Phương pháp phân tích adomian giải gần đúng các phương trình vi phân thường

TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI 2
KHOA TOÁN
*************

ĐỖ THỊ THU HÀ

PHƯƠNG PHÁP PHÂN TÍCH ADOMIAN
GIẢI GẦN ĐÚNG CÁC PHƯƠNG TRÌNH
VI PHÂN THƯỜNG
KHÓA LUẬN TỐT NGHIỆP ĐẠI HỌC
Chuyên ngành: Giải tích

HÀ NỘI – 2018


TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI 2
KHOA TOÁN
*************

ĐỖ THỊ THU HÀ

PHƯƠNG PHÁP PHÂN TÍCH ADOMIAN
GIẢI GẦN ĐÚNG CÁC PHƯƠNG TRÌNH
VI PHÂN THƯỜNG
KHÓA LUẬN TỐT NGHIỆP ĐẠI HỌC
Chuyên ngành: Giải tích

Người hướng dẫn khoa học
PSG.TS KHUẤT VĂN NINH

HÀ NỘI – 2018

ớ ỡ
ữủ tọ ỏ t ỡ s s



P t

ữớ trỹ t t t ữợ

ữợ tr sốt q tr õ ừ
ỗ tớ ụ t ỡ t ổ tr tờ
t t ổ tr rữớ ồ ữ
ở ừ t t
tốt õ õ t q ữ ổ
ũ õ rt ố s tớ
t ỏ õ ổ t tr ọ ỳ
t sõt rt ữủ sỹ õ õ ỵ ừ t ổ
s ồ
t ỡ

ở t


ộ


▲í✐ ❝❛♠ ✤♦❛♥
❊♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣

❡♠ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ t❤➛②

P●❙✳❚❙✳ ❑❤✉➜t ❱➠♥ ◆✐♥❤

✳ ❚r♦♥❣

❦❤✐ ♥❣❤✐➯♥ ❝ù✉✱ ❤♦➔♥ t❤➔♥❤ ❜↔♥ ❦❤â❛ ❧✉➟♥ ♥➔② ❡♠ ✤➣ t❤❛♠ ❦❤↔♦ ♠ët
sè t➔✐ ❧✐➺✉ ✤➣ ❣❤✐ tr♦♥❣ ♣❤➛♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳

P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤
❆❞♦♠✐❛♥ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣
❊♠ ①✐♥ ❦❤➥♥❣ ✤à♥❤ ❦➳t q✉↔ ❝õ❛ ✤➲ t➔✐✿ ✏

✑ ❧➔

❦➳t q✉↔ ❝õ❛ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♥é ❧ü❝ ❤å❝ t➟♣ ❝õ❛ ❜↔♥ t❤➙♥✱ ❦❤æ♥❣
trò♥❣ ❧➦♣ ✈î✐ ❦➳t q✉↔ ❝õ❛ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ◆➳✉ s❛✐ ❡♠ ①✐♥ ❝❤à✉ ❤♦➔♥
t♦➔♥ tr→❝❤ ♥❤✐➺♠✳

❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✽
❙✐♥❤ ✈✐➯♥

✣é ❚❤à ❚❤✉ ❍➔


▼ö❝ ❧ö❝
▼ð ✤➛✉

✶

✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✸

✶✳✶

▼ët sè ❦✐➳♥ t❤ù❝ ✈➲ ❣✐↔✐ t➼❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✶

❇→♥ ❦➼♥❤ ❤ë✐ tö ✈➔ ❦❤♦↔♥❣ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ❧ô②
t❤ø❛

✶✳✷

✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✶✳✶✳✷

❈→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝❤✉é✐ ❧ô② t❤ø❛

✳ ✳ ✳ ✳

✹

✶✳✶✳✸

❑❤❛✐ tr✐➸♥ ❤➔♠ t❤➔♥❤ ❝❤✉é✐ ❧ô② t❤ø❛ ✳ ✳ ✳ ✳ ✳ ✳

✺

▼ët sè ❦✐➳♥ t❤ù❝ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✳ ✳ ✳

✻

✶✳✷✳✶

▼ët sè ❦❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✷✳✷

▼ët sè ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët ❣✐↔✐ ✤÷ñ❝
❜➡♥❣ ❝➛✉ ♣❤÷ì♥❣

✶✳✷✳✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

▼ët sè t➼♥❤ ❝❤➜t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐
♣❤➙♥ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✷✳✹

❇➔✐ t♦→♥ ❈❛✉❝❤②

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✷✳✺

✣à♥❤ ❧þ tç♥ t↕✐ ✈➔ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥
❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾
✶✶

✶✶

✷ P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝
♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣
✶✷
✐✐


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
✷✳✶

P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥

✷✳✷

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❘✐❝❝❛t✐

✷✳✸

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜➟❝ ❝❛♦

✷✳✹

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈î✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉
s✉② ❜✐➳♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸ ❙ü ❤ë✐ tö ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥

