Метод Бернулли

äîâàòåëüíî, C(x) = x3 + C, ãäå C ïðîèçâîëüíàÿ ïîñòîÿííàÿ. Èòàê,
y = xcos+C
x îáùåå ðåøåíèå èñõîäíîãî ëèíåéíîãî íåîäíîðîäíîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ.
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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. Â
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.
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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.
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