Abstract
The paper deals with treating some study methods of the equation integrability of a certain type that are little studied in the theory of differential equations. It is known that a significant part of the differential equations cannot be integrated. Then, to develop methods for their study is, certainly, of scientific interest. The obtained results, formulated as theorems and statements, are of scientific and practical interest because of their importance for applications in modern science.In the paper we present an alternative method for studying the integrability of both linear and nonlinear differential equations of the second order. An introduction of parameters allowed us to develop a study method for the integrability of ordinary differential equations of the second order. We also formulate the theorems describing some General conditions for the integrability of the second-order linear equation and consider special cases of integrability, which arise out of the above facts.Based on the obtained parameter method, some General conditions for the integrability of the nonlinear differential equation of the second order are given, and the consequences of these General conditions are indicated.We have obtained new results related to the construction and development of methods for studying the differential equation to which some types of differential equations are reduced and laid the foundations for a rigorous and systematic study of the introduced special nonlinear differential equation of the second order.
Highlights
Öåëüþ ñòàòüè ÿâëÿåòñÿ ðàçðàáîòêà àïïàðàòà, ìåòîäà èññëåäîâàíèÿ, ñïîñîáñòâóþùåãî ðàçâèòèþ ïðîáëåìíûõ âîïðîñîâ òåîðèè äèôôåðåíöèàëüíûõ óðàâíåíèé
2ν2ν1−1, ñ ïîñëåäóþùèì ïðåîáðàçîâàíèåì, ïîëó÷èì äèôôåðåíöèàëüíîå óðàâíåíèå Áåðíóëëè ðåøåíèå êîòîðîãî èìååò âèä: a
Ââåäåíèåì äîïîëíèòåëüíûõ ôóíêöèé â ðàññìàòðèâàåìûå âèäû äèôôåðåíöèàëüíûõ óðàâíåíèé ìîãóò áûòü ïîëó÷åíû èíòåãðàëüíûå ïðåäñòàâëåíèÿ ðåøåíèé, ïðè ýòîì íà äîïîëíèòåëüíûå ôóíêöèè è êîýôôèöèåíòû ðàññìàòðèâàåìîãî óðàâíåíèÿ íàêëàäûâàåòñÿ óðàâíåíèå ñâÿçè
Summary
Öåëüþ ñòàòüè ÿâëÿåòñÿ ðàçðàáîòêà àïïàðàòà, ìåòîäà èññëåäîâàíèÿ, ñïîñîáñòâóþùåãî ðàçâèòèþ ïðîáëåìíûõ âîïðîñîâ òåîðèè äèôôåðåíöèàëüíûõ óðàâíåíèé. Îïðåäåëÿåòñÿ ñâÿçü ìåæäó ââåäåííûìè ôóíêöèÿìè è êîýôôèöèåíòàìè ðàññìàòðèâàåìûõ äèôôåðåíöèàëüíûõ óðàâíåíèé è äîêàçûâàåòñÿ, ÷òî ðåøåíèå èçó÷àåìîãî óðàâíåíèÿ ïðåäñòàâèìî â âèäå êîíêðåòíîé ôîðìóëû, âûðàæåíèå êîòîðîé ïðåäñòàâëåíî â äàííîé ñòàòüå. Óðàâíåíèå (2) îòíîñèòåëüíî w ÿâëÿåòñÿ ëèíåéíûì äèôôåðåíöèàëüíûì óðàâíåíèåì ïåðâîãî ïîðÿäêà, ðåøåíèå êîòîðîãî w = w1(z) èìååò âèä w = w1(z) = V1−1 c1 + θV1dz , (5) Äèôôåðåíöèàëüíîå óðàâíåíèå (4) îòíîñèòåëüíî w ÿâëÿåòñÿ ëèíåéíûì äèôôåðåíöèàëüíûì óðàâíåíèåì ïåðâîãî ïîðÿäêà, åãî ðåøåíèå w = w2(z) èìååò âèä w = w2(z) = V2−1b(z), (9)
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