Phương pháp WKB để giải phương trình vi phân thường

TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* ĐOÀN THỊ PHƯƠNG NHUNG PHƯƠNG PHÁP WKB ĐỂ GIẢI 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 ************* ĐOÀN THỊ PHƯƠNG NHUNG PHƯƠNG PHÁP WKB ĐỂ GIẢI PHƯƠNG TRÌNH VI PHÂN THƯỜNG KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Giải tích Người hướng dẫn khoa học PSG.TS KHUẤT VĂN NINH HÀ NỘI – 2018 ▲❮■ ❈❷▼ ❒◆ rữợ tr õ tèt ♥❣❤✐➺♣✱ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s tợ P t ữớ t t ữợ õ t t õ ♥➔②✳ ❊♠ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ t♦➔♥ t❤➸ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ ❑❤♦❛ ❚♦→♥✱ ❚r÷í♥❣ ✤↕✐ ❤å❝ s÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷ ✤➣ ❞↕② ❜↔♦ ❡♠ t➟♥ t➻♥❤ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ ❦❤♦❛✳ ◆❤➙♥ ❞à♣ ♥➔② ❡♠ ❝ơ♥❣ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥ ❜➯♥ ❡♠✱ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ ✣♦➔♥ ❚❤à P❤÷ì♥❣ ◆❤✉♥❣ ▲❮■ ❈❆▼ ✣❖❆◆ ❊♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❝❤÷❛ ❤➲ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ❜↔♦ ✈➺ ♠ët ❤å❝ ✈à ♥➔♦✱ ❝→❝ t❤æ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr õ ữủ ró ỗ ố ởt ró r t tr trữợ ♥❤➔ tr÷í♥❣ ✈➲ sü ❝❛♠ ✤♦❛♥ ♥➔②✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ ✣♦➔♥ ❚❤à P❤÷ì♥❣ ◆❤✉♥❣ ▼ư❝ ❧ö❝ ▲❮■ ▼Ð ✣❺❯ ✶ ✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✹ ✶✳✶ ✶✳✷ ❈❤✉é✐ ❧ô② t❤ø❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✶ ❑❤→✐ ♥✐➺♠ ❝❤✉é✐ ❧ơ② t❤ø❛ ✶✳✶✳✷ ❇→♥ ❦➼♥❤ ❤ë✐ tư ❝õ❛ ❝❤✉é✐ ❧ô② t❤ø❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✸ ❑❤❛✐ tr✐➸♥ ❤➔♠ t❤➔♥❤ ❝❤✉é✐ ❧ô② t❤ø❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♥ ✶✳✷✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ❑❤→✐ ♥✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✷ ❇➔✐ t♦→♥ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✸ ❈→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♥ ✶✳✷✳✹ tỗ t t t♦→♥ ❈❛✉❝❤② ✻ ✳ ✳ ✼ ✳ ✳ ✳ ✽ ✷ P❍×❒◆● P❍⑩P ❲❑❇ ●■❷■ ●❺◆ ✣Ĩ◆● ❇⑨■ ❚❖⑩◆ ❑❍➷◆● ◆❍■➍❯ ✶✵ ✷✳✶ ❇➔✐ t♦→♥ ❦❤æ♥❣ ♥❤✐➵✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✷ ❱➼ ❞ö →♣ ❞ö♥❣ ✶✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ P❍×❒◆● P❍⑩P ❲❑❇ ●■❷■ ●❺◆ ✣Ĩ◆● ❇⑨■ ❚❖⑩◆ ❇➚ ◆❍■➍❯ ✶✺ ✸✳✶ ▼ët sè ✤➦❝ tr÷♥❣ r✐➯♥❣ ❝õ❛ ❝→❝ ♥❣❤✐➺♠ ❣➛♥ ❧ỵ♣ ❜✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✶✳✶ ✶✺ ❚➼♥❤ ❝❤➜t t→♥ ①↕ ✈➔ t➼♥❤ ❝❤➜t ♣❤➙♥ t→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ◆❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❲❑❇ t❤❡♦ ❞↕♥❣ ❤➔♠ ♠ơ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✸ ❙ü ♠ð rë♥❣ ❤➻♥❤ t❤ù❝ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❲❑❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✹ ❈→❝ ủ ỵ ữỡ ú ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✸✳✹✳✶ ✸✳✺ ✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆● ◗✉❛♥❣ ❤å❝ ❤➻♥❤ ❤å❝ ✈➔ q✉❛♥❣ ❤å❝ ✈➟t ❧➼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ❙ü ❦➳t ❤ñ♣ ❝→❝ ♣❤➨♣ ❣➛♥ ✤ó♥❣ t✐➺♠ ❝➟♥✿ ◆❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝â ♠ët ✤✐➸♠ ♥❣♦➦t ✸✳✺✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t♦→♥ ♠ët ✤✐➸♠ ♥❣♦➦t ✤ì♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✺ ❑➌❚ ▲❯❾◆ ✹✶ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✹✷ ✐✐ ▲í✐ ỵ t t ❚♦→♥ ❤å❝ ✤÷đ❝ ♣❤→t s✐♥❤ ❞♦ ♥❤✉ ❝➛✉ ❣✐↔✐ q✉②➳t t õ ỗ ố tỹ t ũ ợ sü ♣❤→t tr✐➸♥ ❝õ❛ ♥ë✐ t↕✐ ❚♦→♥ ❤å❝ ✈➔ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ❦❤→❝✱ ❚♦→♥ ❤å❝ ❝❤✐❛ t❤➔♥❤ ❤❛✐ ❧➽♥❤ ✈ü❝✿ ❚♦→♥ ❤å❝ ❧➼ t❤✉②➳t ✈➔ ❚♦→♥ ❤å❝ ù♥❣ ❞ö♥❣✳ ❚r♦♥❣ ❧➽♥❤ ✈ü❝ ❚♦→♥ ù♥❣ ❞ư♥❣ t❤÷í♥❣ ❣➦♣ r➜t ♥❤✐➲✉ ♥❤ú♥❣ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✳ ❱➻ ✈➟② ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ tữớ õ trỏ q trồ tr ỵ tt ❤å❝✳ ❈❤ó♥❣ t❛ ❜✐➳t r➡♥❣ ❝❤➾ ♠ët sè ➼t ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❝â t❤➸ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ❝❤➼♥❤ tr ợ ữỡ tr s✐♥❤ tø ❜➔✐ t♦→♥ t❤ü❝ t✐➵♥ ✤➲✉ ❦❤ỉ♥❣ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝✳ ❉♦ ✈➟②✱ ♠ët ✈➜♥ ✤➲ ✤➦t r❛ ❧➔ t➻♠ ❝→❝❤ ✤➸ ①→❝ ✤à♥❤ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ❳✉➜t ♣❤→t tø ♥❤✉ ❝➛✉ ✤â✱ ❝→❝ ♥❤➔ ❚♦→♥ ❤å❝ ✤➣ t➻♠ r❛ ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤➸ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✳ ❚r♦♥❣ ❜↔♥ ❦❤â❛ ❧✉➟♥ ♥➔② ❡♠ tr➻♥❤ ❜➔② ♠ët ♣❤÷ì♥❣ ♣❤→♣ t✐➺♠ ❝➟♥ ❣✐↔✐ ❜➔✐ t♦→♥ ♥❤✐➵✉ ❦➻ ❞à ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❲❑❇ ❞♦ ❝→❝ ♥❤➔ ✈➟t ❧➼ ●r❡❣♦r ❲❡♥t③❡❧✱ ❍❡♥❞r✐❦ ❑r❛♠❡rs ✈➔ ▲➨♦♥ ❇r✐❧❧♦✇♥ ♣❤→t tr✐➸♥ ✈➔♦ ♥➠♠ ✶✾✷✻✳ ✣➙② ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♠↕♥❤ ❝❤♦ ♣❤➨♣ t❤✉ ✤÷đ❝ ♥❣❤✐➺♠ ①➜♣ ①➾ t♦➔♥ ❝ư❝ ữỡ tr ữợ sỹ ữợ t t P t ❡♠ ❝❤å♥ ✤➲ t➔✐ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✿✏ P❤÷ì♥❣ P❤→♣ ❲❑❇ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✑ ✤➸ t➻♠ ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❲❑❇✳ ✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆● ✷✳ ▼ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ◆❣❤✐➯♥ ❝ù✉ ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❲❑❇ ✈➔♦ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✳ ✸✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ✲ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉✿ P❤÷ì♥❣ ♣❤→♣ ❲❑❇✳ ✲ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ P❤÷ì♥❣ ♣❤→♣ ❲❑❇ ✈➔ ♠ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ❝❤ó♥❣ tr♦♥❣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✳ Pữỡ ự Pữỡ ự t Pữỡ Pữỡ ❤➺ t❤è♥❣ ❤â❛ ✺✳ ❈➜✉ tró❝ ❦❤â❛ ❧✉➟♥ ◆ë✐ ❞✉♥❣ õ ỗ ữỡ ữỡ tự ❜à✳✑ ❈❤÷ì♥❣ ♥➔② ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ❝❤✉é✐ ❧ơ② t❤ø❛✱ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ n, ❜➔✐ t♦→♥ ❈❛✉❝❤②✳ ❈❤÷ì♥❣ ✷ ✏P❤÷ì♥❣ ♣❤→♣ ❲❑❇ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❜➔✐ t♦→♥ ❦❤ỉ♥❣ ♥❤✐➵✉✳✑ ▼ư❝ ✤➼❝❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ ợ t ữỡ ú ❝→❝ ❜➔✐ t♦→♥ ❦❤ỉ♥❣ ♥❤✐➵✉ ✈➔ ♠ët sè ✈➼ ❞ư →♣ ❞ư♥❣✳ ❈❤÷ì♥❣ ✸ ✏P❤÷ì♥❣ ♣❤→♣ ❲❑❇ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❜➔✐ t♦→♥ ❜à ♥❤✐➵✉✳✑ ▼ư❝ ✤➼❝❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ ợ t ữỡ t♦→♥ ❜à ♥❤✐➵✉ ✈➔ ♠ët sè ✈➼ ❞ö →♣ ❞ö♥❣✳ ❑❤â❛ ❧✉➟♥ ♥➔② ✤÷đ❝ tr➻♥❤ ❜➔② tr➯♥ ❝ì sð ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✤÷đ❝ ❧✐➺t ❦➯ tr♦♥❣ ♣❤➛♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ✣â♥❣ ❣â♣ ❝õ❛ ❡♠ t❤➸ ❤✐➺♥ ð ❝❤é✱ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❲❑❇ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤✱ t➻♠ ✤÷đ❝ ♠ët sè ✈➼ ❞ư ♠✐♥❤ ❤å❛ ❝❤♦ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤â✳ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆● ❉♦ t❤í✐ ❣✐❛♥ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ❦❤æ♥❣ ♥❤✐➲✉✱ ❦✐➳♥ t❤ù❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ s❛✐ sât✳ ❊♠ ♠♦♥❣ ữủ sỹ õ õ ỵ qỵ t ổ t ❝↔♠ ì♥✦ ✸ ❈❤÷ì♥❣ ✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② t→❝ ❣✐↔ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ❝❤✉é✐ ❧ơ② t❤ø❛✱ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ❜➔✐ t♦→♥ ❈❛✉❝❤②✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✳ ✶✳✶ ❈❤✉é✐ ❧ô② t❤ø❛ ✶✳✶✳✶ ❑❤→✐ ♥✐➺♠ ❝❤✉é✐ ❧ô② t❤ø❛ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ x0 , a1 , a2 , ✣✐➸♠ +∞ ❈❤✉é✐ ❧ô② t❤ø❛ ❧➔ ♠ët ❤➔♠ ❝â ❞↕♥❣ an (x − x0 )n tr♦♥❣ ✤â n=0 ❧➔ ♥❤ú♥❣ sè t❤ü❝✳ x0 ữủ t ộ ụ tứ ỵ r➡♥❣ ❝❤✉é✐ ❧ơ② t❤ø❛ ❧✉ỉ♥ ❤ë✐ +∞ tư t↕✐ ✤✐➸♠ x = x0 ◆➳✉ ✤➦t y = x − x0 t❤➻ ❝â t❤➸ ✤÷❛ ❝❤✉é✐ ✈➲ ❞↕♥❣ an y n , n=0 ❝❤✉é✐ ❝â t➙♠ t↕✐ y = 0✳ ✶✳✶✳✷ ❇→♥ ❦➼♥❤ ❤ë✐ tư ❝õ❛ ❝❤✉é✐ ❧ơ② t❤ø❛ ✣à♥❤ ❧➼ ❆❜❡❧ ❈❤♦ ❝❤✉é✐ ❧ô② t❤ø❛ +∞ an xn = a0 + a1 x + a2 x2 + n=0 ✹ ✭✶✳✶✮ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣❖⑨◆ ❚❍➚ Pì Pữỡ tr ố ❞ö ✸✳✹ ✈➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ♥➔② sû ❞ö♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✶✵✮ ❧➔ x y(x) ∼ c1 P (x) −1 exp ε x P (t)dt + c2 P (x) −1 exp − ε a ❈❤å♥ P (t)dt a a = 0, t❤➳ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ✈➔ sû ❞ö♥❣ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉✱ t❛ t❤✉ ✤÷đ❝ y(0) ∼ (c1 + c2 ) P (0)− = A, ✭✸✳✷✽✮ P (0) y (0) = − P (0)− P (0)(c1 + c2 ) + (c1 − c2 ) = B ε ◆➳✉ A = 0, B = 1, ✭✸✳✷✾✮ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❝â t❤➸ ✤÷đ❝ ✈✐➳t ♥❤÷ s❛✉ P❤÷ì♥❣ tr➻♥❤ ✭✸✳✷✽✮ trð t❤➔♥❤ c1 + c2 = 0, c1 = −c2 ✭✸✳✸✵✮ P❤÷ì♥❣ tr➻♥❤ ✭✸✳✷✾✮ trð t❤➔♥❤ P (0) (c1 − c2 ) = ε ✭✸✳✸✶✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✸✵✮ ✈➔ ✭✸✳✸✶✮ t❛ t❤✉ ✤÷đ❝ ε c2 = − P (0)− ε c1 = P (0)− , ❉♦ ✤â ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❣✐→ trà ❜❛♥ ✤➛✉ ✤➣ ❝❤♦ ❧➔ x − 14 y(x) ∼ ε(P (x)P (0)) P (t) dtε → ε sinh ❱➼ ❞ư ✸✳✶✵✳ ●✐↔✐ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❲❑❇ ε2 y = (1 − x2 )2 y, y(0) = 0, y (0) = ✷✾ ✭✸✳✸✷✮ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆● ▲í✐ ❣✐↔✐✿ ◆❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❣✐→ trà ❜❛♥ ✤➛✉ ✤➣ ❝❤♦ ❝â t❤➸ ♥❤➟♥ ✤÷đ❝ ❜➡♥❣ ❝→❝❤ t❤➳ P (x) = (1 − x2 )2 , P (0) = ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✸✷✮ t❛ t❤✉ ✤÷đ❝ y(x) ∼ ε − x 2 −1 x (1 − t2 )2 dt, ε sinh ε → 0+ ε y(x) ∼ √ sinh ε − x2 x3 x− , 0+ ủ ỵ ữỡ ú ỵ tt ởt ỵ tt s õ ữủ sỷ ❞ư♥❣ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♠➔ ✤↕♦ ❤➔♠ ❝❛♦ ♥❤➜t ♥❤➙♥ ✈ỵ✐ t❤❛♠ sè ♥❤ä✳ ❚➼♥❤ ❝❤➜t s✉② ỵ tt t ró số ❤↕♥❣ δ tr♦♥❣ ♣❤➨♣ ❣➛♥ ✤ó♥❣ t❤❡♦ ❞↕♥❣ ❤➔♠ ♠ô ✭✸✳✹✮ y(x) ∼ exp δ ❚rø ❦❤✐ S0 ≡ 0, ∞ δ n Sn (x) δ → n=0 ú s ổ tỗ t δ = ❚➼♥❤ ❝❤➜t s✉② ❜✐➳♥ ❝õ❛ ♣❤➨♣ ❣➛♥ ✤ó♥❣ ♥➔② ❝ơ♥❣ ✤÷đ❝ t❤➸ ❤✐➺♥ t❤❡♦ ❝→❝❤ ❦❤â ♠ỉ t↔ ❤ì♥ ✲ ❝❤✉é✐ ❲❑❇ δ n Sn t❤÷í♥❣ ♣❤➙♥ ❦➻✳ ▼➦❝ ❞ò ❝❤✉é✐ ❲❑❇ ♣❤➙♥ ❦➻ ♥❤÷♥❣ ♥â ✈➝♥ ❝â t❤➸ ✤÷❛ r❛ ♠ët ①➜♣ ①➾ r➜t ❝❤➼♥❤ ①→❝ ❝õ❛ y0 (x) ✣➸ ♣❤➨♣ ❣➛♥ ✤ó♥❣ ❲❑❇ ✭✸✳✸✸✮ ❝â ❣✐→ trà tr➯♥ ♠ët ❦❤♦↔♥❣✱ ❝➛♥ ♣❤↔✐ ❝â ❝❤✉é✐ δ n−1 Sn (x) ❧➔ ❝❤✉é✐ t✐➺♠ ❝➟♥ ✤➲✉ t❤❡♦ δ ✸✵ ✈ỵ✐ ♠å✐ x tr♦♥❣ ❦❤♦↔♥❣ ✤â ✈ỵ✐ δ → ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆● ✣✐➲✉ ♥➔② ✤á✐ ❤ä✐ ❝â ❝→❝ ✤✐➲✉ ❦✐➺♥ t✐➺♠ ❝➟♥ s❛✉ S0 (x)δ → , δ S1 (x) S1 (x), δ → 0, δS2 (x) ✭✸✳✸✹✮ ✳ ✳ ✳ δ n Sn+1 (x) δ n−1 Sn (x), δ → ❈→❝ ✤✐➲✉ ❦✐➺♥ ♥➔② t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐ ♠é✐ ❤➔♠ ❤↕♥ ❜ð✐ ❝→❝ ❤➔♠ ❝õ❛ ◆➳✉ ❝❤✉é✐ x Sn+1 (x)/Sn (x)(n = 0, 1, ) tr♦♥❣ ❦❤♦↔♥❣ ✤â✳ δ n−1 Sn (x) t✐➺♠ ❝➟♥ ✤➲✉ ✤è✐ ✈ỵ✐ x ❦❤✐ δ → 0, t❤➻ q✉② t➢❝ ❧♦↕✐ ❜ä sè ❤↕♥❣ tè✐ ÷✉ ❝❤♦ t❛ t❤➜② sü ❧♦↕✐ ❜ä ộ trữợ số ọ t ởt ln y ữủ ợ ợ s số tữỡ ố ♥❤ä tr♦♥❣ ❦❤♦↔♥❣ ❝õ❛ δ N SN +1 (x) ❝❤♦ x ộ t ữợ ♠ô tr♦♥❣ ✭✸✳✸✸✮ ♥➯♥ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ t✐➺♠ ❝➟♥ tr♦♥❣ ✭✸✳✸✹✮ ❦❤æ♥❣ ✤õ ✤↔♠ ❜↔♦ s➩ ❧➔ ♠ët ①➜♣ ①➾ tốt ố ợ ộ ữủ ợt số ❤↕♥❣ y(x), sè ❤↕♥❣ t✐➳♣ t❤❡♦ ♣❤↔✐ ♥❤ä ❤ì♥ δ N SN +1 (x) δ n−1 Sn (x) exp ✈ỵ✐ ♠å✐ x y(x) ❧➔ ♠ët ①➜♣ ①➾ tèt ❝❤♦ tr♦♥❣ ❦❤♦↔♥❣ ✤❛♥❣ ①➨t δ → 1, ✭✸✳✸✺✮ ◆➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ ♥➔② ✤÷đ❝ ❣✐ú t❤➻ δ N SN +1 (x) = + O δ N SN +1 (x) (δ → 0) ❉♦ ✤â s❛✐ sè t÷ì♥❣ ✤è✐ ❣✐ú❛ y(x) ✈➔ ①➜♣ ①➾ ❲❑❇ ♥❤ä N y(x) − exp 1/δ δ n