Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.students.chemport.ru/materials/matan/makaroff_3sem_2004.doc Äàòà èçìåíåíèÿ: Thu Jan 15 18:04:37 2009 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 22:45:47 2012 Êîäèðîâêà: koi8-r

Ðÿäû. Äèôôåðåíöèàëüíûå óðàâíåíèÿ.



À=à1+à2+à3+.= [pic]


Îïðåäåëåíèå: ×èñëîâîé ðÿä - áåñêîíå÷íàÿ óïîðÿäî÷åííàÿ ñóììà ÷èñåë.

Ïðèìåðû ðÿäîâ:

[pic] Ãàðìîíè÷åñêèé ðÿä.

[pic] Äçåòà ôóíêöèÿ Ðèììàíà.

1-1+1-1+1-1+1-1+.

Àn=à1+à2+à3+.+àn - ÷àñòè÷íàÿ ñóììà ðÿäà.
{An}-ïîñëåäîâàòåëüíîñòü ÷àñòè÷íûõ ñóìì.

Îïðåäåëåíèå: ×èñëîâîé ðÿä À ñõîäèòñÿ, åñëè [pic]- ñóììà ñõîäÿùåãîñÿ
÷èñëîâîãî ðÿäÿ. Åñëè[pic], òî ðÿä À ðàñõîäèòñÿ.
1+2+3+4+5+. (ðàñõîäèòñÿ ê áåñêîíå÷íîñòè)

 öåëîì ïî ðÿäàì ñóùåñòâóåò íåñêîëêî òèïîâ çàäà÷:
1) Èññëåäîâàíèå ñõîäèìîñòè ðÿäà.
2) Íàõîæäåíèå ñóììû.

Êðèòåðèé Êîøè ñõîäèìîñòè ðÿäà.
[pic]
( (>0 ( n0 , òàêîå, ÷òî (n>m(n0: |An-Am|<(.
Òîãäà ãîâîðÿò, ÷òî ïîñëåäîâàòåëüíîñòü An - ôóíäàìåíòàëüíà.
An=a1+a2+.+an
Am=a1+a2+.+am, ñëåäîâàòåëüíî, An-Am=am+1+.+an
( (>0 ( n0, òàêîå ÷òî (n>m(n0 => | am+1+.+an |<(.

Ïðèìåð:
[pic] Ãàðìîíè÷åñêèé ðÿä.

Çàôèêñèðóåì (=0.5, m(n0, n=2m

| am+1+.+an |= [pic] =>ðÿä ðàñõîäèòñÿ.
Âñåãî m ñëàãàåìûõ

[pic] - íåîáõîäèìûé ïðèçíàê ñõîäèìîñòè ÷èñëîâîãî ðÿäà.
Äîêàçàòåëüñòâî: n=m+1 ((>0, ( n0 => (n(n0 => |an|<( => [pic]

Ñëåäñòâèå 1


À= [pic] Â=[pic]


Åñëè ( n1:( n(n1 => an=bn, òîãäà A~B (ëèáî îáà ñõîäÿùèåñÿ ëèáî îáà
ðàñõîäÿùèåñÿ)
Äîêàçàòåëüñòâî: n0(n1 | am+1+.+an |=| bm+1+.+bn |

Ñëåäñòâèå 2


A= [pic] B=[pic], Åñëè bn = kan, n(1, k(0, òîãäà A~B.

Åñëè [pic] ñõîäèòñÿ, òî [pic]ñõîäèòñÿ.
Äîêàçàòåëüñòâî: Ïî êðèòåðèþ Êîøè:
| am+1+.+an |˜|am+1|+|am+2|+.+|an|<(

Äîñòàòî÷íûå ïðèçíàêè ñõîäèìîñòè çíàêîïîñòîÿííûõ ðÿäîâ.

Ïðèçíàêè ñðàâíåíèÿ.

1) A= [pic], B=[pic]
[pic]
Äëÿ äîêàçàòåëüñòâà ïðèìåíèì êðèòåðèé Êîøè:
| am+1+.+an | = am+1+am+2+.+an˜bm+1+bm+2+.+bn<(

2) ïðåäåëüíûé
[pic]
Äîêàçàòåëüñòâî: èç ñóùåñòâîâàíèÿ ïðåäåë [pic]ñëåäóþò íåðàâåíñòâà:
[pic]
[pic] òîãäà ïî ïðèçíàêó ñðàâíåíèÿ (1) ðÿä ñõîäèòñÿ.

Ïðèìåð.
((?)= [pic], ðàññìîòðèì êàê âåä¸ò ñåáÿ ýòîò ðÿä â çàâèñèìîñòè îò ?. Ïðè ?=1
ðÿä ðàñõîäèòñÿ (ãàðìîíè÷åñêèé). Áûëî äîêàçàíî ðàíåå.
Äëÿ ?<1 =>[pic] ðàñõîäèòñÿ ïî ïðèçíàêó ñðàâíåíèÿ 1.
Äëÿ ?>1
[pic] òîãäà ïî òåîðåìå î ñðåäíåì âûïîëíÿþòñÿ íåðàâåíñòâà
[pic]
[pic]
[pic][pic]
[pic], ò.å.  - ñõîäèòñÿ, çíà÷èò ïî ïðèçíàêó ñðàâíåíèÿ ((?) ïðè ?>1 òî æå
ñõîäèòñÿ.
Òàêèì îáðàçîì
((?) = [pic]

Ïðèçíàê Äàëàìáåðà.
Ïóñòü [pic], òîãäà
(<1 =>A ñõîäèòñÿ
(>1 =>A ðàñõîäèòñÿ
(=1 =>âîïðîñ î ñõîäèìîñòè îñòà¸òñÿ îòêðûòûì


Äîêàçàòåëüñòâî:

1) (<1, (-(<1, [pic]

Ïåðåìíîæàÿ âñå ýòè íåðàâåíñòâà, ïîëó÷èì: [pic]çíà÷èò
ann0
ò.ê. (+(<1 òî ðÿä ñõîäèòñÿ (ïî ïðèçíàêó ñðàâíåíèÿ 1).
2) (>1 => an+1>an=> an íå ñòðåìèòñÿ ê íóëþ => ðÿä ðàñõîäèòñÿ, ò.ê. íå
âûïîëíÿåòñÿ íåîáõîäèìûé ïðèçíàê ñõîäèìîñòè ðÿäà.

Ïðèçíàê Êîøè (ðàäèêàëüíûé).
Ïóñòü À= [pic], an>0
è ([pic]òîãäà ïðè (<1 À ñõîäèòñÿ, (>1 À ðàñõîäèòñÿ, ïðè (=1 âîïðîñ î
ñõîäèìîñòè ðÿäà îñòà¸òñÿ îòêðûòûì.

Äîêàçàòåëüñòâî:

1)(<1
Âûáåðåì ( : (+(<1 òîãäà èç îïðåäåëåíèÿ ïðåäåëà:
[pic], çíà÷èò an<((+()n, n(n0 ïîëó÷åíà ãåîìåòðè÷åñêàÿ ïðîãðåññèÿ ñ q<1,
ñëåäîâàòåëüíî, ðÿä ñõîäèòñÿ (q=(+().
2)(>1
[pic]
an>1, n (n0 çíà÷èò, [pic], ñëåäîâàòåëüíî, íå âûïîëíåí íåîáõîäèìûé ïðèçíàê
ñõîäèìîñòè ÷èñëîâîãî ðÿäà.
Ïðèçíàê Êîøè áîëåå îáùèé, ÷åì ïðèçíàê Äàëàìáåðà, îäíàêî ïðèìåíÿòü åãî
ñëîæíåå.
Ïðèìåð:
[pic] ðÿä ñõîäèòñÿ (ïî Äàëàìáåðó)

Ïðèçíàê ñðàâíåíèÿ 3.

