ЛЕКЦІЇ ² ВПРАВИ

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