✷✾

✸✺

✹✻

✸✳✶

❙ü ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✻

✸✳✷

❈→❝ ✈➼ ❞ö

✹✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❑➳t ❧✉➟♥

✺✼

❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖

✺✽

✐✐✐


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

▲í✐ ♠ð ✤➛✉

✶✳ ▲þ ❞♦ ❝❤å♥ ✤➲ t➔✐
❚♦→♥ ❤å❝ ❧➔ ♠ët ♠æ♥ ❦❤♦❛ ❤å❝ tü ♥❤✐➯♥ ❣➢♥ ❧✐➲♥ ✈î✐ t❤ü❝ t✐➵♥✳ ❙ü
♣❤→t tr✐➸♥ ❝õ❛ t♦→♥ ❤å❝ ✤÷ñ❝ ✤→♥❤ ❞➜✉ ❜ð✐ ♥❤ú♥❣ ù♥❣ ❞ö♥❣ ❝õ❛ t♦→♥
❤å❝ ✈➔♦ ✈✐➺❝ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ t❤ü❝ t✐➵♥✳ ❚r♦♥❣ t❤ü❝ t✐➵♥ ♥❤✐➲✉
❜➔✐ t♦→♥ ❝õ❛ ❦❤♦❛ ❤å❝✱ ❦ÿ t❤✉➟t ✈➔ ♠æ✐ tr÷í♥❣✱. . . ❞➝♥ ✤➳♥ ✈✐➺❝ ❣✐↔✐
❜➔✐ t♦→♥ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✱ ❝❤➼♥❤ ✈➻ ✈➟②✱ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉
♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✤â♥❣ ♠ët ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ t♦→♥
❤å❝✳
❚r♦♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✱ trø ♠ët sè ♥❤ä ❧î♣ ♣❤÷ì♥❣
tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✤➣ ✤÷ñ❝ ❤å❝✿ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t→❝❤ ❜✐➳♥✱
♣❤÷ì♥❣ tr➻♥❤ ❇❡❝♥♦✉❧❧✐✱ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t✱
♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥✱. . . ❝á♥ ❧↕✐ ♥â✐ ❝❤✉♥❣ ❦❤æ♥❣ t➻♠ ✤÷ñ❝
♥❣❤✐➺♠ ♠ët ❝→❝❤ ❝❤➼♥❤ ①→❝✳ ❉♦ ✈➟②✱ ♠ët ✈➜♥ ✤➲ ✤➦t r❛ ❧➔ t➻♠ ❝→❝❤ ✤➸
①→❝ ✤à♥❤ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✤â✳
❳✉➜t ♣❤→t tø ♥❤✉ ❝➛✉ ♥➔②✱ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✤➣ t➻♠ r❛ ♥❤✐➲✉ ♣❤÷ì♥❣
♣❤→♣ ✤➸ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✳ P❤÷ì♥❣ ♣❤→♣
♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❤ú✉ ❤✐➺✉ ❞➵ →♣
❞ö♥❣ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✳
❱î✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ s➙✉ ❤ì♥ ✈➜♥ ✤➲ ♥➔②✱ ❞÷î✐

P●❙✳❚❙✳ ❑❤✉➜t ❱➠♥ ◆✐♥❤
P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝
♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣
sü ❤÷î♥❣ ❞➝♥ ❝õ❛

❡♠ ✤➣ ♥❣❤✐➯♥ ❝ù✉

✤➲ t➔✐✿ ✏

✑ ✤➸ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❝õ❛ ♠➻♥❤✳

✶


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉
❑❤â❛ ❧✉➟♥ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❣✐↔✐ ❣➛♥
✤ó♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✳

✸✳ ✣è✐ t÷ñ♥❣ ♥❣❤✐➯♥ ❝ù✉

◆❣❤✐➯♥ ❝ù✉ ù♥❣ ❞ö♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❣✐↔✐
❣➛♥ ✤ó♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ♣❤✐ t✉②➳♥✳

✹✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉

❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët✱ ❝➜♣ ❤❛✐ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
❝➜♣ ❝❛♦ ✈î✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ❝❤♦ tr÷î❝✳

✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉

❙÷✉ t➛♠✱ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➔✐ ❧✐➺✉ ❧✐➯♥ q✉❛♥✳
❱➟♥ ❞ö♥❣ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛ ●✐↔✐ t➼❝❤ ✈➔ ▲þ t❤✉②➳t ♣❤÷ì♥❣
tr➻♥❤ ✈✐ ♣❤➙♥✳

P❤➙♥ t➼❝❤✱ tê♥❣ ❤ñ♣ ✈➔ ❤➺ t❤è♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥✳

✻✳ ❈➜✉ tró❝ ✤➲ t➔✐

❑❤â❛ ❧✉➟♥ ❣ç♠ ❜❛ ❝❤÷ì♥❣
❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
❈❤÷ì♥❣ ✷✿ P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝
♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣
❈❤÷ì♥❣ ✸✿ ❙ü ❤ë✐ tö ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥

✷


❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ t❛ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ♣❤÷ì♥❣
tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✈➔ ❝❤✉é✐ ❧ô② t❤ø❛✳ ◆ë✐ ❞✉♥❣ ✤÷ñ❝ t❤❛♠ ❦❤↔♦
tr♦♥❣ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✳

✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ✈➲ ❣✐↔✐ t➼❝❤

✶✳✶✳✶ ❇→♥ ❦➼♥❤ ❤ë✐ tö ✈➔ ❦❤♦↔♥❣ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ❧ô② t❤ø❛
❈❤✉é✐ ❧ô② t❤ø❛ ❧➔ ♠ët ❝❤✉é✐ ❤➔♠ ❝â ❞↕♥❣

∞

an (x − a)n = a0 + a1 (x − a) + . . . + an (x − a)n + . . .
n=0
tr♦♥❣ ✤â

a, an (n = 0, 1, 2, ...)