Ïóñòü À= [pic], Â=[pic], an>0, bn>0.
[pic]
òîãäà, åñëè ðÿä B ñõîäèòñÿ, òî è ðÿä A ñõîäèòñÿ.

Äîêàçàòåëüñòâî:

Ïîñëå ïî÷ëåííîãî ïåðåìíîæåíèÿ
ïîëó÷èì:


òàê êàê
a1/b1=const, è B ñõîäèòñÿ,
òî è À
ñõîäèòñÿ.







Ïðèçíàê Êóììåðà.

Ïóñòü [pic], an>0 (( n(n0), è {bn} ïîñëåäîâ-òü ÷èñåë, bï>0 è

[pic], è ([pic], òîãäà, åñëè ?>0, òî ðÿä ñõîäèòñÿ, åñëè ?<0, òî ðÿä
ðàñõîäèòñÿ, åñëè ?=0, òî âîïðîñ î ñõîäèìîñòè ðÿäà îñòà¸òñÿ îòêðûòûì.
Äîêàçàòåëüñòâî:
1)?>0, âûáåðåì (=?/2 òîãäà, ïî îïðåäåëåíèþ ïðåäåëà,
bn*an/an+1-bn+1>?-(=?/2 (n(n0
îáîçíà÷èì cn=anbn-an+1bn+1>?*an+1/2
è äîêàæåì ñõîäèìîñòü ðÿäà [pic], òàê êàê cn=anbn-an+1bn+1= ?*an+1/2>0, òî
anbn>an+1bn+1[pic], çíà÷èò, {anbn} ìîíîòîííî óáûâàþùàÿ, îãðàíè÷åííàÿ íóë¸ì
ïîñëåäîâàòåëüíîñòü.
Sn = c1+.cn = (a1b1-a2b2)+(a2b2-a3b3)+.+(anbn-an+1bn+1) =
= a1b1-an+1bn+1[pic]
çíà÷èò, ðÿä [pic] ñõîäèòñÿ, [pic] òîæå ñõîäèòñÿ (ïî ïðèçíàêó
ñðàâíåíèÿ 1), ò.ê. ?/2=const, òî è èñõîäíûé ðÿä ñõîäèòñÿ => A ñõîäèòñÿ.
2) ?<0 òîãäà,
bn*an/an+1-bn+1<0 => [pic], [pic] (n(n0, çíà÷èò
ïî óñëîâèþ ðÿä 1/bn - ðàñõîäèòñÿ, à çíà÷èò ïî ïðèçíàêó ñðàâíåíèÿ 3
ðàñõîäèòñÿ è èññëåäóåìûé ðÿä A.

Ñëåäñòâèå 1 (ïðèçíàê Äàëàìáåðà).
Âîçüì¸ì bn=1, òîãäà
[pic] [pic] [pic]
Åñëè ?>0~(<1
?<0~(>1




Ñëåäñòâèå 2 (ïðèçíàê Ðààáå)
Åñëè [pic], òî [pic]
Ïóñòü bn=n
[pic] , [pic] , [pic]
îáîçíà÷èì ?=?+1, òîãäà [pic], çíà÷èò ïðè ?>1 (?>0) ðÿä ñõîäèòñÿ, ïðè ?<1
(?<0) ðàñõîäèòñÿ.

Ñëåäñòâèå 3.
[pic] => A - ðàñõîäèòñÿ.
Ïðèìåíèì ê èññëåäóåìîìó ðÿäó ïðèçíàê Êóììåðà (bn=n*ln(n) (äîêàçàòåëüñòâî
ðàñõîäèìîñòè äàííîãî ðÿäà ñì. íèæå)). Òîãäà
bnan/an+1-bn+1=n*ln(n)*(1+1/n+(n)-(n+1)ln(n+1)=
(n+1)(ln(n)-ln(n+1))+n*ln(n)*(n= -n*ln(1+1/n)-ln(1+1/n)+n*ln(n)*(n[pic]-1,
ò.ê. ïåðâîå èç ñëàãàåìûõ ñòðåìèòñÿ (ïðè n ñòðåìÿùåìñÿ ê áåñêîíå÷íîñòè) ê
-1, âòîðîå ê 0, è 3 ê íóëþ (ò.ê.ln(n)/n-> ê 0).
Èç äîêàçàííûõ âûøå ïðèçíàêîâ Äàëàìáåðà, Ðààáå è ñëåäñòâèÿ 3 ïîëó÷àåì:

Ïðèçíàê Ãàóñà. (áåç äîêàçàòåëüñòâà)
Ïóñòü [pic]
Òîãäà,
1) ?<1 A - ðàñõîäèòñÿ.
2) ?>1 A - ñõîäèòñÿ.
3) ?=1 ?˜1 A - ðàñõîäèòñÿ.
?>1 A - ñõîäèòñÿ. Äîêàçàòåëüñòâî ñëåäóåò èç ñëåäñòâèé 1- 3.

Ïðèìåð:
[pic]

1-?>1 ~ ?<0, A - ñõîäèòñÿ.
1-?<1 ~ ?>0, A - ðàñõîäèòñÿ.
Ïðè ? = 0 ðÿä ñîñòîèò òîëüêî èç íóëåâûõ ñëàãàåìûõ, à ñëåäîâàòåëüíî
ñõîäèòñÿ.

Èíòåãðàëüíûé ïðèçíàê. (Êîøè-Ìàêëîðåíà)
Ïóñòü [pic] - íåïðåðûâíàÿ, íåîòðèöàòåëüíàÿ, ìîíîòîííî óáûâàþùàÿ ôóíêöèÿ,
îïðåäåëåííàÿ ïðè [pic](íà÷èíàÿ ñ íåêîòîðîãî x). Òîãäà ðÿä [pic]~[pic]
Äîêàçàòåëüñòâî:
Ëåììà. Ïóñòü An=a1+.+an - ÷àñòè÷íàÿ ñóììà.Òîãäà ðÿä ñõîäèòñÿ òîãäà,
êîãäà An ìîíîòîííî óáûâàþùàÿ è îãðàíè÷åííàÿ ïîñëåäîâàòåëüíîñòü.
Òîãäà [pic], èëè [pic]. Ïîýòîìó åñëè [pic] ñõîäèòñÿ, òî
[pic]. Òîãäà [pic] [pic] è [pic], [pic] ðÿä ñõîäèòñÿ.
Ïóñòü òåïåðü íàîáîðîò, èçâåñòíî, ÷òî ðÿä ñõîäèòñÿ. Òîãäà [pic]. Âçÿâ
ïðîèçâîëüíîå [pic], âûáåðåì [pic] òàê, ÷òîáû [pic]. Òîãäà [pic]. Çíà÷èò,
[pic] ñõîäèòñÿ.

Ïðèìåð:
[pic]~[pic] => ðÿä ñõîäèòñÿ ïðè ?>1, è ðàñõîäèòñÿ ïðè ?˜1.

Ðàñõîäèìîñòü ðÿäà n*ln(n)
[pic][pic] => ðÿä ðàñõîäèòñÿ.



Çíàêîïåðåìåííûå ðÿäû


Ïóñòü [pic]è ðÿä [pic] ñõîäÿòñÿ îäíîâðåìåííî, òî À òàêæå è ïðè ýòîì
ãîâîðÿò, ÷òî ðÿä A ñõîäèòñÿ àáñîëþòíî.

