äîâàòåëüíî, C(x) = x3 + C, ãäå C ïðîèçâîëüíàÿ ïîñòîÿííàÿ. Èòàê, y = xcos+C x îáùåå ðåøåíèå èñõîäíîãî ëèíåéíîãî íåîäíîðîäíîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ. 3 Ìåòîä Áåðíóëëè Áóäåì èñêàòü ðåøåíèå óðàâíåíèÿ (1.13) â âèäå ïðîèçâåäåíèÿ äâóõ ôóíêöèé, ò.å. ïîëîæèì y = u(x)v(x). Òîãäà y 0 = u 0(x)v(x) + u(x)v 0(x). Ïîäñòàâëÿÿ y è y 0 â óðàâíåíèå (1.13), ïîëó÷èì u 0v + uv 0 + P(x)uv = = f(x), ò.å. v(u 0 + P(x)u) + uv 0 = f(x). Òàê êàê y åñòü ïðîèçâåäåíèå äâóõ ôóíêöèé, òî îäíà èç íèõ ìîæåò áûòü âûáðàíà ïðîèçâîëüíî. Âûáåðåì ôóíêöèþ u(x) òàê, ÷òîáû îíà óäîâëåòâîðÿëà ëèíåéíîìó îäíîðîäíîìó óðàâíåíèþ u 0 + P(x)u = 0. Òîãäà ôóíêöèÿ v(x) äîëæíà óäîâëåòâîðÿòü f(x) äèôôåðåíöèàëüíîìó óðàâíåíèþ uv 0 = f(x) èëè v 0 = u(x) . ÑëåäîâàòåëüR f(x) íî, v = u(x) dx, ãäå u(x) êàêîå-ëèáî ÷àñòíîå ðåøåíèå óðàâíåíèÿ u 0 + + P(x)u = 0. Òàêèì îáðàçîì, îáùåå ðåøåíèå èñõîäíîãî óðàâíåíèÿ èìååò R f(x) dx. âèä y = u(x) u(x) Ðåøèòü óðàâíåíèå y 0 − y tg x = cos3x x . Ïðèìåíèì ìåòîä Áåðíóëëè. Ïîëîæèì y = uv, òîãäà y 0 = = u 0 v + uv 0 ; äàííîå óðàâíåíèå ïðèâîäèòñÿ ê âèäó u 0 v + uv 0 − uv tg x = 3x 3x 0 0 0 = cos x èëè v(u − u tg x) + uv = cos x . Ïîëîæèì u − u tg x = 0. Íàéäåì êàêîå-ëèáî îòëè÷íîå îò íóëÿ ÷àñòíîå ðåøåíèå ýòîãî óðàâíåíèÿ, íàïðèìåð u = cos1 x (ñì. ïðèìåð 14). Äëÿ îòûñêàíèÿ äðóãîé íåèçâåñòíîé ôóíêöèè 3x 1 3x 0 0 2 v(x) èìååì óðàâíåíèå uv 0 = cos x , ò.å. cos x v = cos x , îòñþäà v = 3x è ïîýòîìó v = x3 + C, ãäå C ïðîèçâîëüíàÿ ïîñòîÿííàÿ. Òàêèì îáðàçîì, y = xcos+C x îáùåå ðåøåíèå èñõîäíîãî óðàâíåíèÿ. Ðåøèòü óðàâíåíèå y 0 + y 1+x = 0. Ýòî óðàâíåíèå íå ÿâëÿåòñÿ ëèíåéíûì îòíîñèòåëüíî ôóíêöèè y(x), îäíàêî îòíîñèòåëüíî ôóíêöèè x(y) îíî ÿâëÿåòñÿ ëèíåéíûì.  dy dx +x+y2 = 0, ñàìîì äåëå, dx + y 1+x = 0, îòêóäà (y2 +x) dy+ dx = 0 èëè dy ò.å. xy0 + x = −y2. Ðåøèì ýòî óðàâíåíèå ìåòîäîì Áåðíóëëè, ïîëîæèâ x = = u(y)v(y). Ïîäñòàâëÿÿ x = uv è x 0 = u 0 v+uv 0 â óðàâíåíèå x 0 +x = iy2 , ïîëó÷èì, ÷òî u 0v + uv 0 + uv = −y2 èëè v(u 0 + u) + uv 0 = −y2. Âûáåðåì ôóíêöèþ u(y) òàê, ÷òîáû u 0 + u = 0. Òîãäà ôóíêöèÿ v(y) óäîâëåòâîðÿåò äèôôåðåíöèàëüíîìó óðàâíåíèþ u(v 0) = −y2. Ïåðåïèøåì óðàâíåíèå u 0 + u = 0 â Râèäå du dyR+ u = 0. Ðàçäåëèâ ïåðåìåííûå, ïîëó÷èì du du dy = 0, ò.å. ln |u| + y = ln |C|. Òàêèì u + dy = 0, îòêóäà u + −y îáðàçîì, u = Ce . Òàê êàê íàñ èíòåðåñóåò êàêîå-ëèáî ÷àñòíîå ðåøåíèå, 11 2 Ïðèìåð 16. Ðåøåíèå. 2 2 2 2 3 Ïðèìåð 17. 2 Ðåøåíèå. 2 ïîëîæèì C = 1. Òîãäà u = e−y. Ïîäñòàâèâ u = e−y â óðàâíåíèå u(v 0 ) = R = −y2 , ïîëó÷èì e−y v 0 = −y2 , ò.å. v 0 = −y2 ey , îòêóäà v = − y2 ey dy = = −ey (y2 − 2y + 2) + C (ïðè íàõîæäåíèè ïîñëåäíåãî èíòåãðàëà áûëà äâàæäû ïðèìåíåíà ôîðìóëà èíòåãðèðîâàíèÿ ïî ÷àñòÿì). Èòàê, èìååì u = e−y , v = −ey (y2 − 2y + 2) + C. Òàêèì îáðàçîì, îáùåå ðåøåíèå èñõîäíîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ x = uv = e−y(−ey(y2 − 2y + +2)+C), ò.å. x = −y2 +2y−2+Ce−y , ãäå C ïðîèçâîëüíàÿ ïîñòîÿííàÿ. Óïðàæíåíèÿ äëÿ ñàìîñòîÿòåëüíîé ðàáîòû Ðåøèòü äèôôåðåíöèàëüíûå óðàâíåíèÿ: 1.23. y 0 − y tg x = cosx x . 1.24. (x2 + 3)y 0 + 2xy = x. 1.25. y 0 + sin22x y = cos x. 2y 1.26. y 0 − x+1 = (x + 1)3 . 1.27. x2y 0 + 2xy = ln x. 1.28. y 0 sin x − y = 1 − cos x. Íàéòè ÷àñòíûå ðåøåíèÿ äèôôåðåíöèàëüíûõ óðàâíåíèé, óäîâëåòâîðÿþùèå íà÷àëüíûì óñëîâèÿì: 1.29. ex(y + y 0) = 1, y(0) = 0. 1.30. y 0 − 3yx = exx3, y(1) = e. 1.31. cos x dy + y sin x dx = 0, y(0) = 1. 1.32. y 0 + 2xy = 2x2e−x , y(0) = 0. 1.33. y 0 + x+y1 = 0, y(−1) = 0. 2 2 2 12