❧➔ ❝→❝ ❤➡♥❣ sè✳

❈❤✉é✐ ❧ô② t❤ø❛ ❧✉æ♥ ❧✉æ♥ ❤ë✐ tö t↕✐ ✤✐➸♠
◆➳✉ ♥❣♦➔✐ ✤✐➸♠

x=a

x = a✳

❝❤✉é✐ ♣❤➙♥ ❦ý t❤➻ t❛ ♥â✐ ❝❤✉é✐ ❧ô② t❤ø❛ ♣❤➙♥

❦ý ❦❤➢♣ ♥ì✐✳
◆➳✉ ❝❤✉é✐ ❧ô② t❤ø❛ ❦❤æ♥❣ ♣❤↔✐ ♣❤➙♥ ❦ý ❦❤➢♣ ♥ì✐ t❤➻ tç♥ t↕✐ sè
s❛♦ ❝❤♦ tr♦♥❣ ❦❤♦↔♥❣

(a − R, a + R)
✸

R>0

❝❤✉é✐ ❤ë✐ tö✱ ❝á♥ ♥❣♦➔✐ ✤♦↕♥


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
[a − R, a + R]
❑❤♦↔♥❣

❝❤✉é✐ ♣❤➙♥ ❦ý✳

(a − R, a + R)

✤÷ñ❝ ❣å✐ ❧➔ ❦❤♦↔♥❣ ❤ë✐ tö❀ sè ❞÷ì♥❣

R

✤÷ñ❝

❣å✐ ❧➔ ❜→♥ ❦➼♥❤ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ❧ô② t❤ø❛✳
❇→♥ ❦➼♥❤ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ❧ô② t❤ø❛ ✤÷ñ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝
❈❛✉❝❤② ✲ ❍❛❞❛♠❛r❞

1
= lim n |an |.
R
✣➦❝ ❜✐➺t ❜→♥ ❦➼♥❤ ❤ë✐ tö ❝á♥ ✤÷ñ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝→❝ ❝æ♥❣ t❤ù❝

R = lim

n→+∞

R = lim

an
an+1
1

n→+∞

n

|an |

♥➳✉ ❝→❝ ❣✐î✐ ❤↕♥ tr➯♥ tç♥ t↕✐✳

✶✳✶✳✷ ❈→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝❤✉é✐ ❧ô② t❤ø❛
❛✳ ❚ê♥❣ ❝õ❛ ❝❤✉é✐ ❧ô② t❤ø❛ ❧➔ ♠ët ❤➔♠ ❧✐➯♥ tö❝✱ ❤ì♥ ♥ú❛ ❧➔ ♠ët ❤➔♠
❦❤↔ ✈✐ ✈æ ❤↕♥ tr♦♥❣ ❦❤♦↔♥❣ ❤ë✐ tö ❝õ❛ ♥â ✈➔ t❛ ❝â t❤➸ ✤↕♦ ❤➔♠ tø♥❣
sè ❤↕♥❣ ❝õ❛ ❝❤✉é✐

∞

∞
n

an (x − a)

nan (x − a)n−1 .

=
n=1

n=0

❜✳ ❈â t❤➸ ❧➜② t➼❝❤ ♣❤➙♥ tø♥❣ sè ❤↕♥❣ ❝õ❛ ❝❤✉é✐ ❧ô② t❤ø❛ tr➯♥ ✤♦↕♥

[α, β] ❜➜t

♠å✐

❦ý ♥➡♠ ❤♦➔♥ t♦➔♥ tr♦♥❣ ❦❤♦↔♥❣ ❤ë✐ tö ❝õ❛ ♥â✳ ❍ì♥ ♥ú❛ ✈î✐

x ∈ (a − R, a + R)

t❛ ❝â

✹


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
x

∞

∞
n

an (t − a) dt =
n=0

a

n=0

an
(x − a)n+1 .

n+1

✶✳✶✳✸ ❑❤❛✐ tr✐➸♥ ❤➔♠ t❤➔♥❤ ❝❤✉é✐ ❧ô② t❤ø❛
❛✳ ◆➳✉ ❤➔♠

f (x)

❦❤↔ ✈✐ ✈æ ❤↕♥ tr♦♥❣ ❦❤♦↔♥❣

(a − R, a + R)

❝â t❤➸

❦❤❛✐ tr✐➸♥ ✤÷ñ❝ t❤➔♥❤ ❝❤✉é✐ ❧ô② t❤ø❛ tr♦♥❣ ❦❤♦↔♥❣ ✤â t❤➻ ❝❤✉é✐ ❧ô②
t❤ø❛ ♥➔② ❝❤➼♥❤ ❧➔ ❝❤✉é✐ ❚❛②❧♦r ❝õ❛ ❤➔♠

∞

f (x) =
n=0
◆➳✉

a=0

1 (n)
f (a) .(x − a)n , x ∈ (a − R, a + R) .
n!

t❤➻ ❝❤✉é✐ ❚❛②❧♦r ✤÷ñ❝ ❣å✐ ❧➔ ❝❤✉é✐ ▼❛❝❧❛✉r✐♥

∞

f (x) =
n=0
❜✳ ❍➔♠

f (x)

f (x)

f (n) (0) n
x , x ∈ (−R, R) .
n!