Åñëè [pic]ñõîäèòñÿ, [pic] - ðàñõîäèòñÿ, òî À ñõîäèòñÿ óñëîâíî.

Ïðèçíàê Ëåéáíèöà.

[pic] (ìîíîòîííî ñòðåìèòñÿ ê 0), òîãäà A ñõîäèòñÿ.

Äîêàçàòåëüñòâî:

[pic]

Ò.ê. [pic]

[pic].

[pic], [pic], òî åñòü ïîñëåäîâàòåëüíîñòü ÷àñòè÷íûõ ñóìì A2n óáûâàåò, à
A2n+1 âîçðàñòåò.

[pic]

[pic]

Êàæäàÿ èç ïîñëåäîâàòåëüíîñòåé A2n è A2n+1 îãðàíè÷åíà è

[pic]

Ñëåäîâàòåëüíî, [pic].

[pic] [pic]

Çàìåòèì, ÷òî:

[pic].

Ïðèìåð:

Ðÿä Ëåéáíèöà: [pic] ñõîäèòñÿ óñëîâíî (íåàáñîëþòíî), òàê êàê
ãàðìîíè÷åñêèé ðÿä [pic] ðàñõîäèòñÿ.

[pic]

Ïðèìåð (ðàñõîäÿùèéñÿ çíàêî÷åðåäóþùèéñÿ ðÿä):

[pic] íå ìîíîòîííî: [pic] ðàñõîäèòñÿ.

Âîîáùå, åñëè ðÿä ïðåäñòàâèì â âèäå ñóììû ðÿäîâ:

1) Åñëè îáà ðÿäà ñõîäÿòñÿ, òî èõ ñóììà ñõîäèòñÿ.

2) Åñëè îäèí èç ðÿäîâ ñõîäèòñÿ, à äðóãîé ðàñõîäèòñÿ, òî èõ ñóììà
ðàñõîäèòñÿ.

Ïðèçíàê Äèðèõëå.

Ïóñòü äàí ðÿä:

[pic] [pic] òîãäà [pic] ñõîäèòñÿ.

Äîêàçàòåëüñòâî: Ïî êðèòåðèþ Êîøè: [pic].

[pic] ïî óñëîâèþ [pic]

Èñïîëüçóÿ ïðåîáðàçîâàíèå Àáåëÿ, ïîëó÷èì íåðàâåíñòâî:
[pic]Ñëåäîâàòåëüíî, êðèòåðèé Êîøè âûïîëíåí, ïîýòîìó ðÿä ñõîäèòñÿ.
Èç ïðèçíàêà Äèðèõëå ñëåäóåò ïðèçíàê Ëåéáíèöà:
Åñëè [pic].

Ïðèçíàê Àáåëÿ.

[pic]; [pic]òîãäà [pic]ñõîäèòñÿ

Äîêàçàòåëüñòâî:

[pic] Äîêàçàíî.

Ïðèìåð 1:

[pic]: [pic]

[pic]Äîêàæåì, ÷òî ýòè ðÿäû ñõîäÿòñÿ óñëîâíî:

Äîêàæåì, ÷òî ðÿä [pic] ðàñõîäèòñÿ. Òàê êàê [pic], ðàññìîòðèì ñëåäóþùèé ðÿä:

[pic].

Çíà÷èò, ðÿä[pic]

Ïðèìåð 2:

Ïðè ïðîèçâîëüíîé ïåðåñòàíîâêå ÷ëåíîâ óñëîâíî ñõîäÿùåãîñÿ ðÿäà åãî ñóììà
ìîæåò èçìåíÿòüñÿ:

[pic];

[pic]





Òåîðåìà Ðèìàíà (áåç äîêàçàòåëüñòâà).

Òåîðåìà: Ïóñòü äàí óñëîâíî ñõîäÿùèéñÿ ðÿä[pic]. Òîãäà: [pic]ïåðåñòàíîâêà
ñëàãàåìûõ, òàêàÿ, ÷òî [pic]

Òåîðåìà Äèðèõëå î ïåðåñòàíîâêå ÷ëåíîâ àáñîëþòíî ñõîäÿùåãîñÿ ÷èñëîâîãî ðÿäà.

Òåîðåìà: Ïóñòü ðÿä [pic]ñõîäèòñÿ àáñîëþòíî, [pic]. Òîãäà, äëÿ ëþáîé
ïåðåñòàíîâêè ðÿäà [pic] íîâûé ðÿä ñõîäèòñÿ. Ïðè ýòîì, ðÿä A( ñõîäèòñÿ
àáñîëþòíî è åãî ñóììà ðàâíà ñóììå èñõîäíîãî ðÿäà, òî åñòü A = A(.

Äîêàçàòåëüñòâî:

1) [pic]

[pic]

k - ôèêñ., [pic], òîãäà [pic][pic]

è [pic].

Àíàëîãè÷íî ðàññìàòðèâàåòñÿ ðÿä À, êàê ïîëó÷åííûé ïåðåñòàíîâêîé

÷ëåíîâ [pic]:

[pic].

Äîêàçàíî.

2) an - ïðîèçâîëüíîãî çíàêà. Ïóñòü òîãäà:

[pic][pic] [pic];

[pic]- ñõîäèòñÿ, [pic]- ñõîäèòñÿ, òàê êàê ðÿä A ñõîäèòñÿ àáñîëþòíî
[pic].

Ïðèìåíÿÿ ê [pic] è [pic] ðåçóëüòàò èç 1), ïîëó÷èì ïîëíîå
äîêàçàòåëüñòâî.

Äîêàçàíî.




Ôóíêöèîíàëüíûå ðÿäû

[pic]- ôóíêöèîíàëüíûå ðÿäû, fn(x), f(x) - ôóíêöèè îò [pic], ãäå D - îáëàñòü
ñõîäèìîñòè ðÿäà.

Ïðèìåðû ôóíêöèîíàëüíûõ ðÿäîâ:

1) [pic]- ñòåïåííîé ðÿä

2)[pic] - òðèãîíîìåòðè÷åñêèé ðÿä Ôóðüå



[pic]

Ðàâíîìåðíàÿ ñõîäèìîñòü ôóíêöèîíàëüíîé ïîñëåäîâàòåëüíîñòè è ôóíêöèîíàëüíîãî
ðÿäà.

Îïðåäåëåíèå (ðàâíîìåðíîé ïîñëåäîâàòåëüíîñòè íà ìíîæåñòâå E( D
ôóíêöèîíàëüíîé ïîñëåäîâàòåëüíîñòè): [pic]

Ïðèìåð:

[pic]

Êðèòåðèé Êîøè: [pic]

Îïðåäåëåíèå ðàâíîìåðíîé ñõîäèìîñòè ôóíêöèîíàëüíîãî ðÿäà íà ìíîæåñòâå E:

[pic]



Êðèòåðèé Êîøè:

[pic].

Ñëåäñòâèå. Åñëè [pic]

Ïðèìåðû:

1) [pic]

[pic]

[pic]

Ïðèçíàê ðàâíîìåðíîé ñõîäèìîñòè.

1) Ïðèçíàê Âåéåðøòðàññà (ìàæîðàíòíûé ïðèçíàê)

Ïóñòü äàí ôóíêöèîíàëüíûé ðÿä [pic] è åñëè [pic] - ñõîäèòñÿ, òî
ôóíêöèîíàëüíûé ðÿä [pic] ñõîäèòñÿ ðàâíîìåðíî íà E.

Äîêàçàòåëüñòâî (ïî êðèòåðèþ Êîøè):

[pic], òàê êàê [pic]è [pic]

Ïðèìåðû:

[pic]

[pic]

Ê ðÿäó [pic] - ïðèçíàê Âåéåðøòðàññà íåïðèìåíèì.

