{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T07:35:42Z","timestamp":1769844942794,"version":"3.49.0"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2014,8,30]],"date-time":"2014-08-30T00:00:00Z","timestamp":1409356800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Eng Math"],"published-print":{"date-parts":[[2014,12]]},"DOI":"10.1007\/s10665-014-9696-3","type":"journal-article","created":{"date-parts":[[2014,8,29]],"date-time":"2014-08-29T14:22:40Z","timestamp":1409322160000},"page":"193-205","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["A class of exact solutions of the Li\u00e9nard-type ordinary nonlinear differential equation"],"prefix":"10.1007","volume":"89","author":[{"given":"Tiberiu","family":"Harko","sequence":"first","affiliation":[]},{"given":"Francisco S. N.","family":"Lobo","sequence":"additional","affiliation":[]},{"given":"M. K.","family":"Mak","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2014,8,30]]},"reference":[{"key":"9696_CR1","first-page":"901","volume":"23","author":"A Li\u00e9nard","year":"1928","unstructured":"Li\u00e9nard A (1928) \u00c9tude des oscillations entret\u00e9nues. Revue g\u00e9n\u00e9rale de l\u2019\u00e9lectricit\u00e9 23:901\u2013912","journal-title":"Revue g\u00e9n\u00e9rale de l\u2019\u00e9lectricit\u00e9"},{"key":"9696_CR2","first-page":"946","volume":"23","author":"A Li\u00e9nard","year":"1928","unstructured":"Li\u00e9nard A (1928) \u00c9tude des oscillations entret\u00e9nues. Revue g\u00e9n\u00e9rale de l\u2019\u00e9lectricit\u00e9 23:946\u2013954","journal-title":"Revue g\u00e9n\u00e9rale de l\u2019\u00e9lectricit\u00e9"},{"key":"9696_CR3","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1215\/S0012-7094-42-00928-1","volume":"9","author":"N Levinson","year":"1942","unstructured":"Levinson N, Smith O (1942) A general equation for relaxation oscillations. Duke Math J 9:382\u2013403","journal-title":"Duke Math J"},{"key":"9696_CR4","doi-asserted-by":"crossref","first-page":"978","DOI":"10.1080\/14786442608564127","volume":"2","author":"B Pol van der","year":"1927","unstructured":"van der Pol B (1927) On relaxation-oscillations. Lond Edinb Dublin Philos Mag J Sci 2:978\u2013992","journal-title":"Lond Edinb Dublin Philos Mag J Sci"},{"key":"9696_CR5","volume-title":"Qualitative theory of second order dynamic systems","author":"AA Andronov","year":"1973","unstructured":"Andronov AA, Leontovich EA, Gordon II, Maier AG (1973) Qualitative theory of second order dynamic systems. Wiley, New York"},{"key":"9696_CR6","volume-title":"Nonlinear dynamics and chaos","author":"SH Strogatz","year":"1994","unstructured":"Strogatz SH (1994) Nonlinear dynamics and chaos. Addison-Wesley, Reading"},{"key":"9696_CR7","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1080\/14786441108564652","volume":"6","author":"B Pol van der","year":"1928","unstructured":"van der Pol B, van der Mark J (1928) The heartbeat considered as a relaxation oscillation, and an electrical model of the heart. Lond Edinb Dublin Philos Mag J Sci 6:763\u2013775","journal-title":"Lond Edinb Dublin Philos Mag J Sci"},{"key":"9696_CR8","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1016\/S0006-3495(61)86902-6","volume":"1","author":"F Fitzhugh","year":"1961","unstructured":"Fitzhugh F (1961) Impulses and physiological states in theoretical models of nerve membranes. Biophys J 1:445\u2013466","journal-title":"Biophys J"},{"key":"9696_CR9","first-page":"2061","volume":"50","author":"J Nagumo","year":"1962","unstructured":"Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc Inst Radio Eng 50:2061\u20132070","journal-title":"Proc Inst Radio Eng"},{"key":"9696_CR10","volume-title":"Theory of heart","author":"L Glass","year":"1990","unstructured":"Glass L (1990) Theory of heart. Springer, New York"},{"key":"9696_CR11","doi-asserted-by":"crossref","DOI":"10.1002\/9783527617548","volume-title":"Applied nonlinear