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Ó â³äïîâ³äíîñò³ äî ïî÷àòêîâèõ óìîâ (11.6.19) ñêëàäåìî<br />
ñèñòåìó<br />
( ) ( )<br />
( ) ( )<br />
⎧ ⎪Cy 1 1<br />
x0 + Cy<br />
2 2<br />
x0 = y0<br />
⎨<br />
Cy′ 1 1<br />
x0 Cy′ 2 2<br />
x0 y′<br />
, (11.6.20)<br />
⎪⎩ + =<br />
0<br />
â ÿê³é C 1 ³ C 2 — íåâ³äîì³ ÷èñëà.<br />
Íåâàæêî ïîáà÷èòè, ùî âèçíà÷íèê ö³º¿ ñèñòåìè º âèçíà÷íèêîì<br />
Âðîíñüêîãî. Îñê³ëüêè çà óìîâîþ òåîðåìè ôóíêö³¿<br />
y 1 (x) ³ y 2 (x) — ë³í³éíî íåçàëåæí³ íà (a, b), òî çàâäÿêè òåîðåì³<br />
11.6.3 W(x 0 ) ≠ 0. Òîìó ñèñòåìà (11.6.20) ìຠºäèíèé ðîçâ’ÿçîê,<br />
ÿêèé ìè ïîçíà÷èìî òàê: C = C , C = C .<br />
0<br />
0<br />
ϳäñòàâëÿþ÷è<br />
0<br />
1<br />
1 1<br />
2 2<br />
0<br />
C ³ C â ð³âí³ñòü (11.6.18), îòðèìàºìî øóêàíèé ÷àñòèííèé<br />
2<br />
ðîçâ’ÿçîê ð³âíÿííÿ (11.3.1): y( x) C 0 y ( x) C 0 y ( x)<br />
= + , ÿêèé<br />
1 1 2 2<br />
çàäîâîëüíÿº ïî÷àòêîâ³ óìîâè (11.6.19). Öå ³ îçíà÷àº, ùî ðîçâ’ÿçîê<br />
(11.6.18) º çàãàëüíèì ðîçâ’ÿçêîì ð³âíÿííÿ (11.6.2).<br />
11.7. ˲ͲÉͲ ÄÈÔÅÐÅÍÖ²ÀËÜͲ<br />
вÂÍßÍÍß ÄÐÓÃÎÃÎ ÏÎÐßÄÊÓ<br />
Dz ÑÒÀËÈÌÈ ÊÎÅÔ²Ö²ªÍÒÀÌÈ<br />
11.7.1. Îäíîð³äí³ ð³âíÿííÿ<br />
Çàãàëüíèé âèãëÿä îäíîð³äíîãî ð³âíÿííÿ:<br />
y′′ + py′<br />
+ qy = 0 (x∈R), (11.7.1)<br />
äå p ³ q — ñòàë³.<br />
Çàãàëüíèé ðîçâ’ÿçîê äèôåðåíö³àëüíîãî ð³âíÿííÿ (11.7.1)<br />
ïîâ’ÿçàíèé ç ðîçâ’ÿçàííÿì õàðàêòåðèñòè÷íîãî ð³âíÿííÿ<br />
2<br />
k pk q<br />
+ + = 0 . (11.7.2)<br />
kx<br />
= , äå k — ñòàëà, ùî ï³äëÿãຠâè-<br />
ijéñíî, ïîêëàäåìî y e<br />
kx<br />
çíà÷åííþ. ϳäñòàâèìî y = e äî ð³âíÿííÿ (11.7.1). Âðàõîâóþ÷è,<br />
ùî y′ = ke , y′′ = k e îäåðæèìî, ùî<br />
kx<br />
2 kx<br />
∀x∈R:<br />
2<br />
( )<br />
kx<br />
e k + pk + q = 0 .<br />
kx<br />
kx<br />
Îñê³ëüêè e ≠ 0 ∀x∈R , òî ôóíêö³ÿ y = e áóäå ðîçâ’ÿçêîì<br />
ð³âíÿííÿ (11.7.1) ò³ëüêè ïðè çä³éñíåíí³ âèìîãè<br />
(11.7.2).  çàëåæíîñò³ â³ä âèäó êîðåí³â õàðàêòåðèñòè÷íîãî<br />
ð³âíÿííÿ (11.7.2) áóäóþòüñÿ ð³çí³ âèäè çàãàëüíîãî ðîçâ’ÿçêó<br />
ð³âíÿííÿ (11.7.1), àëå ñòðóêòóðà çàãàëüíîãî ðîçâ’ÿçêó<br />
ð³âíÿííÿ (11.7.1) îäíà é òà ñàìà:<br />
( ) ( )<br />
y = c y x + c y x ,<br />
1 1 2 2<br />
äå y 1 (x) ³ y 2 (x) ë³í³éíî íåçàëåæí³ (ËÍÇ) ðîçâ’ÿçêè ð³âíÿííÿ<br />
(11.7.1), à c 1 ³ c 2 — äîâ³ëüí³ ñòàë³.<br />
Ïîøóê ËÍÇ ðîçâ’ÿçê³â ð³âíÿííÿ (11.7.1) ïîâ’ÿçàíèé ç<br />
ìîæëèâèìè âàð³àíòàìè ðîçâ’ÿçêó õàðàêòåðèñòè÷íîãî ð³âíÿííÿ<br />
(11.7.2). Ö³ âàð³àíòè íàäàþòüñÿ ó íèæ÷å íàâåäåí³é<br />
òàáëèö³.<br />
Ïðèêëàä 11.7.1. Çíàéòè çàãàëüíèé ðîçâ’ÿçîê ð³âíÿííÿ<br />
y′′ − 5y′<br />
+ 6y<br />
= 0.<br />
Ðîçâ’ÿçàííÿ. Õàðàêòåðèñòè÷íå ð³âíÿííÿ, ÿêå â³äïîâ³äíå<br />
äàíîìó äèôåðåíö³àëüíîìó ð³âíÿííþ, ìຠâèãëÿä:<br />
2<br />
k − 5k<br />
+ 6 = 0.<br />
Éîãî êîðåí³ k 1 = 2, k 2 = 3. Ôóíäàìåíòàëüíà (ËÍÇ) ñèñòåìà<br />
2x<br />
3x<br />
÷àñòèííèõ ðîçâ’ÿçê³â: y1<br />
= e , y2<br />
= e . Çàãàëüíèé ðîçâ’ÿçîê<br />
ìຠâèãëÿä:<br />
y = c e + c e .<br />
2x<br />
3x<br />
1 1 2<br />
Ïðèêëàä 11.7.2. Çíàéòè çàãàëüíèé ðîçâ’ÿçîê ð³âíÿííÿ<br />
y′′ − 4y′<br />
+ 4y<br />
= 0.<br />
Ð î ç â ’ ÿ ç à í í ÿ. Õàðàêòåðèñòè÷íå ð³âíÿííÿ<br />
2<br />
k − 4k<br />
+ 4 = 0<br />
ìຠð³âí³ êîðåí³ k 1 = k 2 = 2. Ôóíäàìåíòàëüíà (ËÍÇ) ñèñòåìà<br />
÷àñòèííèõ ðîçâ’ÿçê³â:<br />
y = e , y = xe .<br />
2x<br />
2x<br />
1 2<br />
434 435
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