2) Ïðèçíàê Àáåëÿ - Äèðèõëå.

Ïóñòü äàí ôóíêöèîíàëüíûé ðÿä [pic], x(E( D.

|Ïðèçíàê Àáåëÿ |Ïðèçíàê Äèðèõëå |
|Åñëè: |Åñëè: |
|[pic] |[pic] |
|bn(x) - ìîíîòîííàÿ ïî n |ïî n ìîíîòîííî, ïî ðàâíîìåðíî |
|ïîñëåäîâàòåëüíîñòü ïðè | |
|ôèêñèðîâàííîì x. | |


Òî ðÿä [pic] ñõîäèòñÿ ðàâíîìåðíî íà E. (áåç äîêàçàòåëüñòâà)





Ïðèìåðû:

[pic]

[pic]= f(x) = [pic], x((0; 2() (áåç äîêàçàòåëüñòâà).

[pic]

Òåîðåìà î íåïðåðûâíîñòè ñóììû ôóíêöèîíàëüíîãî ðÿäà.

Òåîðåìà: Ïóñòü[pic]

Äîêàçàòåëüñòâî:

[pic] Äîêàæåì, ÷òî [pic]

[pic]

Äîêàçàíî.



Òåîðåìà îá èíòåãðèðîâàíèè ôóíêöèîíàëüíîãî ðÿäà.

Òåîðåìà: [pic]

Äîêàçàòåëüñòâî:

[pic]


Òåîðåìà äîêàçàíà.

Äèôôåðåíöèðîâàíèå ôóíêöèîíàëüíûõ ðÿäîâ

Òåîðåìà: Ïóñòü [pic]fn(x) > [pic]f(x), x[pic]O(a),
fn'(x) [pic]C(O(a)),
[pic]
Òîãäà f(x)[pic]D(O(a)) è f('(x)=g(x), x[pic]O(a)
Äîêàçàòåëüñòâî (íà îñíîâàíèè òåîðåìû îá èíòåãðèðîâàíèè ôóíêöèîíàëüíîãî
ðÿäà):
[pic]fn'(t)=g(t), t[pic][a,x] - íåïðåðûâíàÿ ôóíêöèÿ, òàê êàê ðÿä
[pic]fn'(t) ðàâíîìåðíî ñõîäèòñÿ íà O(a). Íà îñíîâàíèè òåîðåìû îá
èíòåãðèðîâàíèè ôóíêöèîíàëüíîãî ðÿäà ýòîò ðÿä ìîæíî ïðîèíòåãðèðîâàòü
ïî÷ëåííî.
[pic][pic]
Òåîðåìà äîêàçàíà.

Ñòåïåííûå ðÿäû

Ñòåïåííûìè ðÿäàìè íàçûâàþòñÿ ðÿäû âèäà [pic], ãäå an, x0 -ïîñòîÿííûå, x -
ïåðåìåííàÿ.
Ìû áóäåì ðàññìàòðèâàòü ðÿäû ñ x0 = 0, ò.å. [pic]
1 òåîðåìà Àáåëÿ. Ïóñòü [pic]ñõîäèòñÿ ïðè íåêîòîðîì x0. Òîãäà äëÿ ëþáîãî
(h<[pic] ðÿä [pic]ñõîäèòñÿ ðàâíîìåðíî íà [-h;h]
Äîêàçàòåëüñòâî: Òàê êàê [pic] ñõîäèòñÿ, òî [pic], ãäå M>0 - íåêîòîðàÿ
ïîñòîÿííàÿ.
[pic]
[pic]ñõîäèòñÿ [pic]ïî ïðèçíàêó Âåéåðøòðàññà [pic]
Ñëåäñòâèå: 1) Îáëàñòü ñõîäèìîñòè ñòåïåííîãî ðÿäà D ìîæåò áûòü îäíèì èç
ñëåäóþùèõ ìíîæåñòâ:
D=[pic], ãäå R - ðàäèóñ ñõîäèìîñòè.
Ëþáîé ñòåïåííîé ðÿä ñõîäèòñÿ â òî÷êå x=0.  îñòàëüíûõ ñëó÷àÿõ ðÿä ñõîäèòñÿ
ïðè âñåõ [pic], åñëè R - ðàäèóñ ñõîäèìîñòè (òî÷íàÿ âåðõíÿÿ ãðàíü ìíîæåñòâà
x, äëÿ êîòîðûõ ðÿä ñõîäèòñÿ)-ñóùåñòâóåò. Åñëè òî÷íîé âåðõíåé ãðàíè íåò, òî
ïîëàãàþò [pic] - ðÿä ñõîäèòñÿ íà âñåé ÷èñëîâîé ïðÿìîé.
Ïðèâåä¸ì ïðèìåðû:
[pic]
×òîáû íàéòè ðàäèóñ ñõîäèìîñòè, ìîæíî âîñïîëüçîâàòüñÿ ïðèçíàêàìè ñõîäèìîñòè
çíàêîïîñòîÿííûõ ðÿäîâ Äàëàìáåðà, ëèáî Êîøè.
Ïðèçíàê Äàëàìáåðà:
[pic] [pic] [pic]
[pic]
[pic]

Ïðèçíàê Êîøè: [pic]

[pic] [pic]

Ïðèìå÷àíèå. Åñëè íè îäèí èç óêàçàííûõ ïðåäåëîâ íå ñóùåñòâóåò, òî íóæíî
ïîëîæèòü ðàäèóñ ñõîäèìîñòè ðàâíûì íèæíåìó ïðåäåëó (íàèìåíüøåìó ÷àñòè÷íîìó
ïðåäåëó) âûðàæåíèÿ äëÿ R.
Ïðèìåð: [pic]
[pic] íå ñóùåñòâóåò, íî [pic]=1 => => R = 1
2 òåîðåìà Àáåëÿ: Ðÿä [pic] ñõîäèòñÿ â òî÷êå x=x0 . Òîãäà ðÿä [pic] ñõîäèòñÿ
ðàâíîìåðíî íà îòðåçêå [0;x0] (èëè [x0;0] åñëè x0<0).
Äîêàçàòåëüñòâî: [pic]
[pic]
=> Ïî ïðèçíàêó Àáåëÿ [pic]
Ñëåäñòâèÿ:
1) Íåïðåðûâíîñòü ñóììû ñòåïåííîãî ðÿäà
[pic], D - îáëàñòü ñõîäèìîñòè
[pic]
2) Èíòåãðèðîâàíèå ñóììû ñòåïåííîãî ðÿäà
[pic], D - îáëàñòü ñõîäèìîñòè
[pic]
[pic] - ðàäèóñ ñõîäèìîñòè íå ìåíÿåòñÿ.
2) Äèôôåðåíöèðîâàíèå ñóììû ñòåïåííîãî ðÿäà
[pic]
[pic], ðàäèóñ ñõîäèìîñòè ïðè äèôôåðåíöèðîâàíèå íå ìåíÿåòñÿ.

Ðÿäû Òåéëîðà

Ïðèìåíÿÿ ïîñëåäîâàòåëüíî òåîðåìó î ïî÷ëåííîì äèôôåðåíöèðîâàíèè ñòåïåííîãî
ðÿäà, ïîëó÷èì ñîîòíîøåíèå äëÿ n-ãî êîýôôèöèåíòà ðÿäà.
Ïóñòü [pic], R - ðàäèóñ ñõîäèìîñòè. Òîãäà [pic]
Äîêàçàòåëüñòâî:
[pic]
[pic]
[pic] - êîýôôèöèåíòû ñòåïåííîãî ðÿäà Òåéëîðà

[pic], ò.å. ðÿä Òåéëîðà äëÿ ôóíêöèè f(x) íå âñåãäà ñîâïàäàåò ñ ñàìîé
ôóíêöèåé.

