TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* ĐỖ THỊ THU HÀ PHƯƠNG PHÁP PHÂN TÍCH ADOMIAN GIẢI GẦN ĐÚNG CÁC PHƯƠNG TRÌNH VI PHÂN THƯỜNG KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Giải tích HÀ NỘI – 2018 TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI 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 ▲í✐ ❝↔♠ ì♥ ❊♠ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❱➠♥ P t ữớ trỹ t t t ữợ ữợ tr sốt q tr õ ỗ t❤í✐ ❡♠ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝æ tr♦♥❣ tê ●✐↔✐ t➼❝❤ ✈➔ ❝→❝ t❤➛② ❝æ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ❤♦➔♥ t❤➔♥❤ tèt ❜➔✐ ❦❤â❛ ❧✉➟♥ ♥➔② ✤➸ ❝â ❦➳t q✉↔ ♥❤÷ ♥❣➔② ❤ỉ♠ ♥❛②✳ ▼➦❝ ❞ò ✤➣ ❝â r➜t ♥❤✐➲✉ ❝è ❣➢♥❣✱ s♦♥❣ t❤í✐ ❣✐❛♥ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ❜↔♥ t❤➙♥ ❝á♥ ♥❤✐➲✉ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❦❤ỉ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât r➜t ♠♦♥❣ ✤÷đ❝ sỹ õ õ ỵ t ổ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ ✈➔ ❜↕♥ ✤å❝✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ ✣é ❚❤à ❚❤✉ ❍➔ ▲í✐ ❝❛♠ ✤♦❛♥ ❊♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ r ữợ sỹ ữợ t 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❤➙♥✱ ❦❤ỉ♥❣ trò♥❣ ❧➦♣ ✈ỵ✐ ❦➳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✐ tư ✈➔ ❦❤♦↔♥❣ ❤ë✐ tư ❝õ❛ ❝❤✉é✐ ❧ơ② t❤ø❛ ✶✳✷ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✷ ❈→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝❤✉é✐ ❧ơ② t❤ø❛ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✸ ❑❤❛✐ tr✐➸♥ ❤➔♠ t❤➔♥❤ ❝❤✉é✐ ❧ô② t❤ø❛ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ▼ët sè ❦✐➳♥ t❤ù❝ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✳ ✳ ✳ ✻ ✶✳✷✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✷ ▼ët sè ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët ❣✐↔✐ ✤÷đ❝ ❜➡♥❣ ❝➛✉ ♣❤÷ì♥❣ ✶✳✷✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ▼ët sè t➼♥❤ ❝❤➜t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✹ ❇➔✐ t♦→♥ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tỗ t t t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✶ ✶✶ ✷ P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✶✷ ✐✐ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✷✳✶ P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ✷✳✷ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❘✐❝❝❛t✐ ✷✳✸ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ữỡ tr ợ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ s✉② ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❙ü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ✷✾ ✸✺ ✹✻ ✸✳✶ ❙ü ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✷ ❈→❝ ✈➼ ❞ö ✹✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ✺✼ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✺✽ ✐✐✐ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲í✐ ♠ð ✤➛✉ ✶✳ ỵ t ởt ổ ❦❤♦❛ ❤å❝ tü ♥❤✐➯♥ ❣➢♥ ❧✐➲♥ ✈ỵ✐ t❤ü❝ t✐➵♥✳ ❙ü ♣❤→t tr✐➸♥ ❝õ❛ t♦→♥ ❤å❝ ✤÷đ❝ ✤→♥❤ ❞➜✉ ❜ð✐ ♥❤ú♥❣ ù♥❣ ❞ö♥❣ ❝õ❛ t♦→♥ ❤å❝ ✈➔♦ ✈✐➺❝ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ t❤ü❝ t✐➵♥✳ ❚r♦♥❣ t❤ü❝ t✐➵♥ ♥❤✐➲✉ ❜➔✐ t♦→♥ ❝õ❛ ❦❤♦❛ ❤å❝✱ ❦ÿ t❤✉➟t ✈➔ ♠ỉ✐ 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 số ọ ợ ữỡ tr tữớ ữủ ❤å❝✿ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t→❝❤ ❜✐➳♥✱ ♣❤÷ì♥❣ tr➻♥❤ ❇❡❝♥♦✉❧❧✐✱ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t✱ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥✱ ❝á♥ ❧↕✐ ♥â✐ ❝❤✉♥❣ ❦❤ỉ♥❣ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ♠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✉ ❤✐➺✉ ❞➵ →♣ ❞ư♥❣ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✳ ❱ỵ✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ự s ỡ ữợ P t ◆✐♥❤ P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝ ữỡ tr tữớ sỹ ữợ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐✿ ✏ ✑ ✤➸ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❝õ❛ ♠➻♥❤✳ ✶ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ❑❤â❛ ❧✉➟♥ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✳ ✸✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ ◆❣❤✐➯♥ ❝ù✉ ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ♣❤✐ t✉②➳♥✳ ✹✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët✱ ❝➜♣ ữỡ tr ợ trữợ Pữỡ ự ❙÷✉ t➛♠✱ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➔✐ ❧✐➺✉ ❧✐➯♥ q✉❛♥✳ ❱➟♥ ởt số ữỡ t ỵ t❤✉②➳t ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ P❤➙♥ t➼❝❤✱ tê♥❣ ❤đ♣ ✈➔ ❤➺ t❤è♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥✳ ✻✳ ❈➜✉ tró❝ t õ ỗ ữỡ ữỡ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ✷✿ P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❈❤÷ì♥❣ ✸✿ ❙ü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ✷ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ t❛ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✈➔ ❝❤✉é✐ ❧ơ② t❤ø❛✳ ◆ë✐ ❞✉♥❣ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✳ ✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ✈➲ ❣✐↔✐ t➼❝❤ ✶✳✶✳✶ ❇→♥ ❦➼♥❤ ❤ë✐ tư ✈➔ ❦❤♦↔♥❣ ❤ë✐ tư ❝õ❛ ❝❤✉é✐ ❧ơ② 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❛ ❧✉ỉ♥ ❧✉ỉ♥ ❤ë✐ tư t↕✐ ✤✐➸♠ ◆➳✉ ♥❣♦➔✐ ✤✐➸♠ x=a x = a✳ ❝❤✉é✐ ♣❤➙♥ ❦ý t❤➻ t❛ ♥â✐ ❝❤✉é✐ ❧ơ② t❤ø❛ ♣❤➙♥ ❦ý ❦❤➢♣ ♥ì✐✳ ◆➳✉ ❝❤✉é✐ ❧ơ② tứ ổ ý ỡ t tỗ 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) ✤÷đ❝ ❣å✐ ❧➔ ❦❤♦↔♥❣ ❤ë✐ tư❀ sè ❞÷ì♥❣ R ✤÷đ❝ ❣å✐ ❧➔ ❜→♥ ❦➼♥❤ ❤ë✐ tư ❝õ❛ ❝❤✉é✐ ❧ơ② t❤ø❛✳ ❇→♥ ❦➼♥❤ ❤ë✐ tư ❝õ❛ ❝❤✉é✐ ❧ơ② t❤ø❛ ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ ❈❛✉❝❤② ✲ ❍❛❞❛♠❛r❞ = lim n |an | R ✣➦❝ ❜✐➺t ❜→♥ ❦➼♥❤ ❤ë✐ tư ❝á♥ ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝→❝ ❝æ♥❣ t❤ù❝ R = lim n→+∞ R = lim an an+1 n→+∞ n |an | ♥➳✉ ❝→❝ ❣✐ỵ✐ tr tỗ t t t ỡ ❝õ❛ ❝❤✉é✐ ❧ô② t❤ø❛ ❛✳ ❚ê♥❣ ❝õ❛ ❝❤✉é✐ ❧ô② t❤ø❛ ❧➔ ♠ët ❤➔♠ ❧✐➯♥ tư❝✱ ❤ì♥ ♥ú❛ ❧➔ ♠ët ❤➔♠ ❦❤↔ ✈✐ ✈ỉ ❤↕♥ tr♦♥❣ ❦❤♦↔♥❣ ❤ë✐ tư ❝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✐ tư ❝õ❛ ♥â✳ ❍ì♥ ♥ú❛ ✈ỵ✐ x ∈ (a − R, a + R) t❛ ❝â ✹ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❑❤✐ ✤â t❤❛② ♣❤÷ì♥❣ tr➻♥❤ (2.4.48) ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ (2.4.27) ✈➔ ❝❤✉②➸♥ ✈➳ t❛ ✤÷đ❝ y (x) = y (0) + L−1 + 12x + x2 + x3 − L−1 (y) ✭✷✳✹✳✹✾✮ tr♦♥❣ ✤â x x L−1 + 12x + x2 + x3 = x−1 x + 12x + x2 + x3 dxdx 0 x x4 x5 + dx 3x + 4x + −1 =x x5 x6 =x x +x + + 20 30 x4 x5 =x +x + + 20 30 −1 ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ (2.4.49) trð t❤➔♥❤ y (x) = x2 + x3 + x4 x5 + − L−1 (y) 20 30 ✭✷✳✹✳✺✵✮ ❚❛ ✤➦t y0 = x2 + x3 yk+1 ❚❤❛② k=0 x4 x5 = + − L−1 (Ak ) 20 30 ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ y1 = (2.4.52) ✭✷✳✹✳✺✷✮ t❛ ✤÷đ❝ x4 x5 + − L−1 (A0 ) 20 30 tr♦♥❣ ✤â A0 = N (y0 ) = y0 = x2 + x3 ✹✹ ✭✷✳✹✳✺✶✮ ✭✷✳✹✳✺✸✮ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❉♦ ✤â x x4 x5 + − x−1 y1 = 20 30 x x2 + x3 dxdx = ❚÷ì♥❣ tü t❤❛② x x x x x + − − =0 20 30 20 30 k=1 ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ (2.4.52) ✭✷✳✹✳✺✹✮ t❛ ❝â x4 x5 + − L−1 (A1 ) y2 = 20 30 x x4 x5 = + − x−1 20 30 ữỡ tỹ ợ k=2 x x4 x5 x.0dxdx = + 20 30 ✭✷✳✹✳✺✺✮ t❛ ❝â x4 x5 + − L−1 (A2 ) y3 = 20 30 ✭✷✳✹✳✺✻✮ tr♦♥❣ ✤â x4 x5 A2 = y2 N (y0 ) + y1 N (y0 ) = y2 N (y0 ) = + 2! 