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Äàë³ âàð³þºìî ñòàëó ³ øóêàºìî ðîçâ’ÿçîê ð³âíÿííÿ<br />
(11.31) ó âèãëÿä³<br />
2<br />
y = c( x)<br />
e −x . (11.5.12)<br />
ϳäñòàâëÿþ÷è ôóíêö³þ (11.5.12) ³ ¿¿ ïîõ³äíó äî (11.5.11)<br />
( ( ) −x<br />
)<br />
2 2 −x<br />
y′ = c′<br />
x e − xe 2<br />
c ( x ) , ïåðåéäåìî äî ïðîñòîãî äèôåðåíö³àëüíîãî<br />
ð³âíÿííÿ â³äíîñíî c(x):<br />
2 2<br />
−x<br />
−x<br />
′ = àáî ( ) 1<br />
c( x)<br />
e<br />
e<br />
c′ x = .<br />
Î÷åâèäíî, ùî c(x) =x + c 0 ³ øóêàíèé ðîçâ’ÿçîê áóäå âèãëÿäàòè<br />
òàê:<br />
2 2<br />
−x<br />
−x<br />
y = xe + c e , c = const.<br />
0 0<br />
Çàóâàæåííÿ 1.  íàâåäåíèõ ïðèêëàäàõ ëåãêî ïîáà÷èòè<br />
çàêîíîì³ðí³ñòü: çàãàëüíèé ðîçâ’ÿçîê ë³í³éíîãî íåîäíîð³äíîãî<br />
ð³âíÿííÿ çîáðàæåíî ó âèãëÿä³ ñóìè çàãàëüíîãî ðîçâ’ÿçêó<br />
ë³í³éíîãî îäíîð³äíîãî ð³âíÿííÿ ³ ÷àñòèííîãî ðîçâ’ÿçêó<br />
íåîäíîð³äíîãî ð³âíÿííÿ.<br />
Äðóãèé ìåòîä (ìåòîä Áåðíóëë³ – Ôóð’º) ðîçâ’ÿçàííÿ ð³âíÿííÿ<br />
âèãëÿäó (11.5.1) ïîëÿãຠâ çàì³í³ ôóíêö³¿ y äîáóòêîì<br />
äâîõ äîïîì³æíèõ ôóíêö³é: y = uv. ˳í³éíå ð³âíÿííÿ<br />
ïðè öüîìó çâîäèòüñÿ äî äâîõ ð³âíÿíü ç â³äîêðåìëþâàíèìè<br />
çì³ííèìè â³äíîñíî êîæíî¿ ç äîïîì³æíèõ ôóíêö³é u ³ v.<br />
ijéñíî, ï³äñòàâëÿþ÷è âèðàç äëÿ y ³ y′ äî (11.5.1), îòðèìà-<br />
ºìî<br />
uv ′ + vu ′ + P x uv= Q x ⇒ u′ + P x u v+ vu ′ = Q x .<br />
( ) ( ) ( ( ) ) ( )<br />
Âèáåðåìî ìíîæíèê u òàêèì, ùîá u′ + P( x) u = 0. Òîä³<br />
( )<br />
vu ′ = Q x . Îñòàíí³ äâà ð³âíÿííÿ ëåãêî ðîçâ’ÿçóþòüñÿ:<br />
Pxdx ( )<br />
u = e − ∫<br />
(äèâ. ôîðìóëó (11.5.2), äå c = 1) ³<br />
îñòàòî÷íî<br />
Pxdx ( )<br />
Pxdx ( )<br />
v′ = Q x e<br />
∫<br />
⇒ v = Q x e<br />
∫<br />
∫ dx + c ,<br />
( )<br />
1<br />
( )<br />
−∫ ( ) −∫ ( ) ∫ ( )<br />
( )<br />
Pxdx Pxdx Pxdx<br />
y = c e + e ∫ Q x e dx ,<br />
1<br />
ùî çá³ãàºòüñÿ ç ôîðìóëîþ (11.5.4). Ïðî³ëþñòðóºìî ìåòîä<br />
ïðèêëàäàìè.<br />
Çàóâàæåííÿ 2. Çà äîïîìîãîþ ï³äñòàíîâêè y = uv<br />
( y′ = uv ′ + vu ′ ) ðîçâ’ÿçóºòüñÿ òàêîæ ð³âíÿííÿ Áåðíóëë³<br />
n<br />
y′ + P( x) y = y Q( x)<br />
, ùî â³äð³çíÿºòüñÿ â³ä ë³í³éíîãî òèì, ùî<br />
äî ïðàâî¿ ÷àñòèíè âõîäèòü ìíîæíèêîì ôóíêö³ÿ y â ñòåïåí³,<br />
â³äì³ííîìó â³ä íóëÿ é îäèíèö³.<br />
Ïðèêëàä 11.5.4. Ðîçâ’ÿçàòè ð³âíÿííÿ<br />
y′ − yctg<br />
x = sin x.<br />
Ðîçâ’ÿçàííÿ. Ïîêëàäåìî y = uv, òîä³ y′ = u′ v + uv′<br />
³<br />
ð³âíÿííÿ ïåðåòâîðèòüñÿ äî âèãëÿäó:<br />
+ − = àáî ′ ( ′ ctg )<br />
uv ′ uv′<br />
uvctg<br />
x sin x<br />
uv+ u v − v x = sin x.<br />
Îñê³ëüêè îäíó ç äîïîì³æíèõ ôóíêö³é u àáî v ìîæíà<br />
óçÿòè äîâ³ëüíî, òî âèáåðåìî v ÿê äåÿêèé ÷àñòèííèé ³íòåãðàë<br />
ð³âíÿííÿ v′ − vctg x = 0 .<br />
Òîä³ äëÿ çíàõîäæåííÿ u îòðèìàºìî ùå îäíå ð³âíÿííÿ<br />
uv ′ = sin x.<br />
Ðîçâ’ÿçóºìî ïåðøå ð³âíÿííÿ, âèçíà÷àþ÷è â³äì³ííèé â³ä<br />
íóëÿ ÷àñòèííèé ³íòåãðàë<br />
dv<br />
ctg xdx ln v ln sin x v sin x<br />
v = ⇒ = ⇒ = .<br />
ϳäñòàâëÿþ÷è v äî äðóãîãî ð³âíÿííÿ ³ ðîçâ’ÿçóþ÷è éîãî,<br />
çíàéäåìî u ÿê çàãàëüíèé ³íòåãðàë öüîãî ð³âíÿííÿ<br />
u′ sin x = sin x ⇒ du = dx ⇒ u = x + c .<br />
Îòæå, îñòàòî÷íî<br />
y = uv = ( x+ c)sin<br />
x.<br />
Ïðèêëàä 11.5.5. Ðîçâ’ÿçàòè ð³âíÿííÿ<br />
2 2 3<br />
xyy′ + xy = 1 .<br />
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