dynamics","author":"A Nayfeh","year":"1995","unstructured":"Nayfeh A, Balachandran B (1995) Applied nonlinear dynamics. Wiley, New York"},{"key":"9696_CR12","volume-title":"Mathematical models in biology","author":"L Edelstein-Keshet","year":"1988","unstructured":"Edelstein-Keshet L (1988) Mathematical models in biology. Random House, New York"},{"key":"9696_CR13","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1103\/PhysRevE.49.157","volume":"49","author":"D Poland","year":"1994","unstructured":"Poland D (1994) Loci of limit cycles. Phys Rev E 49:157\u2013165","journal-title":"Phys Rev E"},{"key":"9696_CR14","doi-asserted-by":"crossref","first-page":"3119","DOI":"10.1142\/S0217732395003276","volume":"10","author":"L Salasnich","year":"1995","unstructured":"Salasnich L (1995) Instabilities, point attractors and limit cycles in an inflationary universe. Mod Phys Lett A 10:3119\u20133127","journal-title":"Mod Phys Lett A"},{"key":"9696_CR15","first-page":"873","volume":"112","author":"L Salasnich","year":"1997","unstructured":"Salasnich L (1997) On the limit cycle of an inflationary universe. Nuovo Cimento B 112:873\u2013880","journal-title":"Nuovo Cimento B"},{"key":"9696_CR16","first-page":"41","volume":"5","author":"Z Ran","year":"2009","unstructured":"Ran Z (2009) One exactly soluble model in isotropic turbulence. Adv Appl Fluid Mech 5:41\u201347","journal-title":"Adv Appl Fluid Mech"},{"key":"9696_CR17","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1098\/rspa.1938.0013","volume":"164","author":"T Karman de","year":"1938","unstructured":"de Karman T, Howarth L (1938) On the statistical theory of the isotropic turbulence. Proc R Soc Lond A 164:192\u2013215","journal-title":"Proc R Soc Lond A"},{"key":"9696_CR18","unstructured":"Sen S, Ray AK (2012) Implications of nonlinearity for spherically symmetric accretion. arXiv:1207.1070"},{"key":"9696_CR19","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-8176-4648-6","volume-title":"Classical mechanics: theory and mathematical modeling","author":"E DiBenedetto","year":"2011","unstructured":"DiBenedetto E (2011) Classical mechanics: theory and mathematical modeling. Birkh\u00e4user\/Springer, New York"},{"key":"9696_CR20","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1088\/0951-7715\/3\/4\/004","volume":"3","author":"F Dumortier","year":"1990","unstructured":"Dumortier F, Rousseau C (1990) Cubic Li\u00e9nard equations with linear damping. Nonlinearity 3:1015\u20131039","journal-title":"Nonlinearity"},{"issue":"1","key":"9696_CR21","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/BF02969477","volume":"3","author":"F Dumortier","year":"2000","unstructured":"Dumortier F, Kooij RE, Li CZ (2000) Cubic Li\u00e9nard equations with quadratic damping having two antisaddles. Qual Theory Dyn Syst 3(1):163\u2013209","journal-title":"Qual Theory Dyn Syst"},{"key":"9696_CR22","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/S0010-4655(00)00042-4","volume":"130","author":"ES Cheb-Terrab","year":"2000","unstructured":"Cheb-Terrab ES, Roche AD (2000) Abel ODEs: equivalence and integrable classes. Comput Phys Commun 130:204\u2013231","journal-title":"Comput Phys Commun"},{"key":"9696_CR23","doi-asserted-by":"crossref","first-page":"9743","DOI":"10.1088\/0305-4470\/39\/31\/006","volume":"39","author":"VK Chandrasekar","year":"2006","unstructured":"Chandrasekar VK, Senthilvelan M, Kundu A, Lakshmanan M (2006) A nonlocal connection between certain linear and nonlinear ordinary differential equations\/oscillators. J Phys A 39:9743\u20139754","journal-title":"J Phys A"},{"key":"9696_CR24","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/j.cam.2007.07.024","volume":"219","author":"XG Liu","year":"2008","unstructured":"Liu XG, Tang ML, Martin RR (2008) Periodic solutions for a kind of Li\u00e9nard equation. J Comput Appl Math 219:263\u2013275","journal-title":"J Comput Appl