Ïðèìåð:

[pic] [pic]=> [pic]



Òåîðåìà: Ïóñòü [pic]

è [pic]; òîãäà [pic]

Äîêàçàòåëüñòâî: Ïî ôîðìóëå Òåéëîðà ñ îñòàòêîì â ôîðìå Ëàãðàíæà ïîëó÷èì:

[pic]
Òåîðåìà äîêàçàíà.


Ðÿäû Òåéëîðà äëÿ îñíîâíûõ ýëåìåíòàðíûõ ôóíêöèé


Ïðèâåäåì ðàçëîæåíèÿ îñíîâíûõ ýëåìåíòàðíûõ ôóíêöèé â ðÿä Òåéëîðà.
1) [pic]
[pic], [pic]
=> [pic]
[pic]
[pic][pic]

[pic]
[pic]

[pic]

[pic]

[pic][pic]

[pic]Ïðîèíòåãðèðîâàâ â ïðåäåëàõ îò 0 äî x, ïîëó÷èì:

[pic][pic]

[pic]ñõîäèòñÿ ïðè [pic], â ÷àñòíîñòè:

[pic][pic]

[pic][pic]


Òðèãîíîìåòðè÷åñêèå ðÿäû Ôóðüå

[pic], äàëåå ôóíêöèÿ ïåðèîäè÷åñêàÿ ñ ïåðèîäîì 2?.
Ðÿä Äèðèõëå [pic]ñõîäèòñÿ ïðè âñåõ x.
[pic] [pic]
[pic]

?/2



? 2?

íå÷åòíàÿ ôóíêöèÿ, an=0
[pic]
[pic](signx)
[pic]


Äèôôåðåíöèàëüíûå óðàâíåíèÿ


Îïðåäåëåíèå: Äèôôåðåíöèàëüíûì óðàâíåíèåì íàçûâàåòñÿ óðàâíåíèå âèäà [pic],
ãäå [pic] - ôóíêöèÿ, îïðåäåëåííàÿ â íåêîòîðîé îáëàñòè [pic] ïðîñòðàíñòâà
[pic], [pic] - íåçàâèñèìàÿ ïåðåìåííàÿ, [pic] - ôóíêöèÿ îò [pic], [pic] - åå
ïðîèçâîäíûå.
Îïðåäåëåíèå: Ïîðÿäêîì óðàâíåíèÿ n íàçûâàåòñÿ íàèâûñøèé èç ïîðÿäêîâ
ïðîèçâîäíûõ [pic], âõîäÿùèõ â óðàâíåíèå.
Îïðåäåëåíèå: Ôóíêöèÿ [pic] íàçûâàåòñÿ ðåøåíèåì äèôôåðåíöèàëüíîãî
óðàâíåíèÿ íà ïðîìåæóòêå [pic], åñëè äëÿ âñåõ [pic] èç [pic] âûïîëíÿåòñÿ
ðàâåíñòâî: [pic]. Äèôôåðåíöèàëüíîìó óðàâíåíèþ óäîâëåòâîðÿåò áåñêîíå÷íîå
ìíîæåñòâî ôóíêöèé, íî ïðè íåêîòîðûõ óñëîâèÿõ ðåøåíèå òàêîãî óðàâíåíèÿ
åäèíñòâåííîå.
Îïðåäåëåíèå: Èíòåãðàëüíàÿ êðèâàÿ - ýòî ãðàôèê ðåøåíèÿ äèôôåðåíöèàëüíîãî
óðàâíåíèÿ, ò.å ãðàôèê ôóíêöèè, óäîâëåòâîðÿþùåé ýòîìó óðàâíåíèþ.
Ïðèìåð 1: Ðåøèòü óðàâíåíèå [pic]. Åãî ðåøåíèå: [pic]
îïðåäåëåíî íà [pic]. Îòìåòèì, ÷òî ýòà ïîñòîÿííàÿ - ïðîèçâîëüíàÿ è ðåøåíèå -
íå åäèíñòâåííîå, à èìååòñÿ áåñêîíå÷íîå ìíîæåñòâî ðåøåíèé.
|[pic]ðèñ.1 | |

[pic][pic][pic] Òàêèì îáðàçîì, ñåðèÿ ãðàôèêîâ ïîëó÷åíà ïàðàëëåëüíûì
ïåðåíîñîì íà êîíñòàíòó Ñ.

[pic]ðèñ.2

Ïðèìåð 2:

Âûâåäåì çàêîí äâèæåíèÿ òåëà, áðîøåííîãî ñ íà÷àëüíîé ñêîðîñòüþ V ïîä óãëîì
? ê ãîðèçîíòó.
[pic]
Íî ïî óñëîâèþ y(0) = 0 > C2 = 0 > [pic]
Íàéäåì âðåìÿ ïîäúåìà: [pic]
Íàéäåì âûñîòó ïîäúåìà:[pic]
Äàëüíîñòü ïîëåòà xmax (ïðè y(t) = 0 )
y(t) = 0[pic]
ïðè [pic]

Ïðèìåð 3:

Ðåøèòü óðàâíåíèå [pic]: [pic], [pic], èíòåãðèðóÿ îáå ÷àñòè óðàâíåíèÿ,
ïîëó÷èì
d(lny) = d(lnx) [pic].
Ïîòåíöèèðóÿ îáå ÷àñòè óðàâíåíèÿ, ïîëó÷àåì îáùåå ðåøåíèå y = Cx,
êîòîðîå èçîáðàæàåòñÿ ñåðèåé ëèíåéíûõ èíòåãðàëüíûõ êðèâûõ, ïðîõîäÿùèõ ÷åðåç
òî÷êó (0,0). Ïðè ýòîì èç ãðàôèêà (ðèñ.3) âèäíî, ÷òî ÷åðåç ëþáóþ òî÷êó, íå
ïðèíàäëåæàùóþ (0,0), ïðîõîäèò òîëüêî îäíà èíòåãðàëüíàÿ êðèâàÿ (ðåøåíèå).
[pic]
Ðèñ.3

Ïðèìåð 4:

Ðàññìîòðèì óðàâíåíèå [pic]: [pic] [pic] èíòåãðèðóÿ, ïîëó÷àåì: x2 +
y2 = C = R2 (ðèñ.4)
[pic]- ìíîæåñòâî îêðóæíîñòåé ñ öåíòðîì â íà÷àëå êîîðäèíàò
ðèñ.4
Îïðåäåëåíèå: Îáùåå ðåøåíèå - ìíîæåñòâî ðåøåíèé äèôôåðåíöèàëüíîãî
óðàâíåíèÿ [pic]åñòü ñîâîêóïíîñòü ôóíêöèé F(x, y, C)=0, C((.
Îïðåäåëåíèå: ×àñòíîå ðåøåíèå ïîëó÷àþò ïðè ïîäñòàíîâêå êîíêðåòíîãî
çíà÷åíèÿ êîíñòàíòû â îáùåå ðåøåíèå
Îñîáûå ðåøåíèÿ íå âõîäÿò â îáùèå ðåøåíèÿ ÷åðåç êàæäóþ òî÷êó îñîáîãî
ðåøåíèÿ ïðîõîäèò áîëåå îäíîé èíòåãðàëüíîé êðèâîé.


