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Òîä³<br />
( ) ( )<br />
y2 x ≡λy1 x , λ= const. (11.6.13)<br />
( ) ( )<br />
y′ x ≡λ y′<br />
x . (11.6.14)<br />
2 1<br />
ϳäñòàâèìî (11.6.13) – (11.6.14) ó âèçíà÷íèê Âðîíñüêîãî.<br />
 ðåçóëüòàò³ ìàòèìåìî<br />
( )<br />
y y y λy<br />
= = = 0 ∀ ∈( , ). (11.6.15)<br />
1 2 1 1<br />
W y x a b<br />
y′ 1<br />
y′ 2<br />
y′ 1<br />
λy′<br />
1<br />
Òóò â îñòàííüîìó ëàíöþæêó ð³âíîñòåé ìè ñêîðèñòàëèñÿ<br />
âëàñòèâ³ñòþ âèçíà÷íèêà (äèâ. 2.3.2).<br />
Îòæå, òåîðåìó äîâåäåíî.<br />
Òåîðåìà 11.6.3. ßêùî ðîçâ’ÿçêè y 1 (x) ³ y 2 (x) ËÎÄÐ<br />
(11.6.1) ë³í³éíî íåçàëåæí³ íà (a, b), òî âèçíà÷íèê Âðîíñüêîãî,<br />
ñêëàäåíèé ç íèõ, â³äì³ííèé â³ä íóëÿ íà öüîìó ³íòåðâàë³.<br />
Ä î â å ä å í í ÿ áóäåìî ïðîâîäèòè ìåòîäîì â³ä ñóïðîòèâíîãî.<br />
Ïðèïóñòèìî, ùî ³ñíóº òî÷êà x 0 ∈(a, b), â ÿê³é âðîíñê³àí<br />
ïåðåòâîðþºòüñÿ â íóëü, òîáòî W(x 0 ) = 0. Ñêëàäåìî ñèñòåìó<br />
ð³âíÿíü<br />
⎧α ⎪ 1y1( x0) +α<br />
2y2( x0)<br />
= 0<br />
⎨<br />
⎪⎩ α 1y′ 1( x0) +α 2y′<br />
2( x0)<br />
= , (11.6.16)<br />
0<br />
â ÿê³é α 1 ³ α 2 — íåâ³äîì³ ÷èñëà. Îñê³ëüêè âèçíà÷íèê ö³º¿<br />
ñèñòåìè<br />
( ) ( )<br />
( ) ( )<br />
y x y x<br />
( ) 0<br />
′ ′ ,<br />
1 0 2 0<br />
∆= = W x0<br />
=<br />
y1 x0 y2 x0<br />
òî âîíà ìຠ(äèâ. ï. 3.6.2) íåòðèâ³àëüíèé ðîçâ’ÿçîê â³äíîñíî<br />
α 1 ³ α 2 (òîáòî õî÷à á îäíå ç íèõ â³äì³ííå â³ä íóëÿ).<br />
Ðîçãëÿíåìî òåïåð ôóíêö³þ<br />
( ) ( ) ( )<br />
y x =α y x +α y x ,<br />
1 1 2 2<br />
äå α 1 ³ α 2 — íåòðèâ³àëüí³ ðîçâ’ÿçêè ñèñòåìè (11.6.16).<br />
Çà òåîðåìîþ 11.6.1 öÿ ôóíêö³ÿ º ðîçâ’ÿçêîì ð³âíÿííÿ<br />
(11.6.2). Êð³ì öüîãî, îñê³ëüêè α 1 ³ α 2 — ðîçâ’ÿçêè ñèñòåìè<br />
(11.6.2), òî ôóíêö³ÿ y(x) çã³äíî ç (11.6.16) çàäîâîëüíÿº íóëüîâ³<br />
ïî÷àòêîâ³ óìîâè<br />
( ) ′<br />
0 ( 0)<br />
y x = 0, y x = 0 . (11.6.17)<br />
ßñíî, ùî òàê³ ïî÷àòêîâ³ óìîâè çàäîâîëüíÿº ³ ôóíêö³ÿ<br />
y(x) ≡ 0. Çà òåîðåìîþ ³ñíóâàííÿ òà ºäèíîñò³, ðîçâ’ÿçîê<br />
y(x) ≡ 0 º ºäèíèì ðîçâ’ÿçêîì ð³âíÿííÿ (11.6.2) ç ïî÷àòêîâèìè<br />
óìîâàìè (11.6.17). Îòæå y ( x) y ( x)<br />
1 1 2 2<br />
0<br />
α +α ≡ íà ³íòåðâàë³<br />
(a, b), à öå îçíà÷àº, ùî ôóíêö³¿ y 1 (x) ³ y 2 (x) ë³í³éíî çàëåæí³.<br />
Îòðèìàëè ñóïåðå÷í³ñòü, ÿêà é äîâîäèòü òåîðåìó.<br />
Âñòàíîâèìî òåïåð, çà ÿêèõ óìîâ ôóíêö³ÿ<br />
y( x) = C1y1( x) + C2y2( x)<br />
º çàãàëüíèì ðîçâ’ÿçêîì ËÎÄÐ<br />
(11.6.2).<br />
Òåîðåìà 11.6.4 (ïðî ñòðóêòóðó çàãàëüíîãî ðîçâ’ÿçêó<br />
ËÎÄÐ). ßêùî ôóíêö³¿ y 1 (x) ³ y 2 (x) ë³í³éíî íåçàëåæí³ ðîçâ’ÿçêè<br />
ð³âíÿííÿ (11.6.2), òî ôóíêö³ÿ<br />
( ) ( ) ( )<br />
y x = C y x + C y x (x∈(a, b)), (11.6.18)<br />
1 1 2 2<br />
äå C 1 ³ C 2 — äîâ³ëüí³ ñòàë³, ÿâëÿº ñîáîþ çàãàëüíèé ðîçâ’ÿçîê<br />
ð³âíÿííÿ (11.6.1).<br />
Äîâåäåííÿ. Íàãàäàºìî, ùî íà ï³äñòàâ³ òåîðåìè 11.6.1<br />
ôóíêö³ÿ y = C1y1( x) + C2y2( x)<br />
ïðè áóäü-ÿêèõ çíà÷åííÿõ ñòàëèõ<br />
C 1 ³ C 2 º ðîçâ’ÿçêîì ð³âíÿííÿ (11.6.1). Òåïåð, äëÿ òîãî<br />
ùîá äîâåñòè, ùî öÿ ôóíêö³ÿ º çàãàëüíèì ðîçâ’ÿçêîì, äîñòàòíüî<br />
âñòàíîâèòè, ùî ç íüîãî ìîæíà âèä³ëèòè ÷àñòèíí³ ðîçâ’ÿçêè.<br />
Íåõàé x 0 ∈(a, b) ³<br />
( ) , ( )<br />
y x = y y′ x = y′<br />
(11.6.19)<br />
0 0 0 0<br />
äîâ³ëüí³ ïî÷àòêîâ³ óìîâè.<br />
Ïîêàæåìî, ùî ñòàë³ C 1 ³ C 2 ìîæíà ï³ä³áðàòè òàê, ùî ðîçâ’ÿçîê<br />
âèãëÿäó (11.6.18) ïðè öèõ çíà÷åííÿõ ñòàëèõ ÿâëÿº<br />
ñîáîþ ÷àñòèííèé ðîçâ’ÿçîê, ÿêèé çàäîâîëüíÿº ïî÷àòêîâ³ óìîâè<br />
(11.6.19).<br />
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