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2x<br />
Çàãàëüíèé ðîçâ’ÿçîê ð³âíÿííÿ y e ( c xc )<br />
= + .<br />
1 2<br />
Äèôåðåíö³àëüíå y′′ + py′<br />
+ qy = 0<br />
ð³âíÿííÿ<br />
2<br />
Õàðàêòåðèñòè÷íå k + pk+ q = 0<br />
ð³âíÿííÿ<br />
Êîðåí³ õàðàêòåðè- k1 ≠ k2<br />
k1 = k2<br />
= k k1<br />
ñòè÷íîãî ð³âíÿííÿ k2<br />
Ôóíäàìåíòàëüíà kx 1<br />
e<br />
(ËÍÇ) ñèñòåìà ÷àñ-<br />
x<br />
e k 2<br />
òèííèõ ðîçâ’ÿçê³â<br />
Âèä çàãàëüíîãî<br />
kx 1<br />
y = c1e<br />
+<br />
ðîçâ’ÿçêó<br />
kx 2<br />
ce<br />
2<br />
y<br />
kx<br />
e<br />
kx<br />
xe<br />
kx<br />
= ×<br />
+ ( c1 c2x)<br />
e<br />
e<br />
e<br />
αx<br />
αx<br />
=α+βi<br />
=α−βi<br />
cosβx<br />
sin βx<br />
y = e αx ×<br />
× + × ( c 1<br />
cos β x +<br />
+ c2 sin βx<br />
Ïðèêëàä 11.7.3. Çíàéòè çàãàëüíèé ðîçâ’ÿçîê ð³âíÿííÿ<br />
y′′ + 4y′<br />
+ 13y<br />
= 0.<br />
Ð î ç â ’ ÿ ç à í í ÿ. Õàðàêòåðèñòè÷íå ð³âíÿííÿ<br />
2<br />
k + 4k<br />
+ 13 = 0 ìຠêîìïëåêñíî ñïðÿæåí³ (äèâ. äîä. 1) êîðåí³<br />
k1 =− 2+ 3 i, k2<br />
=−2− 3i<br />
. Ôóíäàìåíòàëüíà (ËÍÇ) ñèñòåìà<br />
÷àñòèííèõ ðîçâ’ÿçê³â<br />
y = e cos3 x, y = e sin 3x.<br />
−2x<br />
−2x<br />
1 2<br />
Çàãàëüíèé ðîçâ’ÿçîê ð³âíÿííÿ òàêèé:<br />
ÂÏÐÀÂÈ<br />
= ( cos3 + sin 3 ) .<br />
−2x<br />
y e c1 x c2<br />
x<br />
Çíàéòè çàãàëüí³ ðîçâ’ÿçêè ð³âíÿíü:<br />
11.28 y′′ − 4y′<br />
+ 3y<br />
= 0; 11.29. y′′ + 5y′<br />
+ 6y<br />
= 0;<br />
)<br />
11.30. y′′ − 2y′<br />
+ y = 0; 11.31. y′′ − 10y′<br />
+ 25y<br />
= 0;<br />
11.32 y′′ − 6y′<br />
+ 9y<br />
= 0; 11.33. y′′ − 2y′<br />
+ 5y<br />
= 0;<br />
2<br />
11.34. y′′ +ω y = 0, ω> 0 ; 11.35. y′′ + y = 0 .<br />
11.7.2. Íåîäíîð³äí³ ð³âíÿííÿ<br />
Çàãàëüíèé âèä íåîäíîð³äíîãî ð³âíÿííÿ:<br />
y′′ + py′<br />
+ qy = f( x)<br />
(x∈R), (11.7.3)<br />
äå p ³ q — ñòàë³; f(x) — â³äîìà íåïåðåðâíà ôóíêö³ÿ.<br />
Íåâàæêî ïîêàçàòè, ùî çàãàëüíèé ðîçâ’ÿçîê ð³âíÿííÿ<br />
(11.7.3) ÿâëÿº ñîáîþ ñóìó ÷àñòèííîãî ðîçâ’ÿçêó íåîäíîð³äíîãî<br />
ð³âíÿííÿ ³ çàãàëüíîãî ðîçâ’ÿçêó â³äïîâ³äíîãî îäíîð³äíîãî<br />
ð³âíÿííÿ (÷èòà÷åâ³ ðåêîìåíäóºòüñÿ öå çä³éñíèòè). ßê<br />
çíàõîäèòè çàãàëüíèé ðîçâ’ÿçîê îäíîð³äíîãî ð³âíÿííÿ áóëî<br />
ïîêàçàíî âèùå. Äëÿ çíàõîäæåííÿ ÷àñòèííîãî ðîçâ’ÿçêó íåîäíîð³äíîãî<br />
ð³âíÿííÿ (11.7.3) ìîæíà çàñòîñóâàòè ìåòîä âàð³àö³¿<br />
äîâ³ëüíèõ ñòàëèõ. Öåé ìåòîä, âçàãàë³ êàæó÷è, çàñòîñîâóºòüñÿ<br />
äî áóäü-ÿêî¿ íåïåðåðâíî¿ ôóíêö³¿ f(x). Îäíàê äëÿ<br />
ð³âíÿíü ç³ ñòàëèìè êîåô³ö³ºíòàìè, ïðàâ³ ÷àñòèíè ÿêèõ ìàþòü<br />
ñïåö³àëüíèé âèãëÿä, ³ñíóº á³ëüø ïðîñòèé ñïîñ³á çíàõîäæåííÿ<br />
÷àñòèííîãî ðîçâ’ÿçêó.<br />
Çàçíà÷èìî ôîðìó, â ÿê³é ñë³ä øóêàòè ÷àñòèííèé ðîçâ’ÿçîê<br />
â çàëåæíîñò³ â³ä âèãëÿäó ïðàâî¿ ÷àñòèíè f(x) äèôåðåíö³àëüíîãî<br />
ð³âíÿííÿ.<br />
αx<br />
I. Ïðàâà ÷àñòèíà ð³âíÿííÿ fx ( ) = e ⋅ Pn<br />
( x)<br />
. Òóò P n (x) —<br />
ìíîãî÷ëåí ñòåïåí³ n, à êîåô³ö³ºíò α â ïîêàçíèêó — ä³éñíå<br />
÷èñëî.  öüîìó âèïàäêó ÷àñòèííèé ðîçâ’ÿçîê íåîäíîð³äíîãî<br />
ð³âíÿííÿ ñë³ä øóêàòè ó âèãëÿä³<br />
αx<br />
r<br />
yr . .<br />
= Q ( )<br />
H n<br />
x e x , (11.7.4)<br />
äå Q n (x) — ìíîãî÷ëåí òîãî ñàìîãî ñòåïåíÿ, ùî ³ ìíîãî÷ëåí<br />
P n (x), àëå ç íåâ³äîìèìè ïîêè ùî êîåô³ö³ºíòàìè, à r — ÷èñëî<br />
êîðåí³â õàðàêòåðèñòè÷íîãî ð³âíÿííÿ, ùî çá³ãàþòüñÿ ç êîåô³ö³ºíòîì<br />
α.<br />
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