Ïðèìåð 5:

[pic] ñì. ðèñ.5 (÷åðåç êàæäóþ òî÷êó íà îñè Îõ ïðîõîäèò äâà ðåøåíèÿ
(äâå èíòåãðàëüíûå êðèâûå): ÷àñòíîå è îñîáîå).
[pic]
Ðèñ.5
Ìîæíî ïîñòðîèòü èíòåãðàëüíóþ êðèâóþ â êàæäîé òî÷êå, èñïîëüçóÿ
ïîíÿòèå î ãåîìåòðè÷åñêîì ñìûñëå ïðîèçâîäíîé: tg? = f(x,y) (ðèñ.6).
Òàêèì îáðàçîì çàäàþò ïîëå íàïðàâëåíèé, ò.å. çàäàþò ïðÿìóþ â êàæäîé
òî÷êå, à ïîòîì ïðîâîäÿò êðèâóþ êàñàòåëüíóþ êî âñåì ïðÿìûì â ýòèõ
òî÷êàõ è ïîëó÷àþò èíòåãðàëüíóþ êðèâóþ (îäíî èç ðåøåíèé).
[pic]ðèñ.6
Ñôîðìóëèðóåì âàæíåéøóþ òåîðåìó.
Òåîðåìà (Î ñóùåñòâîâàíèè è åäèíñòâåííîñòè ðåøåíèÿ çàäà÷è Êîøè
äèôôåðåíöèàëüíîãî óðàâíåíèÿ y'=f(x ,y)):
Ïóñòü [pic] - íåïðåðûâíàÿ ôóíêöèÿ (ðèñ.7) â îáëàñòè[pic], ïðè÷åì [pic] -
òàêæå íåïðåðûâíà â [pic]. Òîãäà ñóùåñòâóåò åäèíñòâåííîå ðåøåíèå y=y(x)
äèôôåðåíöèàëüíîãî óðàâíåíèÿ y'=f(x, y) ñ íà÷àëüíûì óñëîâèåì y(x0)=y0,
(x0,y0) ïðèíàäëåæèò D. Ñëåäîâàòåëüíî, ÷åðåç òî÷êó [pic] ïðîõîäèò òîëüêî
îäíà èíòåãðàëüíàÿ êðèâàÿ.[pic]

Ðèñ.7
(áåç äîêàçàòåëüñòâà).

Ïðèìåð 7:

Ðàññìîòðèì ïîäðîáíåå óðàâíåíèå [pic]:
[pic] [pic]
Òàê êàê ïðîèçâîäíàÿ ôóíêöèè f(y) íåîïðåäåëåíà ïðè ó = 0 (ðàçðûâ âäîëü
îñè Îõ), òî ïðè ó = 0 åñòü åùå îäíî ðåøåíèå (îñîáîå).


Îñíîâíûå òèíû äèôôåðåíöèàëüíûõ óðàâíåíèé

1) Óðàâíåíèÿ ñ ðàçäåëÿþùèìèñÿ ïåðåìåííûìè. Óðàâíåíèÿìè ñ ðàçäåëÿþùèìèñÿ
ïåðåìåííûìè íàçûâàþòñÿ óðàâíåíèÿ âèäà [pic], ãäå [pic] - íåïðåðûâíà íà
íåêîòîðîì [pic], à [pic] íåïðåðûâíà íà [pic], ïðè÷åì [pic] íà [pic]. [pic]
(ìåòîä ðàçäåëåíèÿ ïåðåìåííûõ). Èíòåãðèðóÿ îáå ÷àñòè, ïîëó÷àåì [pic].
Îáîçíà÷àÿ [pic] ëþáóþ ïåðâîîáðàçíóþ äëÿ [pic], à [pic] - ëþáóþ
ïåðâîîáðàçíóþ äëÿ [pic], ïåðåïèøåì ýòî óðàâíåíèå â âèäå íåÿâíî âûðàæåííîé
ôóíêöèè [pic]. Ýòî - îáùåå ðåøåíèå.
Ðàññìîòðèì ïðèìåð òàêîãî óðàâíåíèÿ
[pic] [pic] èíòåãðèðóÿ, ïîëó÷èì [pic].
2) Îäíîðîäíûå óðàâíåíèÿ. Ïîä îäíîðîäíûìè óðàâíåíèÿìè ïîíèìàþòñÿ óðàâíåíèÿ
âèäà [pic]. Äëÿ èõ ðåøåíèÿ òðåáóåòñÿ ñäåëàòü çàìåíó [pic], ïîñëå ÷åãî
ïîëó÷èòñÿ óðàâíåíèå ñ ðàçäåëÿþùèìèñÿ ïåðåìåííûìè: [pic] [pic] [pic]
[pic]
[pic].

Ïðèìåð 1: Ðàññìîòðèì ïàðàáîëè÷åñêîå çåðêàëî. Ðàñïîëîæèì íà÷àëî êîîðäèíàò
â ôîêóñå ïàðàáîëû (ðèñ.8). Òàêîå çåðêàëî èìååò èíòåðåñíîå ñâîéñòâî: ïðè
ïîìåùåíèè èñòî÷íèêà ñâåòà â ôîêóñ çåðêàëà ëó÷è, ðàäèàëüíî ðàñõîäÿùèåñÿ â
ðàçíûå ñòîðîíû , ïîñëå îòðàæåíèÿ ñòàíîâÿòñÿ ïàðàëëåëüíûìè (òàê ïîëó÷àþò
ïëîñêèå ñâåòîâûå âîëíû), ïðè÷åì ïî çàêîíó îòðàæåíèÿ óãîë ïàäåíèÿ ðàâåí
óãëó îòðàæåíèÿ.

[pic]ðèñ.8

[pic] [pic] =>
[pic]
Ââåäåì çàìåíó: y = zx [pic] è ðàññìîòðèì îäèí ñëó÷àé, êîãäà[pic]
Ñîêðàùàÿ íà z, ïîëó÷àåì [pic] èíòåãðèðóåì ðàâåíñòâî:
[pic]
Âîçâîäèì â êâàäðàò z2 - 1 = C2x2 - 2Cxz + z2 [pic]
Òàêèì îáðàçîì, ïîëó÷åíî óðàâíåíèå ïàðàáîëû.

Ïðèìåð 2 (óðàâíåíèå õèìè÷åñêîé ðåàêöèè):
1) [pic] [pic]
Ðàçëîæèì íà ìíîæèòåëè:
[pic] [pic]
ïðè x =a 1=A(b-a )[pic]A=-1/(a-b)
ïðè x = b 1= B(a-b) [pic]B=1/(a-b)

[pic]
 òî÷êå (0,0) ÷àñòíîå ðåøåíèå èñõîäíîãî óðàâíåíèÿ: [pic]

[pic][pic]