20 30 ❙✉② r❛ x x4 x5 y3 = + − x−1 20 30 x x4 x5 x + dxdx 20 30 x4 x5 x6 x7 + − − = 20 30 840 1680 ❱➟② t❛ ❝â ♥❣❤✐➺♠ ①➜♣ ①➾ ❝õ❛ ❜➔✐ t♦→♥ x4 x5 x6 x7 y (x) ≈ x + x + + − − 10 15 840 1680 ✹✺ ✭✷✳✹✳✺✼✮ ❈❤÷ì♥❣ ✸ ❙ü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ t❛ tr➻♥❤ ❜➔② ♠ët sè ✤✐➲✉ ❦✐➺♥ ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✸❪✳ ✸✳✶ ❙ü ❤ë✐ tư ❳➨t ♣❤÷ì♥❣ tr➻♥❤ t♦→♥ tû s❛✉ Ny = f tr♦♥❣ ✤â N ❧➔ t♦→♥ tû t✉②➳♥ t➼♥❤ tø ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❧➔ ♠ët ❤➔♠ sè ①→❝ ✤à♥❤ tr♦♥❣ ❚❛ ❝➛♥ t➻♠ y∈H t❤ä❛ ♠➣♥ ✭✸✳✶✳✶✮ H ✈➔♦ H✱ f H✳ (3.1.1)✳ ❱➔♦ ✤➛✉ ♥➠♠ ✶✾✽✵✱ ❆❞♦♠✐❛♥ ✤➣ ✤÷❛ r❛ ♠ët ❣✐↔✐ ♣❤→♣ ✤➸ ❣✐↔✐ q✉②➳t ✹✻ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✈➜♥ ✤➲ ♥➔②✳ ✣â ❧➔ ❜➡♥❣ ❝→❝❤ ❝♦✐ y ❧➔ tê♥❣ ❝õ❛ ♠ët ❝❤✉é✐ ∞ y= yi ✭✸✳✶✳✷✮ i=0 ✈➔ Ny ❧➔ tê♥❣ ❝õ❛ ❝❤✉é✐ ∞ Ny = An ✭✸✳✶✳✸✮ n=0 P❤÷ì♥❣ ♣❤→♣ ♥➔② ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝→❝ ②➳✉ tè   y0 = f ✭✸✳✶✳✹✮  yn+1 = An (y0 , y1 , , yn ) tr♦♥❣ ✤â An ❧➔ ❝→❝ ✤❛ t❤ù❝ ❝õ❛ y0 ✱ y1 ✱ .✱ yn ❝á♥ ❣å✐ ❧➔ ❝→❝ ✤❛ t❤ù❝ ❆❞♦♠✐❛♥ t❤✉ ✤÷đ❝ ❜ð✐ ❝ỉ♥❣ t❤ù❝ dn An = N n! dλn ∞ λi yi , n = 0, 1, 2, i=0 ✭✸✳✶✳✺✮ λ=0 P❤÷ì♥❣ ♣❤→♣ ①→❝ ✤à♥❤ ♥❤÷ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② t÷ì♥❣ ữỡ ợ Sn = y1 + y2 + + yn ✭✸✳✶✳✻✮ ❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ ♣❤➨♣ ❧➦♣ S0 = Sn+1 = N (y0 + Sn ) ✹✼ ✭✸✳✶✳✼✮ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ õ tốt t ủ ợ ữỡ tr➻♥❤ ❤➔♠ S = N (y0 + S) ✭✸✳✶✳✽✮ ✣➸ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ (3.1.7) ❈❤❡rr✉❛♥❧t sû ỵ t ỵ N ❧➔ t♦→♥ tû tø ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✈➔♦ H ✈➔ y ❧➔ ♠ët ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (3.1.1)✳ ❈❤✉é✐ ∞ yi ❤ë✐ tư i=0 tỵ✐ y ❦❤✐ ∃0 ≤ α < 1, yk+1 ≤ α yk , ∀k ∈ N ∪ {0} ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳ ❱ỵ✐ ♠é✐ i ∈ N ∪ {0} t❛ ✤à♥❤ ♥❣❤➽❛   y   i+1 , yi = yi αi =    0, yi = ∞ q r ỵ tr ộ y ❦❤✐ ≤ αi < 1, i = 0, 1, 2, yi ❤ë✐ tư tỵ✐ ♥❣❤✐➺♠ ❝❤➼♥❤ i=0 ✸✳✷ ❈→❝ ✈➼ ❞ư ❱➼ ❞ư ✸✳✷✳✶✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ y + + x2 y = x4 + 2x3 + 2x2 + 2x + ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ y(0) = 1✳ ●✐↔✐ ✹✽ ✭✸✳✷✳✶✮ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❳➨t