Math"},{"key":"9696_CR25","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1016\/j.cam.2007.11.005","volume":"222","author":"L Zou","year":"2008","unstructured":"Zou L, Chen XW, Zhang WN (2008) Local bifurcations of critical periods for cubic Li\u00e9nard equations with cubic damping. J Comput Appl Math 222:404\u2013410","journal-title":"J Comput Appl Math"},{"key":"9696_CR26","doi-asserted-by":"crossref","first-page":"052901","DOI":"10.1063\/1.3126493","volume":"50","author":"RG Pradeep","year":"2009","unstructured":"Pradeep RG, Chandrasekar VK, Senthilvelan M, Lakshmanan M (2009) Nonstandard conserved Hamiltonian structures in dissipative\/damped systems: nonlinear generalizations of damped harmonic oscillator. J Math Phys 50:052901","journal-title":"J Math Phys"},{"key":"9696_CR27","doi-asserted-by":"crossref","first-page":"082702","DOI":"10.1063\/1.3187783","volume":"50","author":"SN Pandey","year":"2009","unstructured":"Pandey SN, Bindu PS, Senthilvelan M, Lakshmanan M (2009) A group theoretical identification of integrable cases of the Linard-type equation $${\\ddot{x}}+f(x){\\dot{x}}+g(x)=0$$ x \u00a8 + f ( x ) x \u02d9 + g ( x ) = 0 . I. Equations having nonmaximal number of Lie point symmetries. J Math Phys 50:082702","journal-title":"J Math Phys"},{"key":"9696_CR28","doi-asserted-by":"crossref","first-page":"102701","DOI":"10.1063\/1.3204075","volume":"50","author":"SN Pandey","year":"2009","unstructured":"Pandey SN, Bindu PS, Senthilvelan M, Lakshmanan M (2009) A group theoretical identification of integrable equations in the Li\u00e9nard-type equation $${\\ddot{x}}+f(x){\\dot{x}}+g(x)=0$$ x \u00a8 + f ( x ) x \u02d9 + g ( x ) = 0 . II: Equations having maximal Lie point symmetries. J Math Phys 50:102701","journal-title":"J Math Phys"},{"key":"9696_CR29","doi-asserted-by":"crossref","first-page":"062001","DOI":"10.1088\/1751-8113\/43\/6\/062001","volume":"43","author":"D Banerjee","year":"2010","unstructured":"Banerjee D, Bhattacharjee JK (2010) Renormalization group and Li\u00e9nard systems of differential equations. J Phys A 43:062001","journal-title":"J Phys A"},{"key":"9696_CR30","doi-asserted-by":"crossref","first-page":"1402","DOI":"10.1016\/j.physd.2011.06.006","volume":"240","author":"M Messias","year":"2011","unstructured":"Messias M, Gouveia MRA (2011) Time-periodic perturbation of a Li\u00e9nard equation with an unbounded homoclinic loop. Physica D 240:1402\u20131409","journal-title":"Physica D"},{"key":"9696_CR31","doi-asserted-by":"crossref","first-page":"485","DOI":"10.3934\/dcdsb.2008.10.485","volume":"10","author":"IA Garcia","year":"2008","unstructured":"Garcia IA, Gin\u00e9 J, Llibre J (2008) Li\u00e9nard and Riccati differential equations related via Lie algebras. Discr Contin Dyn Syst B 10:485\u2013494","journal-title":"Discr Contin Dyn Syst B"},{"key":"9696_CR32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4064\/dm479-0-1","volume":"479","author":"JF Carinena","year":"2011","unstructured":"Carinena JF, de Lucas J (2011) Lie systems: theory, generalisations, and applications. Dissertationes Mathematicae (Rozprawy Matematyczne) 479:1\u2013162","journal-title":"Dissertationes Mathematicae (Rozprawy Matematyczne)"},{"key":"9696_CR33","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1016\/j.aml.2010.01.004","volume":"23","author":"J Gin\u00e9","year":"2010","unstructured":"Gin\u00e9 J, Grau M (2010) Weierstrass integrability of differential equations. Appl Math Lett 23:523\u2013526","journal-title":"Appl Math Lett"},{"key":"9696_CR34","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1016\/j.jmaa.2010.11.005","volume":"377","author":"J Gin\u00e9","year":"2011","unstructured":"Gin\u00e9 J, Llibre J (2011) Weierstrass integrability in Li\u00e9nard differential systems. J Math Anal Appl 377:362\u2013369","journal-title":"J Math Anal Appl"},{"key":"9696_CR35","volume-title":"Handbook of exact solutions for ordinary differential