❦❤↔ ✈✐ ✈æ ❤↕♥ tr♦♥❣

δ

✲ ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠

x=a

❝â t❤➸

❦❤❛✐ tr✐➸♥ ✤÷ñ❝ t❤➔♥❤ ❝❤✉é✐ ❚❛②❧♦r tr♦♥❣ ❧➙♥ ❝➟♥ ✤â ♥➳✉ tç♥ t↕✐ ♠ët
sè

M >0

s❛♦ ❝❤♦

f (n) (x) ≤ M, n = 0, 1, 2, . . . ; ∀x ∈ (a − δ, a + δ) .
❝✳ ❈→❝ ❦❤❛✐ tr✐➸♥ ❧ô② t❤ø❛ ❝ì ❜↔♥

xn
x2
+ ... +
+ . . . , x ∈ (−∞, +∞)
• e =1+x+
2!
n!
x

• sinx = x −

x3 x5
x2n−1
+ − . . . + (−1)n−1
+ . . . , x ∈ (−∞, +∞)
3! 5!
(2n − 1)!

2n
x2 x4
n x
• cosx = 1 −
+
− . . . + (−1)
+ . . . , x ∈ (−∞, +∞)
2!

4!
(2n)!

✺


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

n
x2 x3
n−1 x
• ln (1 + x) = x −
+
− . . . + (−1)
+ . . . , −1 < x ≤ 1
2
3
n

• (1 + x)m = 1+mx+

m (m − 1) . . . (m − n + 1) n
m (m − 1) 2
x +. . .+
x
2!
n!

+ . . . , −1 < x < 1

✶✳✷ ▼ët sè ❦✐➳♥ t❤ù❝ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
t❤÷í♥❣

✶✳✷✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠
P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❧➔ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❜✐➳♥ ✤ë❝ ❧➟♣
❝➛♥ t➻♠

y = f (x)

x✱

❤➔♠

✈➔ ❝→❝ ✤↕♦ ❤➔♠ ❝→❝ ❝➜♣ ❝õ❛ ♥â✳

◆â✐ ❝→❝❤ ❦❤→❝✱ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ✤↕♦ ❤➔♠ ❤♦➦❝ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠
❝➛♥ t➻♠ ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳
P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❝â ❞↕♥❣

F (x, y, y , y , . . . , y (n) ) = 0
tr♦♥❣ ✤â

x

❧➔ ❜✐➳♥ ✤ë❝ ❧➟♣❀

y

❧➔ ❤➔♠ ❝➛♥ t➻♠ ✈➔ ♥❤➜t t❤✐➳t ♣❤↔✐ ❝â

✤↕♦ ❤➔♠ ✭✤➳♥ ❝➜♣ ♥➔♦ ✤â✮ ❝õ❛ ➞♥
❤➔♠ sè

y ✭y

❧➔ ❤➔♠ sè ❝õ❛

✭✶✳✷✳✶✮

y ❀ y , y , . . . , y (n)

❧➔ ❝→❝ ✤↕♦ ❤➔♠ ❝õ❛

x✮✳

❈➜♣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ ✤↕♦ ❤➔♠ ❝➜♣ ❝❛♦ ♥❤➜t ❝â ♠➦t tr♦♥❣ ♣❤÷ì♥❣
tr➻♥❤✳
❍➔♠ sè
t❤❛②

y = ϕ(x)

✤÷ñ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

y = ϕ(x)✱ y = ϕ (x), . . . , y (n) = ϕ(n) (x)

t❤➻ ♣❤÷ì♥❣ tr➻♥❤
❍➔♠ sè

(1.2.1)

(1.2.1)

✈➔♦ ♣❤÷ì♥❣ tr➻♥❤

♥➳✉

(1.2.1)

trð t❤➔♥❤ ✤ç♥❣ ♥❤➜t t❤ù❝✳

y = ϕ(x, c) (c ∈ R)

❝â ✤↕♦ ❤➔♠ t❤❡♦ ❜✐➳♥

✻

x

✤➳♥ ❝➜♣

n

✤÷ñ❝


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
❣å✐ ❧➔ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

✰ ▼å✐

(x, y) ∈ D ✭D

❣✐↔✐ r❛ ✤è✐ ✈î✐

✰ ❍➔♠
❦❤➢♣

♥➳✉

❧➔ ♠✐➲♥ ①→❝ ✤à♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✮ t❛ ❝â t❤➸

c✿ c = ψ(x, y)✳

y = ϕ(x, c)
D✱

(1.2.1)

✈î✐ ♠å✐

t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤

(1.2.1)

❦❤✐

(x, y)

❝❤↕②

c ∈ R✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ n
P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ n ❝â ❞↕♥❣

F (x, y, y , y , . . . , y (n) ) = 0
❤❛②

y (n) = f x, y, y , . . . , y (n−1)
tr♦♥❣ ✤â x ❧➔ ❜✐➳♥ ✤ë❝ ❧➟♣❀ y ❧➔ ❤➔♠ ❝➛♥ t➻♠❀ y , y , . . . , y (n) ❧➔ ❝→❝ ✤↕♦
❤➔♠ ❝õ❛ ❤➔♠ sè y ✭y ❧➔ ❤➔♠ sè ❝õ❛ x✮✳

✶✳✷✳✷ ▼ët sè ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët ❣✐↔✐ ✤÷ñ❝ ❜➡♥❣
❝➛✉ ♣❤÷ì♥❣
❛✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝â ❜✐➳♥ sè ♣❤➙♥ ❧②