Ïðèìåð 3:
Íàéäåì çàêîí Ò(t) îñòûâàíèÿ êèïÿùåé âîäû äî êîìíàòíîé òåìïåðàòóðû
(têîìí=200) è âðåìÿ äîñòèæåíèÿ 400, åñëè äî 600 âîäà îñòûâàåò çà 20 ìèí.
Èçâåñòíî, ÷òî ìãíîâåííàÿ ñêîðîñòü îñòûâàíèÿ ëèíåéíî çàâèñèò îò ðàçíèöû Ò è
têîìí.
Ñîñòàâèì äèôôåðåíöèàëüíîå óðàâíåíèå: [pic] [pic] [pic]
[pic] [pic]
Ïðèìåð 4:
Íàéòè êîëè÷åñòâî ñîëè â ðàñòâîðå ÷åðåç âðåìÿ t, åñëè èçâåñòíî, ÷òî
èçíà÷àëüíî áûëî 10 êã ñîëè â 100 ë âîäû, íî êàæäóþ ìèíóòó â ðåçåðâóàð
ïîñòóïàåò 20 ë âîäû, à âûëèâàåòñÿ 30 ë ðàñòâîðà.
Vð-ðà(t) = 100 + 30t -20t = 100 + 10t x(0)=0, x(t)- êîëè÷åñòâî
ñîëè
[pic] ; [pic]; [pic]
[pic] [pic]; ïðè t = 0 x(0)=10[pic] C = 1000
[pic]
Ïðèìåð 5:
Íàéòè òî÷íûé çàêîí ðàäèîàêòèâíîãî ðàñïàäà, åñëè t0-ïåðèîä
ïîëóðàñïàäà., à x0- íà÷àëüíîå êîëè÷åñòâî. Ïðè÷åì èçâåñòíî, ÷òî
ìãíîâåííàÿ ñêîðîñòü ðàñïàäà ëèíåéíî çàâèñèò îò ìãíîâåííîãî êîëè÷åñòâà
âåùåñòâà.
[pic] [pic] [pic] [pic]
Çàäàíî, ÷òî õ (0)=õ0; x(t0)= õ0/2[pic]; [pic]
Òàêèì îáðàçîì, çàêîí ðàñïàäà: [pic]




Ëèíåéíîå äèôôåðåíöèàëüíîå óðàâíåíèå 1-ãî ïîðÿäêà.

Òàêèå óðàâíåíèÿ â îáùåì âèäå ìîãóò áûòü ïðåäñòàâëåíû êàê: [pic]
Ïóñòü y = UV, ãäå U, V- íåêîòîðûå ôóíêöèè îò õ, òîãäà ïîäñòàâëÿÿ, ïîëó÷àåì:
[pic]
Âûáåðåì V(x) òàê, ÷òîáû îíà óäîâëåòâîðÿëà óñëîâèþ: [pic]
[pic] Áåðåì ëþáóþ ôóíêöèþ, óäîâëåòâîðÿþùóþ ýòîìó óðàâíåíèþ, íàïðèìåð, V =
V(x) è ïîäñòàâëÿåì â èñõîäíîå óðàâíåíèå [pic]
Ïðèìåð:

[pic] Çàìåíà: y = UV [pic]
Òàê êàê èùåì îäíî ëþáîå ðåøåíèå, òî ïðè èíòåãðèðîâàíèè íå íàäî äîáàâëÿòü
êîíñòàíòó: [pic] Ïîäñòàâèì â èñõîäíîå óðàâíåíèå: [pic]
Ñëåäîâàòåëüíî, [pic]
Ýòîò ìåòîä ïðèìåíèì è äëÿ íåëèíåéíîãî óðàâíåíèÿ: [pic], ãäå ê- êîíñòàíòà
Ïóñòü y = UV, ãäå U, V- íåêîòîðûå ôóíêöèè îò õ, òîãäà ïîäñòàâëÿÿ ïîëó÷àåì
[pic]
Âûáåðåì V(x) òàê, ÷òîáû îíà óäîâëåòâîðÿëà óñëîâèþ: [pic]
[pic] Áåðåì ëþáóþ ôóíêöèþ, óäîâëåòâîðÿþùóþ ýòîìó óðàâíåíèþ, íàïðèìåð, V =
V(x) è ïîäñòàâëÿåì â èñõîäíîå óðàâíåíèå [pic] èç ïîñëåäíåãî óðàâíåíèÿ
èíòåãðèðîâàíèåì íàõîäèì U, à çàòåì óæå çíàÿ V(x) íàõîäèì ó.

Ïðèìåð:


Ðåøèì äèôôåðåíöèàëüíîå óðàâíåíèå, îïèñûâàþùåå ïðîõîæäåíèå ïî öåïè
ïåðåìåííîãî òîêà, ÷òîáû íàéòè çàâèñèìîñòü ìãíîâåííîé ñèëû òîêà îò âðåìåíè
i(t).

[pic]L - èíäóêòèâíîñòü, R - ñîïðîòèâëåíèå
[pic]
Ñäåëàåì çàìåíó ïåðåìåííûõ: [pic] è ïîäñòàâèì
[pic]
Ãäå: ?=[pic]
[pic]
[pic]
[pic] [pic]
Âñåãäà ìîæíî ââåñòè ?0 (ñîáñòâåííàÿ ÷àñòîòà): [pic] [pic]
[pic][pic] Ïðè áîëüøèõ t ñòðåìèòñÿ ê íóëþ [pic]

Óðàâíåíèå Êëåðî: [pic]. Ââîäÿ ïàðàìåòð [pic], ïîëó÷àåì [pic]. [pic] èëè
[pic]. Òîãäà, åñëè [pic], òî [pic] è [pic] - ýòî îáùåå ðåøåíèå óðàâíåíèÿ
Êëåðî (ïðÿìûå ëèíèè). Åñëè æå [pic], òî [pic]. Òîãäà [pic] - îñîáîå ðåøåíèå
(ïðîâåðÿåòñÿ ïîäñòàíîâêîé â èñõîäíîå óðàâíåíèå) .

Ïðèìåð:

[pic] Îáùåå ðåøåíèå óðàâíåíèÿ áóäåò: [pic]; îñîáîå ðåøåíèå : 0=x + 2C
[pic] Ïðîâåðèì, ÷òî ïîñëåäíÿÿ ôóíêöèÿ äåéñòâèòåëüíî ÿâëÿåòñÿ ðåøåíèåì
èñõîäíîãî óðàâíåíèÿ: [pic]

Äèôôåðåíöèàëüíîå óðàâíåíèå n-íîãî ïîðÿäêà
[pic]
Îáùåå ðåøåíèå â íåÿâíîì âèäå (äîëæíî ñîäåðæàòü n ïðîèçâîëüíûõ íåçàâèñèìûõ
ïîñòîÿííûõ): [pic]
Ëèáî îáùåå ðåøåíèå ìîæåò áûòü íàéäåíî â ÿâíîì âèäå: [pic]

Ïðèìåð:

[pic]Åñëè çàäàòü íà÷àëüíûå óñëîâèÿ: y (0) = y0 , V(0)=V0 , òî V0=Ñ1 y0=Ñ2
[pic]
×òîáû ðåøèòü çàäà÷ó Êîøè äëÿ äèôôåðåíöèàëüíîãî óðàâíåíèÿ 1-ãî ïîðÿäêà:
[pic], ò.å. íàéòè ôóíêöèþ-ðåøåíèå (èíòåãðàëüíóþ êðèâóþ), ïðîõîäÿùóþ ÷åðåç
äàííóþ òî÷êó, äîñòàòî÷íî çàäàòü 1 óñëîâèå: y(õ0)= y0.

Òåîðåìà: Ïóñòü ôóíêöèÿ [pic] îïðåäåëåíà è íåïðåðûâíà â îáëàñòè [pic].
Ïóñòü [pic] íåïðåðûâíû â [pic]. Òîãäà çàäà÷à Êîøè, ñîñòîÿùàÿ â íàõîæäåíèè
ðåøåíèÿ óðàâíåíèÿ [pic] ñ íà÷àëüíûìè óñëîâèÿìè [pic] (ãäå òî÷êè [pic]
ïðèíàäëåæàò îáëàñòè [pic]) èìååò, ïðèòîì åäèíñòâåííîå ðåøåíèå y = y(x), â
îêðåñòíîñòè x=x0. (áåç äîêàçàòåëüñòâà).