equations","author":"AD Polyanin","year":"2003","unstructured":"Polyanin AD, Zaitsev VF (2003) Handbook of exact solutions for ordinary differential equations. Chapman & Hall\/CRC, Boca Raton\/Washington"},{"key":"9696_CR36","volume-title":"Differentialgleichungen: L\u00f6sungsmethoden und L\u00f6sungen","author":"E Kamke","year":"1959","unstructured":"Kamke E (1959) Differentialgleichungen: L\u00f6sungsmethoden und L\u00f6sungen. Chelsea, New York"},{"key":"9696_CR37","first-page":"301","volume":"10","author":"A Chiellini","year":"1931","unstructured":"Chiellini A (1931) Sull\u2019integrazione dell\u2019equazione differenziale $$y^{\\prime } +Py^ 2 +Qy^ 3 =0$$ y \u2032 + P y 2 + Q y 3 = 0 . Bollettino dell Unione Matematica Italiana 10:301\u2013307","journal-title":"Bollettino dell Unione Matematica Italiana"},{"key":"9696_CR38","first-page":"59","volume":"16","author":"I Bandic","year":"1961","unstructured":"Bandic I (1961) Sur le crit\u00e8re d\u2019int\u00e9grabilit\u00e9 de l\u2019\u00e9quation diff\u00e9rentielle g\u00e9n\u00e9ralis\u00e9e de Li\u00e9nard. Bollettino dell Unione Matematica Italiana 16:59\u201367","journal-title":"Bollettino dell Unione Matematica Italiana"},{"key":"9696_CR39","doi-asserted-by":"crossref","first-page":"1395","DOI":"10.1016\/S0898-1221(01)00104-3","volume":"41","author":"MK Mak","year":"2001","unstructured":"Mak MK, Chan HW, Harko T (2001) Solutions generating technique for Abel-type nonlinear ordinary differential equations. Comput Math Appl 41:1395\u20131401","journal-title":"Comput Math Appl"},{"key":"9696_CR40","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/S0898-1221(01)00274-7","volume":"43","author":"MK Mak","year":"2002","unstructured":"Mak MK, Harko T (2002) New method for generating general solution of Abel differential equation. Comput Math Appl 43:91\u201394","journal-title":"Comput Math Appl"},{"key":"9696_CR41","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1016\/S0898-1221(03)90147-7","volume":"46","author":"T Harko","year":"2003","unstructured":"Harko T, Mak MK (2003) Relativistic dissipative cosmological models and Abel differential equation. Comput Math Appl 46:849\u2013853","journal-title":"Comput Math Appl"},{"key":"9696_CR42","doi-asserted-by":"crossref","first-page":"1234","DOI":"10.1016\/j.physleta.2013.04.024","volume":"377","author":"SC Mancas","year":"2013","unstructured":"Mancas SC, Rosu HC (2013) Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations. Phys Lett A 377:1234\u20131238","journal-title":"Phys Lett A"},{"key":"9696_CR43","unstructured":"Mancas SC, Rosu HC (2013) Integrable Ermakov\u2013Pinney equations with nonlinear Chiellini \u2018damping\u2019. arXiv:1301.3567"},{"key":"9696_CR44","doi-asserted-by":"crossref","first-page":"101","DOI":"10.13189\/ujam.2013.010209","volume":"1","author":"T Harko","year":"2013","unstructured":"Harko T, Lobo FSN, Mak MK (2013) A Chiellini type integrability condition for the generalized first kind Abel differential equation. Univers J Appl Math 1:101\u2013104","journal-title":"Univers J Appl Math"}],"container-title":["Journal of Engineering Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10665-014-9696-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10665-014-9696-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10665-014-9696-3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,15]],"date-time":"2022-04-15T22:54:59Z","timestamp":1650063299000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10665-014-9696-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,8,30]]},"references-count":44,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2014,12]]}},"alternative-id":["9696"],"URL":"https:\/\/doi.org\/10.1007\/s10665-014-9696-3","relation":{},"ISSN":["0022-0833","1573-2703"],"issn-type":[{"value":"0022-0833","type":"print"},{"value":"1573-2703","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,8,30]]}}}