✰ P❤÷ì♥❣ tr➻♥❤

✰ P❤÷ì♥❣ tr➻♥❤

✰ P❤÷ì♥❣ tr➻♥❤

⇔

dy

= f (x) ❝â ♥❣❤✐➺♠ y = f (x)dx + c✳
dx
dy
dy
= f (y) ❝â ♥❣❤✐➺♠
= x + c✳
dx
f (y)
M1 (x).N1 (y)dx + M2 (x).N2 (y)dy = 0

M1 (x)
N2 (y)
dx +
dy = 0 ((N1 (y).M2 (x) = 0)
M2 (x)
N1 (y)
✼


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
❜✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët t❤✉➛♥ ♥❤➜t
P❤÷ì♥❣ tr➻♥❤

dy
= f (x, y)
dx

❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ♥➳✉

f (tx, ty) = tk .f (x, y) (t > 0)
✣➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② t❛ ✤➦t

u=

y
x

s❛✉ ✤â ✤÷❛ ✈➲ ✈✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣

tr➻♥❤ ✈✐ ♣❤➙♥ ❝â ❜✐➳♥ sè ♣❤➙♥ ❧②✳
❝✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤÷❛ ✤÷ñ❝ ✈➲ ❞↕♥❣ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t
❝➜♣ ♠ët

dy
=f
dx
✰ ◆➳✉

✰ ◆➳✉

c = c1 = 0

t❤➻

c = 0 ✱ c1 = 0 ✱

ax + by + c
a1 x + b1 y + c1

(1.2.2)
a

✭✶✳✷✳✷✮

❧➔ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❝➜♣ ♠ët✳

b

=0

t❤➻ ✤➦t

a1 b 1


 x=x +α
1
 y =y +β

✈î✐

α, β

1

❧➔ ❤➡♥❣ sè✳

❑❤✐ ✤â t❛ ✤÷ñ❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤✉➛♥ ♥❤➜t ❝➜♣ ♠ët ✤è✐ ✈î✐

x1 ✱

y1 ✳
❞✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ♠ët
❉↕♥❣ tê♥❣ q✉→t

dy
+ P (x).y = Q(x)
dx
✰ ◆➳✉

Q(x) = 0

t❤➻ ♣❤÷ì♥❣ tr➻♥❤

(1.2.3)

✭✶✳✷✳✸✮

❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥

t➼♥❤ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t ❝➜♣ ♠ët✳

✰ ◆➳✉

Q(x) = 0

t❤➻ ♣❤÷ì♥❣ tr➻♥❤

(1.2.3)

❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥

t➼♥❤ t❤✉➛♥ ♥❤➜t ❝➜♣ ♠ët✳

✰ ❈æ♥❣ t❤ù❝ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

✽


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
y = e−

P (x)dx

.

Q(x).e

P (x)dx

dx + c

❡✳ P❤÷ì♥❣ tr➻♥❤ ❇❡❝♥✉❧❧✐
❉↕♥❣ tê♥❣ q✉→t

dy

+ P (x).y = Q(x).y α
dx
✰ ◆➳✉

α=1

t❤➻ ♣❤÷ì♥❣ tr➻♥❤

(1.2.4)

✭✶✳✷✳✹✮

❧➔ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t

❝➜♣ ♠ët✳

✰ ◆➳✉

α=0

t❤➻ ♣❤÷ì♥❣ tr➻♥❤

(1.2.4)

❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❦❤æ♥❣ t❤✉➛♥

♥❤➜t ❝➜♣ ♠ët✳

✰ ◆➳✉

α = 0✱ α = 1

✤â ✤➦t

z = y 1−α

t❤➻ t❛ ❝❤✐❛ ❝↔ ❤❛✐ ✈➳ ❝õ❛

(1.2.4)

❝❤♦

yα

s❛✉

✈➔ ✤÷❛ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ t❤✉➛♥

♥❤➜t✳

✶✳✷✳✸ ▼ët sè t➼♥❤ ❝❤➜t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
t✉②➳♥ t➼♥❤
❳➨t ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t ❝➜♣

n

y (n) + p1 (x).y (n−1) + . . . + pn (x).y = 0.
✣➸ ✤ì♥ ❣✐↔♥ ❝→❝❤ ✈✐➳t ✈➲ s❛✉✱ t❛ ❦þ ❤✐➺✉

L(y) = y (n) + p1 (x).y (n−1) + . . . + pn (x).y

✾

✭✶✳✷✳✺✮


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

L(y) ✤÷ñ❝ ❣å✐ ❧➔ t♦→♥ tû ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤✳ ❚ø ❦þ ❤✐➺✉ tr➯♥✱

tr♦♥❣ ✤â

♣❤÷ì♥❣ tr➻♥❤

(1.2.5)

✤÷ñ❝ ✈✐➳t ❞÷î✐ ❞↕♥❣

L(y) = 0.
❚♦→♥ tû

L(y)

✰ ✣è✐ ✈î✐

✭✶✳✷✳✻✮

❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉

y1 (x)✱ y2 (x)

❦❤↔ ✈✐

n

❧➛♥ ❧✐➯♥ tö❝ t❛ ❝â

L(y1 + y2 ) = L (y1 ) + L (y2 )
✰ ✣è✐ ✈î✐ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝

n

❧➛♥

y(x)

✈➔ ❤➡♥❣ sè

c

❜➜t ❦ý t❛ ❝â

L(cy) = cL(y)

❉ü❛ ✈➔♦ t➼♥❤ ❝❤➜t ❝õ❛ t♦→♥ tû

L t❛ s✉② r❛ ❝→❝ t➼♥❤ ❝❤➜t ✈➲ t➟♣ ♥❣❤✐➺♠

❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t ❝➜♣

✰ ◆➳✉

y(x)

❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

n

(1.2.5)

t❤➻

❤➡♥❣ sè tò② þ ❝ô♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

✰ ◆➳✉
t❤➻

y1 (x)✱ y2 (x)

c.y(x)

✈î✐

c

❧➔

(1.2.5)✳

❧➔ ❤❛✐ ♥❣❤✐➺♠ ❜➜t ❦ý ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

(1.2.5)

y(x) = y1 (x) + y2 (x) ❝ô♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (1.2.5)✳

✰ ◆➳✉

y1 (x), y2 (x), . . . , yk (x) ❧➔ ❝→❝ ♥❣❤✐➺♠ ❜➜t ❦ý ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

(1.2.5) t❤➻ y(x) = c1 y1 (x) + c2 y2 (x) + . . . + ck yk (x) ❝ô♥❣ ❧➔ ♥❣❤✐➺♠
❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

(1.2.5)✳

✶✵




õ tốt ồ

t
sỷ tứ ữỡ tr

(1.2.1)

t r ữủ ữỡ tr ố ợ

t

y (n) = f x, y, y , . . . , y (n1)
tr õ

f

tr



D Rn+1

(x0 , y0 , y0 , . . . , y0 (n1) ) D

sỷ
õ t



y (n) = f x, y, y , . . . , y (n1) , x, y, y , . . . , y (n1) D

y (x0 ) = y0 , y (x0 ) = y , . . . , y (n1) (x0 ) = y (n1)
0
0
ồ t t tr


(n1)

y (x0 ) = y0 , y (x0 ) = y 0 , . . . , y (n1) (x0 ) = y0

ồ



ỵ tỗ t t ừ t

sỷ tr

G Rn+1

st t

(n1)

(x0 , y0 , y0 , . . . , y0
ữỡ tr

f (x, u1 , u2 , . . . , un )

tỗ t t

tọ

(n1)

y0 , . . . , y (n1) (x0 ) = y0

tử tọ

u1 u2 , . . . , un õ ợ t ý tr

) G

(1.1.7)



y = y(x)

y(x0 ) = y0 , y (x0 ) =



t õ



ừ

(x0 h, x0 + h)

ừ

x0


❈❤÷ì♥❣ ✷
P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥

❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐
♣❤➙♥ t❤÷í♥❣
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ t❛ tr➻♥❤ ❜➔② ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❆❞♦♠✐❛♥ ✈➔
♣❤÷ì♥❣ ♣❤→♣ ❆❞♦♠✐❛♥ ❝↔✐ ❜✐➯♥ ✤➸ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ❈❛✉❝❤② ✤è✐ ✈î✐
♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷ñ❝ t❤❛♠
❦❤↔♦ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✹❪✱ ❬✺❪✳

✷✳✶ P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥
P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❜✐➸✉ ❞✐➵♥ ♥❣❤✐➺♠ ❞÷î✐ ❞↕♥❣ ❝❤✉é✐
✈æ ❤↕♥✱ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❝õ❛ ❝❤✉é✐ ❝â t❤➸ ✤÷ñ❝ ①→❝ ✤à♥❤ ♠ët ❝→❝❤ ❞➵
❞➔♥❣ t❤æ♥❣ q✉❛ q✉❛♥ ❤➺ tr✉② ❤ç✐✳ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ❝❤➾ ①→❝ ✤à♥❤ ✤÷ñ❝
❤ú✉ ❤↕♥ t❤➔♥❤ ♣❤➛♥ t❤➻ t❛ t❤✉ ✤÷ñ❝ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤✳ ❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ①➨t ❞↕♥❣ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ ✈✐ ♣❤➙♥ ♣❤✐ t✉②➳♥ ♥❤÷ s❛✉

✶✷


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

Fy = f
tr♦♥❣ ✤â

F

❧➔ ♠ët ❤➔♠ ♣❤✐ t✉②➳♥✱

❚❛ ✈✐➳t ❧↕✐ ♣❤÷ì♥❣ tr➻♥❤

(2.1.1)

y

✈➔

✭✷✳✶✳✶✮

f

❧➔ ❝→❝ ❤➔♠ sè ❝õ❛

x✳

❞÷î✐ ❞↕♥❣

Ly + Ry + N y = f

✭✷✳✶✳✷✮

tr♦♥❣ ✤â

✰

L

❧➔ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤ ❦❤↔ ♥❣❤à❝❤✱ ❧➔ ♠ët ♣❤➛♥ ❤♦➦❝ t♦➔♥

❜ë ♣❤➛♥ t✉②➳♥ t➼♥❤ ❝õ❛

F

✰

R

❧➔ ♣❤➛♥ t✉②➳♥ t➼♥❤ ❝á♥ ❧↕✐ ❝õ❛ t♦→♥ tû

✰

N

❧➔ ♣❤➛♥ ♣❤✐ t✉②➳♥ t➼♥❤ ❝õ❛

❚→❝ ✤ë♥❣ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦

L−1

F

F

✈➔♦ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

(2.1.2)

t❛ ❝â

L−1 Ly = L−1 f − L−1 Ry − L−1 N y.

❱➼ ❞ö✱ ♥➳✉

L

❧➔ ♠ët t♦→♥ tû ✈✐ ♣❤➙♥ ❜➟❝ ♠ët t❤➻ t♦→♥ tû

✭✷✳✶✳✸✮

L−1

❧➔ t♦→♥

tû t➼❝❤ ♣❤➙♥ ✤÷ñ❝ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝

x

L−1 =

(.)dx.