sü ❤ë✐ tư t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ❆❞♦♠✐❛♥ t✐➯✉ ❝❤✉➞♥ ✣➦t y = Ly tr♦♥❣ ✤â L= d dx x L−1 (.) = (.)dx ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ (3.2.1) trð t❤➔♥❤ Ly = x4 + 2x3 + 2x2 + 2x + − + x2 y ✭✸✳✷✳✷✮ L−1 (3.2.2) ❚→❝ ✤ë♥❣ t♦→♥ tû ❦❤↔ ♥❣❤à❝❤ ✈➔♦ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❛ ❝â y (x) = y (0) + L−1 x4 + 2x3 + 2x2 + 2x + − L−1 + x2 y ✭✸✳✷✳✸✮ ❚❛ ✤➦t y0 = y (0) + L−1 x4 + 2x3 + 2x2 + 2x + yn+1 = −L−1 tr♦♥❣ ✤â An + x2 An , n ≥ ❧➔ ❝→❝ ✤❛ t❤ù❝ ❆❞♦♠✐❛♥ ❝õ❛ sè ❤↕♥❣ ♣❤✐ t✉②➳♥ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ A0 = y02 A1 = 2y0 y1 A2 = 2y0 y2 + y12 ✳✳ ✳ ✹✾ ✭✸✳✷✳✹✮ y2 ✤÷đ❝ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚❛ ❝â x5 x4 y0 = + + x + x2 + 2x + y1 = −L−1 =− + x2 A0 = −L−1 + x2 y02 x13 x12 167 11 19 10 497 − − x − x − x 325 60 3300 150 1620 311 68 32 7 − x8 − x − x − x − x − x − 2x2 − x 315 45 15 3 y2 = −L−1 + x2 A1 = −L−1 + x2 2y0 y1 = 19 20 697 19 89699 18 x21 + x + x + x 34125 39000 313500 11583000 + 2951813 16 13716251 15 12170033 14 2336929 17 x + x + x + x 98455500 46332000 91216125 37837800 + 7722409 13 4267002 12 292627 11 162161 10 2269 x + x + x + x + x 12162150 3742200 155925 56700 567 + 6373 88 61 82 25 x + x + x + x + x + x + x2 1260 15 10 15 ∞ yi ❉➵ ❞➔♥❣ t❤➜② r➡♥❣ ♥❣❤✐➺♠ i= y(x) = x + 1✳ ❦❤ỉ♥❣ ❤ë✐ tư ✤➳♥ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ◆❣♦➔✐ r❛ ❜➡♥❣ ❝→❝❤ t➼♥❤ αi t❛ ❝â α0 = y1 = 1.575091504 > y0 α1 = y2 = 2.924439213 > y1 α2 = y3 = 1.588190598 > y2 ✺✵ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ Ð ✤➙②✱ αi ✤➲✉ ợ ỡ t ữỡ t t❤➻ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❦❤ỉ♥❣ ❤ë✐ tư✳ ✣➸ t✐➳♣ tư❝✱ t❛ ①➨t ♣❤÷ì♥❣ tr➻♥❤ ▲❛♥❡ ✲ ❊♠❞❡♥ ✤÷đ❝ ①➙② ❞ü♥❣ ♥❤÷ s❛✉✿   y + y + F (x, y) = g (x) , < x ≤ x  y (0) = A, y (0) = B ✭✸✳✷✳✺✮ ❱➼ ❞ư ✸✳✷✳✷✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ s❛✉ y + y = 110x8 x ✭✸✳✷✳✻✮ ✈ỵ✐ ❣✐→ trà ❜❛♥ ✤➛✉ y (0) = 0, y (0) = 0✳ ❳➨t sü ❤ë✐ tư ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜✐➳t ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❧➔ y (x) = x10 ✳ ●✐↔✐ ❳➨t sü ❤ë✐ tư t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ❆❞♦♠✐❛♥ t✐➯✉ ❝❤✉➞♥ ✣➦t Ly = y (3.2.6) ❦❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ trð t❤➔♥❤ Ly = 110x8 − y x ❱➻ ✈➟② t♦→♥ tû L−1 ✭✸✳✷✳✼✮ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝ x x L−1 (.) = (.)dxdx ❚→❝ ✤ë♥❣ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ L−1 ✈➔♦ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (3.2.7) t❛ ❝â y (x) = L−1 110x8 − L−1 y x ✺✶ + y (0) + xy (0) ✭✸✳✷✳✽✮ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣➦t y0 = y (0) + xy (0) + L−1 110x8 ✭✸✳✷✳✾✮ y ,k ≥ x k yk+1 = −L−1 ✭✸✳✷✳✶✵✮ ❚ø ✤â t❛ t➼♥❤ ✤÷đ❝ x x y0 = L−1 110x8 = 110x8 dxdx = 0 x y1 = −L−1 y x 22 110 x dxdx = − x10 x 81 y x y3 = −L−1 x 220 44 10 x dxdx = x x 729 729 = y x x =− x y2 = −L−1 11 10 x x x =− 440 −88 10 x dxdx = x x 6561 6561 ✳✳ ✳ ❉♦ ✤â✱ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t õ t ữủ ữợ ộ ữ s❛✉ −11 yn (x) = ∞ n=0 −2 n+1 x10 ✣â ❧➔ ♠ët ❝➜♣ sè ♥❤➙♥ ❤ë✐ tö ✈➲ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ◆❣♦➔✐ r❛ t❛ ❝â α0 = y1 = 0.2222222223 < y0 α1 = y2 = 0.2222222222 < y1 α2 = y3 = 0.2222222222 < y2 ✺✷ ✭✸✳✷✳✶✶✮ y (x) = x10 ✳ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❳➨t sü ❤ë✐ tư t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ❆❞♦♠✐❛♥ ❝↔✐ ❜✐➯♥ ✣➦t Ly = y + y x tr♦♥❣ ✤â L = x−2 t❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ (3.2.6) d d x2 dx dx t❛ ❝â Ly = 110x8 ❑❤✐ ✤â✱ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ L−1 ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ x L−1 (.) = x x−2 ❚→❝ ✤ë♥❣ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ ✭✸✳✷✳✶✷✮ L−1 x2 (.)dxdx ✭✸✳✷✳✶✸✮ ✈➔♦ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (3.2.12) t❛ ❝â y (x) = L−1 110x8 + y (0) + xy (0) ❚✐➳♥ ữ trữợ t t ữủ y0 (x) = x10 y1 (x) = y2 (x) = ✳✳ ✳ ◆❤÷ ✈➟② t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ❆❞♦♠✐❛♥ ❝↔✐ ❜✐➯♥ t❛ t❤✉ ✤÷đ❝ ♥❣❛② ♥❣❤✐➺♠ ✺✸ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❝❤➼♥❤ ①→❝ ❝õ❛ ❜➔✐ t♦→♥ y (x) = x10 ✳ ◆❣♦➔✐ r❛✱ t❛ ❝â αi = < 1, i = 0, 1, 2, ❱➼ ❞ư ✸✳✷✳✸✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ s❛✉ y + y + y = x5 + 30x3 x ✭✸✳✷✳✶✹✮ ✈ỵ✐ ❣✐→ trà ❜❛♥ ✤➛✉ y (0) = 0, y (0) = 0✳ ❳➨t sü ❤ë✐ tư ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜✐➳t ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❧➔ y (x) = x5 ✳ ●✐↔✐ ❳➨t sü ❤ë✐ tư t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ❆❞♦♠✐❛♥ t✐➯✉ ❝❤✉➞♥ ✣➦t Ly = y (3.2.14) ❦❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ trð t❤➔♥❤ Ly = x5 + 30x3 − y − y x ❱➻ ✈➟② t♦→♥ tû L−1 ✭✸✳✷✳✶✺✮ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝ x x L−1 (.) = (.)dxdx ❚→❝ ✤ë♥❣ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ L−1 ✈➔♦ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (3.2.15) t❛ ❝â y (x) = L−1 x5 + 30x3 + y (0) + xy (0) − L−1 y + y x ❈❤ó♥❣ t❛ ❝â ✤÷đ❝ ♠è✐ q✉❛♥ ❤➺ ♥❤÷ s❛✉    y0 = y (0) + xy (0) + L−1 x5 + 30x3  −1  y + y ,n ≥ yn+1 = −L x ✺✹ ✭✸✳✷✳✶✻✮ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚ø ✤â t❛ t➼♥❤ ✤÷đ❝ y0 = y1 = y2 = y3 = x7 x + 42 11 x9 −3 x − x − 252 3024 7 25 x11 x + x + x + 216 36288 332640 −3 179 271 137 x13 11 x − x − x − x − 16 9072 435456 19958400 51891840 ✭✸✳✷✳✶✼✮ ✈➔ α0 = 0.5178432480 < α1 = 0.5118817363 < ✭✸✳✷✳✶✽✮ α2 = 0.5079364636 < ❚❤❡♦ ❦➳t q✉↔ t❤✉ ✤÷đ❝ tr♦♥❣ ❤ë✐ tư ✤➳♥ (3.2.17)✱ (3.2.18) t❤➻ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ y (x) = x5 ✳ ❳➨t sü ❤ë✐ tư t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ❆❞♦♠✐❛♥ ❝↔✐ ❜✐➯♥ ✣➦t Ly = y + y x tr♦♥❣ ✤â L = x−2 ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ (3.2.14) d d x2 dx dx t❛ ❝â Ly = x5 + 30x3 ✺✺ ✭✸✳✷✳✶✾✮ ✣➱ ❚❍➚ ❚❍❯ ❍⑨ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ tr♦♥❣ ✤â x L−1 (.) = x x−2 ❚→❝ ✤ë♥❣ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ L−1 x2 (.)dxdx ✭✸✳✷✳✷✵✮ ✈➔♦ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (3.2.19) t❛ ❝â ❉♦ ✤â t❛ ❝â y (x) = y (0) + L−1 x5 + 30x3 ✭✸✳✷✳✷✶✮   y = y (0) + L−1 x5 + 30x3  y = −L−1 (y ) , n ≥ ✭✸✳✷✳✷✷✮ n+1 n ❚ø ✤â t❛ t➼♥❤ ✤÷đ❝ −x7 x9 x9 x11 x7 − , y2 = + y0 = x + , y1 = 56 56 5040 5040 665280 ✭✸✳✷✳✷✸✮ ✈➔ α0 = 0.01521198194 < α1 = 0.009843639085 < ✭✸✳✷✳✷✹✮ ✳✳ ✳ ❚ø ❦➳t q✉↔ t❤✉ ✤÷đ❝ ð (3.2.20)✱ (3.2.21) s✉② r❛ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ y (x) = x5 ✳ ❇➡♥❣ ❝→❝❤ s♦ s→♥❤ ❣✐ú❛ ❝→❝ tử tợ t q t ữủ t❛ ❝â t❤➸ t❤➜② r➡♥❣ tè❝ ✤ë ❤ë✐ tö ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❆❞♦♠✐❛♥ ❝↔✐ ❜✐➯♥ ❝❛♦ ❤ì♥ tè❝ ✤ë ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t✐➯✉ ❝❤✉➞♥✳ ✺✻ ❑➳t ❧✉➟♥ P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❚r➯♥ ✤➙② ❧➔ t♦➔♥ ❜ë ♥ë✐ ❞✉♥❣ ❦❤â❛ ❧✉➟♥ ✈➲ ✤➲ t➔✐ ✏ ✑✳ ❚r♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔②✱ ❡♠ ✤➣ tr➻♥❤ ❜➔②✿ ✰ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✈➔ ❝❤✉é✐ ❧ơ② t❤ø❛✳ ✰ P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥ ✈➔ ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤â ✤➸ ❣✐↔✐ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ♣❤✐ t✉②➳♥ ❦❤✐ ❦❤ỉ♥❣ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ♥â✳ ✰ ✣✐➲✉ ❦✐➺♥ ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❆❞♦♠✐❛♥✳ ▼➦❝ ❞ò ✤➣ ❤➳t sù❝ ❝è ❣➢♥❣✱ ❞♦ sü ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥✱ ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ♥➯♥ ❦❤♦→ ❧✉➟♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❊♠ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü q✉❛♥ t➙♠ õ õ ỵ ổ ❜↕♥ ✤➸ ❦❤â❛ ❧✉➟♥ ♥➔② ✤÷đ❝ ✤➛② ✤õ ✈➔ ❝❤➼♥❤ rữợ t tú õ ởt ♥ú❛ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤è✐ ✈ỵ✐ ❝→❝ ❚❤➛②✱ ❈ỉ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥✱ ✤➦❝ ❜✐➺t ❧➔ t❤➛② ❣✐→♦ P●❙✳❚❙✳ ❑❤✉➜t ❱➠♥ ◆✐♥❤ ❣✐ó♣ ✤ï ❡♠ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ỡ t t ữợ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬❆❪ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ P P ỵ tt ♣❤➛♥ ✶✱ ❈ì sð ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ◆❳❇ ●✐→♦ ❞ö❝ ❱✐➺t ◆❛♠✳ ❬✷❪ ❚r➛♥ ✣ù❝ ▲♦♥❣✱ ◆❣✉②➵♥ ✣➻♥❤ ❙❛♥❣✱ ❍♦➔♥❣ ◗✉è❝ ❚♦➔♥ ✭✷✵✵✻✮✱ ●✐→♦ tr➻♥❤ ❣✐↔✐ t➼❝❤ t➟♣ ✷✱ ❬❇❪ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❨✳ ❈❤❡rr❛✉❧t ♠❡t❤♦❞✱ ✭✷✵✵✻✮✱❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❆❞♦♠✐❛♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱♣♣✳✶✵✶✲✶✵✻✳ ❬✹❪ ❨✳◗✳ ❍❛s❛♥ ❛♥❞ ▲✳▼✳❩❤✉ ✭✷✵✵✽✮✱ ▼♦❞✐❢✐❡❞ ❆❞♦♠✐❛♥ ❞❡❝♦♠♣♦s✐✲ t✐♦♥ ♠❡t❤♦❞ ❢♦r s✐♥❣✉❧❛r ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠s ✐♥ t❤❡ s❡❝♦♥❞ ♦r❞❡r ♦r❞✐♥❛r② ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❙✉r✈❡②s ✐♥ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✳✸✱♣♣✳✶✽✸✲✶✾✸✳ ❬✺❪ ❚✳❙✳▲✳ ❘❛❞❤✐❦❛✱ ❚✳❑✳❱✳ ■②❡♥❣❛r✱ ❚✳ ❘❛❥❛r❛♥✐ ✭✷✵✶✺✮✱ ❆♣♣r♦①✐✲ ♠❛t❡ ❛♥❛❧②t✐❝❛❧ ♠❡t❤♦❞s ❢♦r s♦❧✈✐♥❣ ♦r❞✐♥❛r② ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛✲ t✐♦♥s✱ ❈❘❈ Pr❡ss ✐s ❛♥ ✐♠♣r✐♥t ♦❢ t❤❡ ❚❛②❧♦r ❛♥❞ ❋r❛♥❝✐s ●r♦✉♣✱ ❛♥ ■♥❢♦r♠❛ ❜✉s✐♥❡ss✱ ❆ ❈❤❛♣♠❛♥ ❛♥❞ ❍❛❧❧ ❜♦♦❦✳ ✺✽ ... TOÁN ************* ĐỖ THỊ THU HÀ PHƯƠNG PHÁP PHÂN TÍCH ADOMIAN GIẢI GẦN ĐÚNG CÁC PHƯƠNG TRÌNH VI PHÂN THƯỜNG KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Giải tích Người hướng dẫn khoa học PSG.TS