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Ðîçâ’ÿçàííÿ. Õàðàêòåðèñòè÷íå ð³âíÿííÿ ìຠêîðåí³<br />
k 1 =2i, k 2 =–2i, òîìó çàãàëüíèé ðîçâ’ÿçîê îäíîð³äíîãî ð³âíÿííÿ<br />
ìຠñòðóêòóðó<br />
y ç.o. 1 2<br />
= c cos2x + c sin 2x<br />
.<br />
×àñòèííèé æå ðîçâ’ÿçîê, ó â³äïîâ³äíîñò³ äî ôîðìóëè<br />
(11.7.9), (α = 0, k 1 = β = 2), ñë³ä øóêàòè ó âèãëÿä³<br />
y ÷.í. = x(A cos2x + B sin2x).<br />
ϳäñòàâëÿþ÷è äî ð³âíÿííÿ (11.7.12) ³ ïðèð³âíþþ÷è êîåô³ö³ºíòè<br />
ïðè cos 2x ³ sin2x îäåðæóºìî ñèñòåìó ð³âíÿíü äëÿ<br />
âèçíà÷åííÿ A ³ B: 4B = 1, –4A = 0, çâ³äêè A = 0, B = 1/4.<br />
Òàêèì ÷èíîì, çàãàëüíèé ðîçâ’ÿçîê ð³âíÿííÿ (11.7.12) âèãëÿäàº<br />
òàê:<br />
1<br />
y ç.í. = c 1<br />
cos2x + c 2<br />
sin 2x + xsin 2x<br />
.<br />
4<br />
Çàóâàæåííÿ 2. Îòðèìàí³ ðîçâ’ÿçêè ïðè x → +∞, îïèñóþ÷è<br />
êîëèâàííÿ, íåîáìåæåíî çðîñòàþòü.<br />
ÂÏÐÀÂÈ<br />
Çíàéòè çàãàëüí³ ðîçâ’ÿçêè ð³âíÿíü:<br />
x<br />
4x<br />
11.36. y′′ + y = e ;<br />
11.37. y′′ − 4y′<br />
= 4 e ;<br />
x<br />
11.38 y′′ + y = sin5 x;<br />
11.39. y′′ + y′<br />
= xe ;<br />
x<br />
11.40. y′′ + 7y′<br />
+ 20 y = e ; 11.41. y′′ + 9y = cos3 x;<br />
3x<br />
11.42. y′′ − 6y′<br />
+ 9 y = e ; 11.43. y′′ + 3y′<br />
+ y = 3cos2 x;<br />
11.44. y′′ + y′<br />
= xsin x;<br />
11.45. y′′ + y = cos x + sin5 x.<br />
Âêàç³âêà. Ó âïðàâ³ 11.44 íåîáõ³äíî øóêàòè äâà âèäè<br />
÷àñòèííèõ ðîçâ’ÿçê³â.<br />
11.8. ÄÈÔÅÐÅÍÖ²ÀËÜͲ вÂÍßÍÍß ÂÈÙÈÕ<br />
ÏÎÐßÄʲÂ, ÙÎ ÄÎÏÓÑÊÀÞÒÜ<br />
ÇÍÈÆÅÍÍß ÏÎÐßÄÊÓ<br />
11.8.1. гâíÿííÿ n-ãî ïîðÿäêó âèäó y (n) = f(x)<br />
Âîíè ðîçâ’ÿçóþòüñÿ ïîñë³äîâíèì ³íòåãðóâàííÿì. À ñàìå:<br />
ïîìíîæóþ÷è îáèäâ³ ÷àñòè ð³âíÿííÿ íà dx ³ ³íòåãðóþ÷è,<br />
îäåðæóºìî ð³âíÿííÿ (n – 1)-ãî ïîðÿäêó:<br />
( n−1)<br />
( ) ( )<br />
y = ∫ f x dx + c1 = ϕ<br />
1<br />
x + c1.<br />
Ïîò³ì, ïîâòîðþþ÷è òó ñàìó ïðîöåäóðó, îäåðæóºìî ð³âíÿííÿ<br />
(n – 2)-ãî ïîðÿäêó:<br />
( n−2)<br />
( ) ( )<br />
y = ϕ x dx + c dx + c = ϕ x + c x + c<br />
∫ 1 ∫ 1 2 2 1 2<br />
³ ò.ä.<br />
ϳñëÿ n-êðàòíîãî ³íòåãðóâàííÿ îäåðæóºìî ³íòåãðàë y<br />
öüîãî ð³âíÿííÿ ó âèãëÿä³ ÿâíî¿ ôóíêö³¿ â³ä x ³ n äîâ³ëüíèõ<br />
ñòàëèõ:<br />
n<br />
n−1 n−2<br />
( )<br />
y =ϕ x + c x + c x + K + c .<br />
1 1<br />
11.8.2. гâíÿííÿ äðóãîãî ïîðÿäêó:<br />
à) f( x, y′ , y′′ ) = 0 ³ á) ( y, y′ , y′′ ) = 0, ùî íå ì³ñòÿòü ÿâíî ôóíêö³¿<br />
y àáî àðãóìåíòó x, ïåðåòâîðÿòüñÿ â ð³âíÿííÿ 1-ãî ïîðÿäêó<br />
çà äîïîìîãîþ ï³äñòàíîâêè y′ = p (çâ³äêè y′′ = dp/<br />
dx —<br />
dp<br />
äëÿ ð³âíÿííÿ à) àáî y′′ = p dy<br />
— äëÿ ð³âíÿííÿ á).<br />
2<br />
Ïðèêëàä 11.8.1 Ðîçâ’ÿçàòè ð³âíÿííÿ y′′′ = 60x<br />
.<br />
Ð î ç â ’ ÿ ç à í í ÿ. Ïîìíîæóþ÷è îáèäâ³ ÷àñòè ð³âíÿííÿ íà<br />
dx ³ ïîò³ì ³íòåãðóþ÷è, îäåðæóºìî ð³âíÿííÿ 2-ãî ïîðÿäêó:<br />
2 3<br />
ydx ′′′ = 60 xdx, y′′<br />
= 20x + c.<br />
1<br />
Äàë³ çà òèì ñàìèì ñïîñîáîì îäåðæóºìî ð³âíÿííÿ ïåðøîãî<br />
ïîðÿäêó ³ ïîò³ì øóêàíó ôóíêö³þ — çàãàëüíèé ³íòåãðàë<br />
ð³âíÿííÿ:<br />
n<br />
( )<br />
ydx ′′ = 20 xdx+ cdx, y′ = 5 x + cx+ c, ydx ′ = 5 x + cx+<br />
c dx,<br />
3 4 4<br />
1 1 2 1 2<br />
442 443
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