Ëèíåéíûå äèôôåðåíöèàëüíûå óðàâíåíèÿ

Ëèíåéíûì äèôôåðåíöèàëüíûì óðàâíåíèåì íàçûâàåòñÿ óðàâíåíèå âèäà:
[pic] (1)
Ïðè q=0, óðàâíåíèå íàçûâàåòñÿ îäíîðîäíûì, q(0 íåîäíîðîäíûì.
y(+py=q -äèôôåðåíöèàëüíîå óðàâíåíèå ïåðâîãî ïîðÿäêà.
Îáîçíà÷èì ëåâóþ ÷àñòü óðàâíåíèÿ (1) ïðè q(x)=0 L(y)=>L(y)=0.
Îòìåòèì äâà ñâîéñòâà L(y).
1) L(y1+y2)=L(y1)+L(y2)
2) L(Cy)=CL(y) => ìíîæåñòâî ðåøåíèé ëèíåéíî îäíîðîäíîãî äèôôåðåíöèàëüíîãî
L(y) = 0 åñòü ëèíåéíîå ïðîñòðàíñòâî.

Ëèíåéíàÿ çàâèñèìîñòü ôóíêöèé
Ôóíêöèè y1,.,yn íàçûâàþòñÿ ëèíåéíî çàâèñèìûìè, åñëè ( ?1,.,?n
(|?1|+.+|?n|(0) òàêèå ÷òî [pic] ñîîòâåòñòâåííî ôóíêöèè íàçûâàþòñÿ ëèíåéíî
íåçàâèñèìûìè åñëè íå óäîâëåòâîðÿþò óðàâíåíèþ (1) ïðè ëþáîì [pic].
Ìíîæåñòâî ðåøåíèé n-ìåðíîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ îáðàçóþò
áàçèñ, ñîñòîÿùèé èç ëèíåéíî íåçàâèñèìûõ ôóíêöèé.

Îïðåäåëèòåëü Âðîíñêîãî.
[pic]
Òåîðåìà 1:
Åñëè ôóíêöèè y1(x),.,yn(x)(âñå ôóíêöèè è èõ ïðîèçâîäíûå íåïðåðûâíû è
ñóùåñòâóþò äî n-1 ãî ïîðÿäêà) ëèíåéíî çàâèñèìû, òî [pic]=0.
Äîêàçàòåëüñòâî:
Òàê êàê ôóíêöèè ëèíåéíî çàâèñèìû, òî [pic]ïîñëå äèôôåðåíöèèðîâàíèÿ
ïîëó÷èì:
[pic] [pic],
[pic] Ýòà ñèñòåìà èìååò íåíóëåâîå ðåøåíèå (
êîãäà îïðåäåëèòåëü ýòîé ñèñòåìû ðàâåí 0. À ýòîò îïðåäåëèòåëü è åñòü
îïðåäåëèòåëü Âðîíñêîãî.
Åñëè W(0, òî ôóíêöèè ëèíåéíî íåçàâèñèìû.
Ïðèìåð 1:
Ïîêàæåì, ÷òî åñëè îïðåäåëèòåëü ðàâåí íóëþ, òî ôóíêöèè
íåîáÿçàòåëüíî ëèíåéíî çàâèñèìû. [pic], [pic] [pic]

Ðàññìîòðèì ïðîèçâîëüíóþ òî÷êó x0>0.
Òåîðåìà 2: Ïóñòü [pic] ðåøåíèÿ óðàâíåíèÿ [pic] è [pic] [pic] òîãäà [pic]
Ñëåäñòâèå: åñëè [pic]õîòÿ áû â îäíîé òî÷êå (a,b) òîãäà ( x ((a,b) W(x)(0 è
ôóíêöèè [pic] ëèíåéíî íåçàâèñèìû.
Äîêàçàòåëüñòâî:
Òàê êàê W=0 â x0, òî [pic] òàê êàê îïðåäåëèòåëü =0 òî åãî ñòîëáöû ëèíåéíî
çàâèñèìû (èõ ëèíåéíàÿ êîìáèíàöèÿ ðàâíà 0). Çíà÷èò
[pic]
Ðàññìîòðèì ôóíêöèþ y=[pic] L(y)=0, ò.å. y- ðåøåíèå äèôôåðåíöèàëüíîãî
óðàâíåíèÿ. Ñ óñëîâèÿìè: [pic]
Ðåøåíèå [pic] óäîâëåòâîðÿåò (1) íî íà÷àëüíûì óñëîâèÿì óäîâëåòâîðÿåò òîëüêî
îäíî ðåøåíèå (ïî òåîðåìå Êîøè î ñóùåñòâîâàíèè åäèíñòâåííîãî ðåøåíèÿ). y(x)
( 0 è [pic] - ëèíåéíî çàâèñèìû => W = 0.
Ïðèìåð 1:
y(n)=0 Ðåøåíèÿ:y1=1, y2=x, y3=x2,.,yn=xn-1
[pic]
ñëåäîâàòåëüíî, ôóíêöèè ëèíåéíî íåçàâèñèìû.
Ïðèìåð 2:
[pic] Çíà÷èò ôóíêöèè sinx è cosx ëèíåéíî íåçàâèñèìû.
Ïðèìåð 3:
[pic], [pic]
[pic]
[pic]





Ôóíäàìåíòàëüíàÿ ñèñòåìà ðåøåíèé ëèíåéíîãî


îäíîðîäíîãî óðàâíåíèÿ



[pic]

Îïðåäåëåíèå: Ëþáûå n ëèíåéíî íåçàâèñèìûõ ðåøåíèé ëèíåéíîãî îäíîðîäíîãî
äèôôåðåíöèàëüíîãî óðàâíåíèÿ n-íîãî ïîðÿäêà íàçûâàåòñÿ ôóíäàìåíòàëüíîé
ñèñòåìîé ðåøåíèé ýòîãî óðàâíåíèÿ. Èç äîêàçàííûõ âûøå òåîðåì ñëåäóåò:

Òåîðåìà: Ðåøåíèÿ [pic] óðàâíåíèÿ îáðàçóþò ôóíäàìåíòàëüíóþ ñèñòåìó ðåøåíèé
ýòîãî óðàâíåíèÿ òîãäà è òîëüêî òîãäà, êîãäà èõ îïðåäåëèòåëü Âðîíñêîãî [pic]
îòëè÷åí îò 0 õîòÿ áû â îäíîé òî÷êå [pic].

Òåîðåìà: Äëÿ ëþáîãî ëèíåéíîãî îäíîðîäíîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ
ñóùåñòâóåò ôóíäàìåíòàëüíàÿ ñèñòåìà åãî ðåøåíèé.

Äîêàçàòåëüñòâî: Ïóñòü

[pic]

Òîãäà îïðåäåëèòåëü Âðîíñêîãî çàïèøåòñÿ òàê:

[pic]

Ñèñòåìà ôóíêöèé [pic] ðåøåíèé äèôôåðåíöèàëüíîãî óðàâíåíèÿ

L(y) = 0 ëèíåéíî íåçàâèñèìà, ïîýòîìó îíà îáðàçóåò ôóíäàìåíòàëüíóþ ñèñòåìó
ðåøåíèé:

[pic]

[pic]

Òåîðåìà: Ïóñòü y(x) - ëþáîå ðåøåíèå äèôôåðåíöèàëüíîãî óðàâíåíèÿ L(y) = 0 è
[pic] Òîãäà

[pic]Äîêàçàòåëüñòâî: Ðàññìîòðèì ñèñòåìó ëèíåéíûõ óðàâíåíèé îòíîñèòåëüíî
íåèçâåñòíûõ [pic]:

[pic]