{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:05Z","timestamp":1740134525948,"version":"3.37.3"},"reference-count":10,"publisher":"Wiley","license":[{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"name":"Development and Promotion of Science and Talents Project","award":["580010"],"award-info":[{"award-number":["580010"]}]},{"DOI":"10.13039\/501100004704","name":"National Research Council of Thailand","doi-asserted-by":"publisher","award":["580010"],"award-info":[{"award-number":["580010"]}],"id":[{"id":"10.13039\/501100004704","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004071","name":"Khon Kaen University","doi-asserted-by":"publisher","award":["580010"],"award-info":[{"award-number":["580010"]}],"id":[{"id":"10.13039\/501100004071","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2015]]},"abstract":"<jats:p>We investigate an analytic solution of the second-order differential equation with a state derivative dependent delay of the form<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u2033<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>p<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>+<\/mml:mo><mml:mi>b<\/mml:mi><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>. Considering a convergent power series<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>g<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>of an auxiliary equation<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:msup><mml:mrow><mml:mi>\u03b3<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:msup><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u2033<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03b3<\/mml:mi><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:msup><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mo stretchy=\"false\">[<\/mml:mo><mml:mi>g<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mrow><mml:mi>\u03b3<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>-<\/mml:mo><mml:mi>p<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>g<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03b3<\/mml:mi><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">]<\/mml:mo><mml:mi>\u03b3<\/mml:mi><mml:msup><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03b3<\/mml:mi><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u2032<\/mml:mi><mml:mi>\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>g<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:mi>\u03b3<\/mml:mi><mml:msup><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03b3<\/mml:mi><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:msup><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u2033<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>with the relation<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>p<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>+<\/mml:mo><mml:mi>b<\/mml:mi><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mi>g<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03b3<\/mml:mi><mml:msup><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>, we obtain an analytic solution<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>. Furthermore, we characterize a polynomial solution when<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mi>p<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>is a polynomial.<\/jats:p>","DOI":"10.1155\/2015\/904679","type":"journal-article","created":{"date-parts":[[2015,10,5]],"date-time":"2015-10-05T21:05:16Z","timestamp":1444079116000},"page":"1-8","source":"Crossref","is-referenced-by-count":0,"title":["Analytic Solutions of a Second-Order Functional Differential Equation with a State Derivative Dependent Delay"],"prefix":"10.1155","volume":"2015","author":[{"given":"Jiraphorn","family":"Somsuwan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Keaitsuda Maneeruk","family":"Nakprasit","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"year":"1963","key":"1"},{"year":"1977","key":"4"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(84)90150-5"},{"issue":"4","key":"8","doi-asserted-by":"crossref","first-page":"471","DOI":"10.11650\/twjm\/1500406123","volume":"1","year":"1997","journal-title":"Taiwanese Journal of Mathematics"},{"issue":"1","key":"6","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1515\/dema-2006-0111","volume":"39","year":"2006","journal-title":"Demonstratio Mathematica"},{"issue":"51","key":"11","first-page":"1","volume":"2009","year":"2009","journal-title":"Electronic Journal of Differential Equations"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1007\/bf03322694"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1155\/2009\/145213"},{"issue":"2","key":"9","doi-asserted-by":"crossref","first-page":"273","DOI":"10.4064\/cm-79-2-273-281","volume":"79","year":"1999","journal-title":"Colloquium Mathematicum"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/7\/1\/009"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2015\/904679.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2015\/904679.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2015\/904679.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,9]],"date-time":"2020-09-09T06:18:13Z","timestamp":1599632293000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2015\/904679\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015]]},"references-count":10,"alternative-id":["904679","904679"],"URL":"https:\/\/doi.org\/10.1155\/2015\/904679","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2015]]}}}