muhammad","family":"Safdar","sequence":"first","affiliation":[{"name":"School of Mechanical and Manufacturing Engineering (SMME), National University of Sciences and Technology, Campus H-12, Islamabad 44000, Pakistan"}]},{"given":"Asghar","family":"Qadir","sequence":"additional","affiliation":[{"name":"School of Natural Sciences (SNS),National University of Sciences and Technology, Campus H-12, Islamabad 44000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1170-6159","authenticated-orcid":false,"given":"Muhammad Umar","family":"Farooq","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences &amp; Humanities, College of E &amp; ME, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2019,9,17]]},"reference":[{"key":"ref_1","first-page":"187","article-title":"Klassifikation und integration von gewohnlichen differentialgleichungen zwischen x, y, die eine gruppe von transformationen gestatten I, II, II and IV","volume":"8","author":"Lie","year":"1883","journal-title":"Archiv. 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Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/9\/1180\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:21:03Z","timestamp":1760188863000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/9\/1180"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9,17]]},"references-count":15,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2019,9]]}},"alternative-id":["sym11091180"],"URL":"https:\/\/doi.org\/10.3390\/sym11091180","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,9,17]]}}}