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11.6.2. ˳í³éí³ îäíîð³äí³ äèôåðåíö³àëüí³ ð³âíÿííÿ<br />
äðóãîãî ïîðÿäêó<br />
Ñïî÷àòêó â³äçíà÷èìî îäíó âàæëèâó âëàñòèâ³ñòü ë³í³éíèõ<br />
îäíîð³äíèõ äèôåðåíö³àëüíèõ ð³âíÿíü (ËÎÄÐ) ó âèãëÿä³ òåîðåìè.<br />
Òåîðåìà 11.6.1. Íåõàé ôóíêö³¿ y 1 (x) ³ y 2 (x) — ðîçâ’ÿçêè<br />
ËÎÄÐ<br />
Òîä³ ôóíêö³ÿ<br />
( ) ( ) 0<br />
y′′ + p x y′<br />
+ q x y = (x ∈ (a, b)). (11.6.2)<br />
( ) ( )<br />
y = C y x + C y x<br />
(11.6.3)<br />
1 1 2 2<br />
ïðè áóäü-ÿêèõ çíà÷åííÿõ ñòàëèõ C 1 òà C 2 òàêîæ º ðîçâ’ÿçêîì<br />
ð³âíÿííÿ (11.6.2).<br />
Ä î â å ä å í í ÿ. Ç ð³âíîñò³ (11.6.2) ïîñë³äîâíî çíàõîäèìî:<br />
( ) ( )<br />
y′ = C y′ x + C y′<br />
x , (11.6.4)<br />
1 1 2 2<br />
( ) ( )<br />
y′′ = C y′′ x + C y′′<br />
x<br />
(11.3.5)<br />
1 1 2 2<br />
³ ï³äñòàâëÿºìî ¿õ â ë³âó ÷àñòèíó ð³âíÿííÿ (11.6.2). Çàâäÿêè<br />
ë³í³éí³é ñòðóêòóð³ âèðàç³â (11.6.3) – (11.6.5) â³äíîñíî yi<br />
( x ),<br />
y′ ( x)<br />
, y′′ ( x)<br />
, i = 1,2 é âëàñòèâîñòÿì ïîõ³äíèõ îòðèìàºìî,<br />
i<br />
ùî<br />
i<br />
( ) ( ) ( ) ( ) ( )<br />
C1 ⎡⎣y′′ 1<br />
x + p x y′<br />
1<br />
x + q x y1<br />
x ⎤⎦<br />
+<br />
( ) ( ) ( ) ( ) ( )<br />
+ C2 ⎡⎣y′′ 2<br />
x + p x y′<br />
2<br />
x + q x y2<br />
x ⎤⎦<br />
≡<br />
≡ C ⋅ 0+ C ⋅0 ≡ 0. (11.6.6)<br />
1 2<br />
Ïðè äîâåäåíí³ òîòîæíîñò³ (11.6.6) ìè âðàõóâàëè, ùî y 1 (x)<br />
³ y 2 (x) º ðîçâ’ÿçêè ð³âíÿííÿ (11.6.2). Òàêèì ÷èíîì, ôóíêö³ÿ<br />
y, ÿêà ïîáóäîâàíà çà ôîðìóëîþ (11.6.3), º ðîçâ’ÿçêîì ð³âíÿííÿ<br />
(11.6.2).<br />
Òåîðåìó äîâåäåíî.<br />
Ôóíêö³ÿ âèäó y C y ( x) C y ( x)<br />
= + ç äîâ³ëüíèìè ñòàëèìè<br />
1 1 2 2<br />
º ðîçâ’ÿçêîì ð³âíÿííÿ (11.6.2). Ïðèðîäíî âèïëèâຠïèòàííÿ,<br />
÷è º öåé ðîçâ’ÿçîê çàãàëüíèì ðîçâ’ÿçêîì ð³âíÿííÿ (11.3.2).<br />
Äëÿ òîãî ùîá â³äïîâ³ñòè íà òàêå çàïèòàííÿ, ðîçãëÿíåìî â<br />
íàñòóïíîìó ïóíêò³ òàê³ ïîíÿòòÿ, ÿê ë³í³éíà çàëåæí³ñòü òà<br />
íåçàëåæí³ñòü ôóíêö³é.<br />
11.6.3. ˳í³éíî çàëåæí³ òà íåçàëåæí³ ôóíêö³¿<br />
íà ³íòåðâàë³ (a, b)<br />
Îçíà÷åííÿ 11.6.1. Ôóíêö³¿ y i (x), i = 1,2 íàçèâàþòüñÿ ë³í³éíî<br />
íåçàëåæíèìè (ËÍÇ) íà ³íòåðâàë³ (a, b), ÿêùî òîòîæí³ñòü<br />
( ) ( )<br />
α<br />
1y1 x + α2y2 x ≡ 0, α = const, i = 1,2 (11.6.7)<br />
i<br />
âèêîíóºòüñÿ ò³ëüêè ïðè α<br />
i<br />
= 0 , i = 1, 2 ; ó ïðîòèâíîìó ðàç³<br />
âîíè íàçèâàþòüñÿ ë³í³éíî çàëåæíèìè.<br />
Çàóâàæåííÿ. Íå âèêëþ÷àºòüñÿ âèïàäîê, êîëè a =–∞, a<br />
b = ∞.<br />
Íàâåäåìî íèçêó ïðèêëàä³â.<br />
Ïðèêëàä 11.6.1. Äîâåñòè, ùî ôóíêö³¿<br />
kx<br />
( ) ( ) ( )<br />
1 kx<br />
= , = 2<br />
, ∈ , ≠<br />
y x e y x e x R k k (11.6.8)<br />
1 2 1 2<br />
ë³í³éíî íåçàëåæí³.<br />
Äîâåäåííÿ áóäåìî ïðîâîäèòè ìåòîäîì â³ä ñóïðîòèâíîãî.<br />
Ïðèïóñòèìî, ùî ìຠì³ñöå òîòîæí³ñòü (11.6.7) çà óìîâè,<br />
ùî õî÷à á îäíå ³ç ÷èñåë α 1 , α 2 íå äîð³âíþº íóëþ. Íàïðèêëàä,<br />
α 2 ≠ 0. Òîä³ òîòîæí³ñòü (11.6.7) ç óðàõóâàííÿì (11.6.8)<br />
ìîæíà ïåðåïèñàòè òàê:<br />
( k k ) x<br />
e − α<br />
≡− α<br />
. (11.6.9)<br />
2 1 1<br />
Îòðèìàëè ñóïåðå÷í³ñòü, îñê³ëüêè ë³âà ÷àñòèíà òîòîæíîñò³<br />
(11.6.9) çì³íþºòüñÿ, à ïðàâà ÷àñòèíà º êîíñòàíòà. Îòæå,<br />
α 2 = 0. Àíàëîã³÷íî äîâîäèòüñÿ, ùî ³ α 1 = 0. Öå îçíà÷àº, ùî<br />
ôóíêö³¿ (11.6.8) ë³í³éíî íåçàëåæí³.<br />
2<br />
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