✭✷✳✶✳✹✮

0
❉♦ ✤â

L−1

❧➔ ♠ët t♦→♥ tû t➼❝❤ ♣❤➙♥ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐

x

L−1 (Ly) =

x

y dx = y (x) − y (0) .

Lydx =
0

0

✶✸

✭✷✳✶✳✺✮


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
❚❤❛② ♣❤÷ì♥❣ tr➻♥❤

(2.1.5)

(2.1.3)

✈➔♦ ♣❤÷ì♥❣ tr➻♥❤

✈➔ ❝❤✉②➸♥ ✈➳ t❛

✤÷ñ❝

y (x) = y (0) + L−1 f − L−1 Ry − L−1 N y.

✭✷✳✶✳✻✮

✣➦t

y0 = y (0) + L−1 f
❑❤✐ ✤â t❤❛②

y0

✈➔♦ ♣❤÷ì♥❣ tr➻♥❤

(2.1.6)

t❛ ❝â

y (x) = y0 − L−1 Ry − L−1 N y.
❚÷ì♥❣ tü✱ ♥➳✉

L

✭✷✳✶✳✼✮

❧➔ ♠ët t♦→♥ tû ✈✐ ♣❤➙♥ ❜➟❝ ❤❛✐ t❤➻ t♦→♥ tû

L−1

❧➔

t♦→♥ tû t➼❝❤ ♣❤➙♥ ❤❛✐ ❧î♣ ✤÷ñ❝ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝

L−1 =
❉♦ ✤â

L−1

(.)dx1 dx2 .

✭✷✳✶✳✽✮

❧➔ ♠ët t♦→♥ tû t➼❝❤ ♣❤➙♥ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐

x

x

x

L−1 (Ly) =

x

(Ly (x))dxdx =
0

0

y (x)dxdx.
0

✭✷✳✶✳✾✮

0

❚ø ✤â t❛ t➼♥❤ ✤÷ñ❝

L−1 Ly = y(x) − y(0) − xy (0).
❑❤✐ ✤â t❤❛② ♣❤÷ì♥❣ tr➻♥❤

✭✷✳✶✳✶✵✮

(2.1.10) ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ (2.1.3) ✈➔ ❝❤✉②➸♥

✈➳ t❛ ❝â

y(x) = y(0) + xy (0) + L−1 (f ) − L−1 (Ry) − L−1 (N y) .

✶✹

✭✷✳✶✳✶✶✮


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
✣➦t

y0 = y(0) + xy (0) + L−1 (f )
❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤

(2.1.11)

trð t❤➔♥❤

y(x) = y0 − L−1 (Ry) + L−1 (N y) .
❈❤♦

y(x)

✭✷✳✶✳✶✷✮

✤÷ñ❝ ❜✐➸✉ ❞✐➵♥ ❜ð✐ ❝❤✉é✐

∞

y(x) =

yn

✭✷✳✶✳✶✸✮

n=0
✈➔ sè ❤↕♥❣ ♣❤✐ t✉②➳♥

N (y)

A0 ✱ A1 ✱ A2 ✱. . .

❆❞♦♠✐❛♥

❧➔ ♠ët ❝❤✉é✐ ✈æ t➟♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤❛ t❤ù❝

♥❤÷ s❛✉

∞

N (y) =

An

✭✷✳✶✳✶✹✮

n=0
tr♦♥❣ ✤â

An

✤÷ñ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝

∞

1 dn
An =
N
n! dλn
❚❤❛② ❝→❝ ♣❤÷ì♥❣ tr➻♥❤

λi yi

, n = 0, 1, 2, . . .

i=0

✭✷✳✶✳✶✺✮

λ=0

(2.1.13)✱ (2.1.14)

✈➔♦ ♣❤÷ì♥❣ tr➻♥❤

(2.1.12)

t❛

✤÷ñ❝

∞

∞

yn = y0 − L
n=0

−1

R

∞

yn
n=0

−1

−L

An .
n=0

◆❤÷ ✈➟② t❛ ❝â

y0 = y (0) + xy (0) + L−1 (f ) ,

✶✺

✭✷✳✶✳✶✻✮


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
y1 = −L−1 (R (y0 )) − L−1 (A0 ) ,
y2 = −L−1 (R (y1 )) − L−1 (A1 ) ,
✳✳
✳

yn+1 = −L−1 (R (yn )) − L−1 (An ) .
●✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② t❛ s➩ t➻♠ ✤÷ñ❝

∞

❑❤✐ ✤â

yk (x)

y(x) =

❈❤ó þ✿

y0 (x)✱ y1 (x)✱ y2 (x)✱. . .

❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥✳

k=0

N (y)

◆➳✉

❧➔ ♠ët t♦→♥ tû ♣❤✐ t✉②➳♥ t❤➻ ❝→❝ ✤❛ t❤ù❝ ❆❞♦♠✐❛♥

❝â t❤➸ ✤÷ñ❝ t➼♥❤ ♥❤÷ s❛✉

A0 = N (y0 ) ,
A1 = y1 N (y0 ) ,
A2 = y2 N (y0 ) +

1 2
y N (y0 ) ,
2! 1

A3 = y3 N (y0 ) + y1 y2 N (y0 ) +

1 3
y N (y0 ) ,
3! 1

✳✳
✳

P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ✤➸ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
✤÷ñ❝ ♠✐♥❤ ❤å❛ ❜➡♥❣ ❝→❝ ✈➼ ❞ö s❛✉✳

❱➼ ❞ö ✷✳✶✳✶✳ ❙û ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❣✐↔✐ ♣❤÷ì♥❣
tr➻♥❤ ✈✐ ♣❤➙♥

dy
+ y3 = 3
dx
✈î✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ y(0) = 0✳

✶✻

✭✷✳✶✳✶✼✮

✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
●✐↔✐
❚❛ ✤➦t

Ly =
❚ø ♣❤÷ì♥❣ tr➻♥❤

dy
, N y = y 3 , f = 3.
dx

✭✷✳✶✳✶✽✮

(2.1.6)

y(x) = y(0) + L−1 (f ) − L−1 (Ry) − L−1 (N y)
t❤❛②

f, N y

✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ t❛ ❝â

x

3dx − L−1 y 3 = y(0) + 3x − L−1 y 3 .

y(x) = y(0) +

✭✷✳✶✳✶✾✮

0
∞
❱î✐

y(x) = y0 +

yk (x)

❧➔ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ t❤➻ tø

k=1
♣❤÷ì♥❣ tr➻♥❤

(2.1.19)

✣➸ t➻♠ r❛ ❝→❝

s✉② r❛

y0 = 3x✳

yk (x) , k = 1, 2, . . .✱

tr÷î❝ t✐➯♥ t❛ ♣❤↔✐ ①→❝ ✤à♥❤ ✤÷ñ❝

❝→❝ ✤❛ t❤ù❝ ❆❞♦♠✐❛♥

A0 = N (y0 ) = (3x)3 = 27x3

❚ø ✤â t❛ t➼♥❤ ✤÷ñ❝

y1 = −L−1 (R (y0 )) − L−1 (A0 ) = −L−1 (A0 )
x

=−

x

27x3 dx =

A0 dx = −
0

0

❙✉② r❛ ✤❛ t❤ù❝ ❆❞♦♠✐❛♥

✶✼

−27 4
x
4


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
A1 = y1 N (y0 ) = y1 .3y02 =

−27 4
−729 6
.x .3.(3x)2 =
x
4
4

❚ø ✤â t❛ t➼♥❤ ✤÷ñ❝

x

y2 = −L−1 (R (y1 ))−L−1 (A1 ) = −L−1 (A1 ) = −

729 7
−729 6
x dx =
x
4
28

0
❱➟② t❛ ❝â ♥❣❤✐➺♠ ①➜♣ ①➾ ❝õ❛ ❜➔✐ t♦→♥

n

yk (x) = 3x −

y(x) =
k=0

27 4 729 7
x +
x + ...
4
28

❱➼ ❞ö ✷✳✶✳✷✳ ❙û ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❣✐↔✐ ♣❤÷ì♥❣
tr➻♥❤ ✈✐ ♣❤➙♥

dy
= −y 2 + 1
dx

✭✷✳✶✳✷✵✮

✈î✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ y(0) = 0✳
●✐↔✐
❚❛ ✤➦t

Ly =
❙û ❞ö♥❣ ♣❤÷ì♥❣ tr➻♥❤

dy
, N y = y 2 , f = 1.
dx

(2.1.6)

y(x) = y(0) + L−1 (f ) − L−1 (Ry) − L−1 (N y)
t❤❛②

f, N y

✈➔♦ t❛ ❝â

x

dx − L−1 y 2 = y(0) + x − L−1 y 2 .

y(x) = y(0) +
0

✶✽

✭✷✳✶✳✷✶✮


✣➱ ❚❍➚ ❚❍❯ ❍⑨

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
n
❱î✐

yk (x)

y(x) = y0 +

❧➔ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ t❤➻ tø

k=1

♣❤÷ì♥❣ tr➻♥❤

(2.1.21)

✣➸ t➻♠ ①→❝ ✤à♥❤ ❝→❝

s✉② r❛

y0 = x ✳

yk (x) , k = 1, 2, . . .✱

tr÷î❝ t✐➯♥ t❛ ♣❤↔✐ ①→❝ ✤à♥❤

✤÷ñ❝ ❝→❝ ✤❛ t❤ù❝ ❆❞♦♠✐❛♥

A0 = N (y0 ) = x2
❚ø ✤â t❛ t➼♥❤ ✤÷ñ❝

x

y1 = −L

−1

(A0 ) = −

x

A0 dx = −

0

x3
x dx = −
3
2

0

❙✉② r❛ ✤❛ t❤ù❝ ❆❞♦♠✐❛♥

−x3
2
A1 = y1 N (y0 ) = y1 .2y0 =
.2x = − x4
3
3
❚ø ✤â t❛ t➼♥❤ ✤÷ñ❝

x

2
2
− x4 dx = x5
3
15

y2 = −L−1 (A1 ) = −
0
❙✉② r❛ ✤❛ t❤ù❝ ❆❞♦♠✐❛♥

1
2
1 −x3
A2 = y2 N (y0 ) + y12 N (y0 ) = x5 .2x + .
2!
15
2
3
❚ø ✤â t❛ t➼♥❤ ✤÷ñ❝

x

y3 = −L−1 (A2 ) = −
0

✶✾

17 6
−17 7
x dx =
x
45
315

2

.2 =

17 6

x
45