Sn (x) n=0 ∼ δ N Sn (x) , y(x) δ→0 ❈↔ ✤✐➲✉ ❦✐➺♥ ✭✸✳✸✹✮ ✈➔ ✭✸✳✸✺✮ ✤➲✉ ♣❤↔✐ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ✸✳✹✳✶ ◗✉❛♥❣ ❤å❝ ❤➻♥❤ ❤å❝ ✈➔ q✉❛♥❣ ❤å❝ ✈➟t ❧➼ ❛✳ ◗✉❛♥❣ ❤å❝ ❤➻♥❤ ❤å❝ ◆➳✉ ❝❤➾ ①➨t sè ❤↕♥❣ ✤➛✉ t✐➯♥ ❝õ❛ ❝❤✉é✐ t❤➻ ❦➳t q✉↔ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ✤÷đ❝ ❣å✐ ❧➔ q✉❛♥❣ ❤å❝ ❤➻♥❤ ❤å❝ ✈➔ ❝â t❤➸ ✤÷đ❝ ✈✐➳t ♥❤÷ s❛✉ ✸✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ y(x) ∼ e S0 (x) δ ✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆● ❜✳ ◗✉❛♥❣ ❤å❝ ✈➟t ❧➼ ◆➳✉ ①➨t ❤❛✐ sè ❤↕♥❣ ✤➛✉ t✐➯♥ ❝õ❛ ❝❤✉é✐ ❲❑❇ t❤➻ ❦➳t q✉↔ ❝õ❛ ♥❣❤✐➺♠ ✤÷đ❝ ❣å✐ ❧➔ q✉❛♥❣ ❤å❝ ✈➟t ❧➼ ✈➔ ❝â t❤➸ ✤÷đ❝ ✈✐➳t ♥❤÷ s❛✉ y(x) ∼ e S0 (x) + S1 (x) , ú ỵ ú ❝õ❛ q✉❛♥❣ ❤å❝ ✈➟t ❧➼ t❤➸ ❤✐➺♥ t➼♥❤ ❝❤➜t t✐➺♠ ❝➟♥ ❝❛♦ ♥❤➜t ❝õ❛ y(x), tr♦♥❣ ❦❤✐ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ q ỗ sỹ t ♥❤❛♥❤ ♥❤➜t t❤❡♦ tø♥❣ t❤➔♥❤ ♣❤➛♥ ❝õ❛ t➼♥❤ ❝❤➜t tèt ♥❤➜t✳ ❝✳ ▼ët sè ✈➼ ❞ö →♣ ❞ö♥❣ ❱➼ ❞ö ✸✳✶✶✳ ❚➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ sè ❆✐r② ✈ỵ✐ x + t ữỡ tr r y = xy Schră odinger ✈ỵ✐ P (x) = x ✈➔ ε = ❉♦ ✤â✱ 3/2 , S1 = − ln x, S2 = ± x−3/2 t❛ ❝â S0 = ± x 48 ❚❛ t❤➜② r➡♥❣ t❤➟♠ ❝❤➼ ❦❤✐ ε = t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ t✐➺♠ ữỡ tr S2 t r ợ x → +∞ ε S1 S0 /ε, εS2 (x → +∞) ❚❤ü❝ ✈➟②✱ ❝❤➾ ❝➛♥ ✤÷❛ r❛ ❝→❝ t➼♥❤ ❝❤➜t tèt ♥❤➜t ❝õ❛ ❝→❝ ♥❣❤✐➺♠ ❝❤♦ x → +∞ 2x3/2/3 y(x) ∼ c± x−1/4 e± c± ✈➝♥ ✤÷đ❝ ❣✐ú ❦❤ỉ♥❣ ✤ê✐ t❤➻ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ q✉❛♥❣ ❤å❝ ✈➟t ❧➼ ❧➔ ú ữỡ tr r ợ tr õ tứ ✭✸✳✾✮✈➔ ✭✸✳✶✷✮ 1± −3/2 x , 48 ❧➔ ❤➡♥❣ số ú ỵ r tố ụ e 2x3 / 2/3 ❜✐➳♥ t❤✐➯♥ ♥❤❛♥❤ ❧➜② tø ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ q✉❛♥❣ ❤å❝ ❤➻♥❤ ❤å❝✳ ❱➼ ❞ö ✸✳✶✷✳ ❚➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ sè ❤➻♥❤ trư ♣❛r❛❜♦❧✐❝ ✈ỵ✐ ✸✷ x → +∞ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣❖⑨◆ ❚❍➚ Pì ỵ tt ụ sỷ ữủ ợ y = x −ν − Ð ✤➙② ♠➦❝ ❞ò x → +∞ ✈➻ S1 x ợ ữỡ tr trử r y = ♥❤÷♥❣ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ q✉❛♥❣ ❤å❝ ❤➻♥❤ ❤å❝ ✈➝♥ ✤ó♥❣ ✈ỵ✐ (x → +∞) ❈→❝ ②➳✉ tè ❝õ❛ t➼♥❤ tø P (x) = x − ν − ♥❤÷ S0 /ε , εS2 q✉❛♥❣ ❤å❝ ❤➻♥❤ ❤å❝ ✈➟t ❧➼ ✤÷đ❝ x S0 = ± ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ s❛✉ x P (t)dt ∼ ± ∼± t 1− 2ν + t2 x2 − ν+ dt ln x , x → +∞, tr♦♥❣ ✤â t❛ sû ❞ö♥❣ ♠ð rë♥❣ ♥❤à t❤ù❝ ❦➳t ❤đ♣ ✈ỵ✐ 1 S1 = − lnP ∼ − ln 4 x (x → +∞), t õ ú q t ỵ y ∼ c± eS0 +S1   c x−ν−1 ex2 /4 , + ∼  c− xν e−x2 /4 , x → + ∞ ✣â ❧➔ ♥❤ú♥❣ t➼♥❤ ❝❤➜t tèt ♥❤➜t ❝õ❛ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➻♥❤ trư ♣❛r❛❜♦❧✐❝✳ ❱➼ ❞ư ✸✳✶✸✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ εy + (1 + x)y = 0, y(0) = 0, y(1) = ▲í✐ s ữỡ tr ợ ữỡ tr Schră odinger P (x) = (1 + x) õ sû ❞ư♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✽✮✱ t❤✉ ✤÷đ❝ S0 (x) = −(1 + x) ✸✸ t❛ ❝â ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆● x (1 + x) −(1 + t)dt = ± 2i ⇒ S0 (x) = ± x0 ❉♦ ✤â✱ ♥❣❤✐➺♠ q✉❛♥❣ ❤å❝ ❤➻♥❤ ❤å❝ ❧➔ y(x) = exp ± 2i (1 + x) 3ε , √ (1 + x) sin 32i ε y(x) = sin 2√3 ε 2 , 2 ε= 24 (nπ)2 , ✭✸✳✸✻✮ ❚ø ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✻✮ t❤✉ ✤÷đ❝ S1 (x) = −1 ln P (x) t ỵ õ t ữủ t ữ s❛✉ y(x) ≈ e S0 (x) ε + S1 (x), ε → ❉♦ ✤â✱ ♥❣❤✐➺♠ tê♥❣ q✉→t ✤÷đ❝ ✈✐➳t ữợ s x x y(x) C1 P (x) exp ε P (t)dt + C2 P −1 (x) exp − ε a tr♦♥❣ ✤â P (x) = −(1 + x) a ❉♦ ✤â✱ ♥❣❤✐➺♠ q✉❛♥❣ ❤å❝ ✈➟t ❧➼ ❧➔ y(x) = P (t)dt, √ (1 + x) 2 sin 32i ε (1 + x) sin 3√2 ε 2 , ε= 2 94 (n2 , π ) ✸✳✺ ❙ü ❦➳t ❤đ♣ ❝→❝ ♣❤➨♣ ❣➛♥ ✤ó♥❣ t✐➺♠ ❝➟♥✿ ◆❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝â ♠ët ✤✐➸♠ ♥❣♦➦t ❈❤ó♥❣ t❛ ✤➣ ❜✐➳t tr♦♥❣ ▼ư❝ ✸✳✹ ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ t❤❡♦ ❞↕♥❣ ❤➔♠ ♠ơ ố ợ ữỡ tr Schră odinger ổ ú tr ♠✐➲♥ ❧➙♥ ❝➟♥ ❝õ❛ ♠ët ✤✐➸♠ ♥❣♦➦t✳ ❚r♦♥❣ t❤ü❝ t➳✱ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ q✉❛♥❣ ❤å❝ ✈➟t ❧➼ tr♦♥❣ ✭✸✳✶✵✮ s✉② ❜✐➳♥ t↕✐ ♠ët ✤✐➸♠ ♥❣♦➦t✳ ❚✉② ♥❤✐➯♥✱ t❛ t❤➜② r➡♥❣ ❝â ♠ët ♣❤÷ì♥❣ ♣❤→♣ tê♥❣ q✉→t ❞ü❛ tr➯♥ ✸✹ ❑❤â❛ tốt Pì ữỡ ♣❤→♣ ♠ð rë♥❣ t✐➳♣ ❝➟♥✱ ✤➸ ①➙② ❞ü♥❣ ♠ët ❣➛♥ ✤ó♥❣ t♦➔♥ ❝ư❝ ❝❤♦ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝â ✤✐➸♠ ♥❣♦➦t✳ ❈→❝❤ t✐➳♣ ❝➟♥ ♥➔② r➜t ❣✐è♥❣ ✈ỵ✐ ữỡ ữủ sỷ tr ỵ tt ợ õ ỗ t ủ ữỡ ✤ó♥❣ ❲❑❇ ❦❤→❝ ♥❤❛✉ tr♦♥❣ ❝→❝ ♠✐➲♥ ❝â ❣✐→ trà t÷ì♥❣ ù♥❣✳ ❚r♦♥❣ ♠ư❝ ♥➔② t❛ s➩ ❜➢t ✤➛✉ ❜➡♥❣ ✈✐➺❝ ①➨t ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♠➔ ❝❤➾ ❝â ♠ët ✤✐➸♠ ♥❣♦➦t✳ ❈ư t❤➸ ❧➔✱ ❡♠ s➩ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ε2 y = P (x)y, tr♦♥❣ ✤â P (x) y(+∞) = 0, ✭✸✳✸✼✮ ❧➔ ❤➔♠ sè ❧✐➯♥ tö❝ ♠➔ ✤✐ q✉❛ ✤✐➸♠ ❦❤ỉ♥❣ ♠ët ❧➛♥✳ ✣➸ ✤ì♥ ❣✐↔♥ ❤â❛✱ t❛ ❧➜② ✤✐➸♠ ♥❣♦➦t ♥➡♠ t↕✐ ✤✐➸♠ ❣è❝✿ P (0) = ✸✳✺✳✶ ❇➔✐ t♦→♥ ♠ët ✤✐➸♠ ♥❣♦➦t ✤ì♥ ❈❤ó♥❣ t❛ ❜➢t ✤➛✉ ♣❤➙♥ t➼❝❤ ❝❤✐ t✐➳t ❜➔✐ t♦→♥ ♠ët ✤✐➸♠ ♥❣♦➦t✱ tr♦♥❣ ✤â ✿ P (x) ∼ ax(x → 0) P (x) ❞÷ì♥❣ ❦❤✐ x ❚❛ ❣✐↔ t❤✐➳t r➡♥❣ ❞÷ì♥❣ ✈➔ ➙♠ ❦❤✐ ±∞.P (x) = sinh x , x + x3 x P (x) ❝â ❤➺ sè ❣â❝ ❞÷ì♥❣ t↕✐ ➙♠✳ ❚❛ ❝ơ♥❣ ❣✐↔ t❤✐➳t r➡♥❣ P (x) ❜➟❝ ♥❤➜t x = 0(a > 0) P (x) ✈➔ x−2 , x → t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ♥➔②✳ ✣è✐ ✈ỵ✐ ❜➔✐ t♦→♥ ❦❤ỉ♥❣ ✤✐➸♠ ✤ì♥✲ ❜➔✐ t♦→♥ ♠ët ✤✐➸♠ ♥❣♦➦t t❛ t✐➳♥ ❤➔♥❤ ♣❤➙♥ t➼❝❤ ♥❤÷ s❛✉✳ ❈❤✐❛ trư❝ x |x| ✈ỵ✐ 1, ♠✐➲♥ III t❤➔♥❤ ❜❛ ♠✐➲♥✿ ♠✐➲♥ x0 ❚r♦♥❣ ♠✐➲♥ ❤å❝ ✈➟t ❧➼ tr♦♥❣ ✭✸✳✶✶✮ ✤➲✉ ✤÷đ❝ ❣✐ú ❧↕✐✳ ❱✐➺❝ ❣✐ỵ✐ ❤↕♥ ❜↔♦ ①➜♣ ①➾ q✉❛♥❣ ❤å❝ ✈➟t ❧➼ ❧➔ ❤ñ♣ ❧➺✳ ❚r♦♥❣ ♠✐➲♥ ✈➻ ❝â ♠ët ✤✐➸♠ ♥❣♦➦t t↕✐ tr♦♥❣ ♠✐➲♥ ❧➙♥ ❝➟♥ ❝õ❛ x = 0, x = 0, I ✈➔ II I ✈➔ P (x) ε2/3 , x III ♠✐➲♥ II ✈ỵ✐ t❤➻ ①➜♣ ①➾ q✉❛♥❣ x−2 , |x| → ∞ ✤↔♠ t❤➻ ①➜♣ ①➾ ❲❑❇ ❦❤æ♥❣ ❤đ♣ ❧➼ ♥❤÷♥❣ ❝â t❤➸ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ tr♦♥❣ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❝→❝ ❤➔♠ ❆✐r② ε2 y = axy ❚❛ t❤➜② r➡♥❣ ❝→❝ ♠✐➲♥ II ✈➔ II III õ sỹ ỗ ♥❤❛✉ tr♦♥❣ ✤â ❝↔ ①➜♣ ①➾ ❲❑❇ ✈➔ ❆✐r② ✤➲✉ ❤ñ♣ ❧➺✳ ✣✐➲✉ ♥➔② ❝❤♦ ♣❤➨♣ ❦➳t ❤ñ♣ t✐➺♠ ❝➟♥ ❝→❝ ♥❣❤✐➺♠ tr♦♥❣ ❝→❝ ♠✐➲♥ ❦❤→❝ ♥❤❛✉✳ ❚ø sü ❦➳t ❤đ♣ ♥➔②✱ t❛ t❤✉ ✤÷đ❝ ❜❛ ❝ỉ♥❣ t❤ù❝ tø ✤â t↕♦ t❤➔♥❤ ♠ët ①➜♣ ①➾ t♦➔♥ ❝ö❝ ❝❤♦ ♥❣❤✐➺♠ ❝õ❛ ✭✸✳✸✼✮✳ ❙❛✉ ✤â✱ ❦➳t ❤đ♣ ❜❛ ❝ỉ♥❣ t❤ù❝ ♥➔② t❤➔♥❤ ♠ët ❜✐➸✉ t❤ù❝ ❞✉② ♥❤➜t ✸✺ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❧➔ ♠ët ①➜♣ ①➾ ❤ñ♣ ❧➼ ❝❤♦ ✣❖⑨◆ Pì y(x) ợ x t y(x) ữủ ợ ởt sè ❝❤✉♥❣ ✈➻ ❝❤➾ →♣ ❞ö♥❣ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❞✉② ♥❤➜t ①➨t ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❦❤→❝ ✤➸ ❝❤✉➞♥ ❤â❛ y(+∞) = y(x) ❱✐➺❝ ♣❤➙♥ t➼❝❤ ♠➔ t❛ ✈ø❛ t❤ü❝ ❤✐➺♥ ❧➔ ♣❤➙♥ t➼❝❤ ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ♠✐➲♥ q✉❛♥❣ ❤å❝ ✈➟t ❧➼ ❝❤♦ y(x) I ❳➜♣ ①➾ tr♦♥❣ ♠✐➲♥ ♥➔② ❝â ❞↕♥❣  yI (+∞) = ❧➼ ✈➔ rã r➔♥❣ t❤ä❛ ♠➣♥ ❜ð✐  x yI (x) = C[P (x)]−1/4 exp − ✣✐➲✉ ❦✐➺♥ ❜✐➯♥ ❉♦ ✤â✱ ε P (t)dt ✭✸✳✸✾✮ ✤➣ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ❧♦↕✐ ❜ä ❝→❝ ♥❣❤✐➺♠ q✉❛♥❣ ❤å❝ ✈➟t yI (x) tr♦♥❣ ✭✸✳✸✾✮✳ tũ ỵ ữợ t➼❝❤ ♣❤➙♥ tr♦♥❣ ✭✸✳✸✾✮ ♥➡♠ t↕✐ ✤✐➸♠ ♥❣♦➦t x = 0, sü ❧ü❛ ❝❤å♥ ❧➔ ❦❤ỉ♥❣ ❝➛♥ t❤✐➳t ♥❤÷♥❣ ♥â ✤ì♥ ❣✐↔♥ ❤â❛ ❝→❝ ❜✐➸✉ t❤ù❝ ①✉➜t ❤✐➺♥ s❛✉ ✤â tr♦♥❣ ♣❤➙♥ t➼❝❤ ❝õ❛ ❝❤ó♥❣ t❛✳ ❈➛♥ ①→❝ ✤à♥❤ ♠✐➲♥ ❤ñ♣ ❧➼ ❝õ❛ ①➜♣ ①➾ ✭✸✳✸✾✮✳ ❈â ❤❛✐ ✤✐➲✉ ❦✐➺♥ ❤ñ♣ ❧➼ ❝õ❛ q✉❛♥❣ ❤å❝ ✈➟t ❧➼ ♠➔ t❛ s✉② r❛ tø ♠ö❝ ✸✳✹ ❧➔ ❱➻ P (x) = ✈ỵ✐ x=0 ✈➔ S0 /ε S1 εS2 (ε → 0+) x−2 , |x| → ∞ P (x) ❈❤ó♥❣ t❛ ❝❤➢❝ ❝❤➢♥ r➡♥❣ ✈ỵ✐ ❜à ❝❤➦♥ tø ✤✐➸♠ ❣è❝ t❤➻ sü ❦❤→❝ ❜✐➺t ❣✐ú❛ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ε ✈ỵ✐ ε → 0+ ❑❤✐ x ♥❤ä t❤➻ P (x) ∼ ax ✈➻ ✈➟② 1(ε → 0+) εS2 ✈➔ y(x) ✈➔ yI (x) x ❧➔ ❞♦ ❜➟❝ S0 (x) ∼ ±2/3a1/2 x3/2 (x → 0+), ❉♦ ✤â✱ ✤✐➲✉ ❦✐➺♥ ❤ñ♣ ❧➼ ❝õ❛ ①➜♣ ①➾ q✉❛♥❣ ❤å❝ ✈➟t ❧➼ ✤÷đ❝ t❤ä❛ ♠➣♥ ♥➳✉ x ε2/3 , ε → 0+, ✭✸✳✹✵✮ ❍➺ t❤ù❝ ♥➔② ữợ t t ❝❤✉②➸♥ s❛♥❣ ♣❤➙♥ t➼❝❤ ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ♠✐➲♥ II ●✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✭✸✳✸✽✮✱ ❡♠ sû ❞ư♥❣ ♣❤➨♣ t❤➳ t = ε−2/3 a1/3 x, ❱ỵ✐ ✤✐➲✉ ❦✐➺♥ t, ữỡ tr ợ yI I ❧➔ d2 yII /dt2 = tyII , t❤➜② r➡♥❣ ✤➙② ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❆✐r②✳ ◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❧➔ tê ❤ñ♣ t✉②➳♥ ✸✻ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆● t➼♥❤ ❝õ❛ ❝→❝ ❤➔♠ sè ❆✐r② yII (x) = DAi(ε−2/3 a1/3 x) + EBi(ε−2/3 a1/3 x), tr♦♥❣ ✤â ❳➜♣ ①➾ D ✈➔ yII (x) E ❧➔ ❝→❝ ❤➡♥❣ sè ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ sü ❦➳t ❤đ♣ t✐➺♠ ❝➟♥ ✈ỵ✐ x yI (x) ❧➔ ✤ó♥❣ ♠✐➵♥ ❧➔ x ❱➻ ❝❤➾ ❦❤✐ ✭✸✳✹✷✮ ε → 0+, 1, ♥❤ä ✈➔ ❝â t❤➸ t❤❛② t❤➳ P (x) = ax, t❤ù❝ tr♦♥❣ ✭✸✳✹✸✮ ①→❝ ✤à♥❤ ❜✐➯♥ tr➯♥ ❝õ❛ ♠✐➲♥ ❑➳t ❤đ♣ ✭✸✳✹✵✮ ✈➔ ✭✸✳✹✸✮ t❛ t❤➜② ❤đ♣ ❧➼✱ ❝ư t❤➸ ❧➔ ε2/3 x yI (x) ❞♦ ✤â ❝â ✤÷đ❝ ✭✸✳✸✽✮ tø ✭✸✳✸✼✮✳ ❍➺ II yII (x) tr♦♥❣ ✭✸✳✹✸✮ tr♦♥❣ ✭✸✳✹✷✮ ❝â ♠ët ♠✐➲♥ ❝❤✉♥❣ 1(ε → 0+) ❚r♦♥❣ ♠✐➲♥ ỗ yI (x) yII (x) ú ợ ữỡ tr ố ❞♦ ✤â ❝❤ó♥❣ ♣❤↔✐ ❝â sü ❦➳t ❤đ♣ t✐➺♠ ❝➟♥✳ ❚✉② ♥❤✐➯♥ t❛ t❤➜② r➡♥❣ yI ✈➔ yII ❣✐è♥❣ ♥❤❛✉ r➜t ➼t ♥➯♥ ❝➛♥ ♣❤➙♥ t➼❝❤ ♥❤✐➲✉ ❤ì♥ ✤➸ ❝❤ù♥❣ r ú tỹ sỹ ũ ủ r ỗ ❝❤➨♦✱ x ♥❤ä ✈➻ ✈➟② P (x) ①➜♣ ①➾ x P (t)dt ∼ 32 a1/2 x3/2 , x → + a−1/4 x−1/4 (x → 0+) ✈➔ ✣➛✉ t✐➯♥ t❛ ①➨t [P (x)]−1/4 ∼ yI (x) ax ❱➻ ✈➟②✱ ❉♦ ✤â✱ 1/2 x3/2 /3ε yI (x) ∼ Ca−1/4 x−1/4 e−2a ◆❤÷ ✤➣ ❜✐➳t ①➜♣ ①➾ ❲❑❇ ❦❤ỉ♥❣ ✤ó♥❣ trứ ữợ tở sè ●✐↔ sû✱ P (x) ∼ ax ∼ bx2 (x → 0) x P (t)dt ∼ x ∼ ,x → + ε2/3 (ε → 0+) x P (x) ❑❤✐ ✤â √ √ at + bt2 dt at + bt 2a dt b ∼ a1/2 x3/2 + √ x5/2 , a ✸✼ x → 0+ ✭✸✳✹✹✮ ❚✉② ♥❤✐➯♥✱ ❣✐ỵ✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ Pì t ữủ ❝➛♥ ♣❤↔✐ ❣✐↔ sû ① ✤õ ♥❤ä ✤➸ t❤➻ t❛ ❝â ✤✐➲✉ ❦✐➺♥ ♠✐➲♥ ε2/3 ❚ø ✤â ✭✸✳✹✹✮ ❧➔ ✤ó♥❣ ✈➔ ❜à ❣✐ỵ✐ ❤↕♥ ❜ð✐ ε2/5 (ε → 0+) x ❚✐➳♣ t❤❡♦ ①➨t ε2/5 (ε → 0+) x √ exp bx5/2 / 5ε a ∼ (ε → 0+) yII (x) r ỗ ữợ t số ❆✐r② ❜ð✐ ❝→❝ t➼♥❤ ❝❤➜t t✐➺♠ ❝➟♥ ❝❛♦ ♥❤➜t ❝õ❛ ú ợ ố số ợ ữỡ ổ tự ✤ó♥❣ ❧➔ 3/2 Ai(t) ∼ √ t−1/4 e−2t /3 , π 3/2 Bi(t) ∼ √ t−1/4 e2t /3 , π t → +∞, t → +∞ ❈→❝ ①➜♣ ①➾ ♥➔② ❝â t❤➸ ✤÷đ❝ sû ❞ư♥❣ ♥➳✉ ❝→❝ ✤è✐ sè ❝õ❛ ❝→❝ ❤➔♠ sè ❆✐r② tr♦♥❣ ✭✸✳✹✷✮ ❧ỵ♥✳ ❱➻ ✈➟② yII (x) ∼ √ a−1/12 ε1/6 x−1/4 π −2a1/2 x3/2 /3ε De ✭✸✳✹✺✮ ❑➳t ❧✉➟♥ ♥➔② ❧➔ ✤ó♥❣ ♥➳✉ ❤❛✐ ✤✐➲✉ ❦✐➺♥ ✤÷đ❝ t❤ä❛ rữợ t ợ 0+ s ữỡ tr➻♥❤ ❆✐r② ✭✸✳✸✽✮ ❧➔ ♠ët ①➜♣ ①➾ tèt ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✭✸✳✸✼✮✳ ❚❤ù ❤❛✐✱ ✈✐➺❝ sû ❞ö♥❣ ❣➛♥ ✤ó♥❣ t✐➺♠ ❝➟♥ ❝❤♦ ❝→❝ ❤➔♠ sè ❆✐r② ✤á✐ ❤ä✐ t = 2/3 a1/3 x ợ tữỡ ữỡ x ε2/3 ✈ỵ✐ ε → 0+ ❉♦ ✤â✱ ♠✐➲♥ ❤đ♣ ❧➼ ❝õ❛ ✭✸✳✹✺✮ ❧➔ ε2/3 x 1, (ε → 0+) ❚❤➜② r➡♥❣ ✭✸✳✸✾✮ ✈➔ ✭✸✳✹✷✮✱ ✭✸✳✹✹✮ ✈➔ ✭✸✳✹✺✮ ❝â ❝ò♥❣ ❞↕♥❣ ❤➔♠ sè ❞♦ ✤â ❝â t❤➸ ❦➳t ❤đ♣ ✈ỵ✐ ♥❤❛✉✳ ▼➔ ✭✸✳✹✹✮ ✈➔ ✭✸✳✹✺✮ ❝â ♠ët ♠✐➲♥ ❝❤✉♥❣ ❤ñ♣ ❧➼ ♠➔ q✉❛ ✤â ❝â t❤➸ ①↔② r❛ sü ❦➳t ❤ñ♣ s❛✉ ε2/3 x ε2/5 , (ε → 0+) ✭✸✳✹✻✮ ✭✸✳✹✹✮ t ủ ợ tr ỗ ✤à♥❤ ✤÷đ❝ ❝→❝ ❤➡♥❣ sè ✈➔ D E √ D = π(aε)−1/6 C, ✭✸✳✹✼✮ E = ✭✸✳✹✽✮ ✸✽ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆● ◆❤÷ t❛ ✤➣ ❜✐➳t sü ❦➳t ❤đ♣ t✐➺♠ ❝➟♥ ♣❤↔✐ ✤÷đ❝ t❤ü❝ ❤✐➺♥ tr♦♥❣ ♠ët ♠✐➲♥ ♠➔ ♣❤↕♠ ε → ✈✐ ❝õ❛ ♥â trð ♥➯♥ ✈ỉ ❤↕♥ ✈ỵ✐ t❤❛♠ số ỗ ữớ ữ ♣❤↕♠ ♥❣✉②➯♥ t➢❝ ♥➔②✳ ❚✉② ♥❤✐➯♥✱ ❜✐➳♥ ❦➳t ❤đ♣ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❧➔ t ✤÷đ❝ ❝❤♦ ❜ð✐ ✭✸✳✹✶✮✳ ❇✐➳♥ ♥➔② ♣❤ò ❤đ♣ ✈ỵ✐ ♠✐➲♥ t❤ü❝ sü trð ♥➯♥ ✈ỉ ❤↕♥ ✈ỵ✐ x ♠➔ ✤ó♥❣ ❤ì♥ ε−4/15 (ε → 0+) t ✈➔ ♥â ε → ❇➙② ❣✐í ❜➔✐ t♦→♥ ✤➣ ✤÷đ❝ ❣✐↔✐ q✉②➳t ♠ët ♥û❛✳ ❙ü ❦➳t ❤đ♣ t✐➺♠ ❝➟♥ ❣✐ú❛ ♠✐➲♥ II ✤➣ ✤÷đ❝ sû ❞ư♥❣✳ ❚✐➳♣ t❤❡♦ s➩ ♣❤➙♥ t➼❝❤ ♠✐➲♥ t➻♠ t❤➜② tr♦♥❣ ♠✐➲♥ III I ✈➔ ✈➔ ♣❤ò ❤đ♣ ✈ỵ✐ ♥❣❤✐➺♠ ❝❤➾ II ❳➜♣ ①➾ q✉❛♥❣ ❤å❝ ✈➟t ❧➼ tr♦♥❣ ♠✐➲♥ III ❧➔ ♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❤❛✐ ❜✐➸✉ t❤ù❝ ❲❑❇ ❞❛♦ ✤ë♥❣ ♥❤❛♥❤   i ε yIII (x) = F [−P (x)] −1/4 exp  −P (t)dt x   + G [−P (x)] −1/4 exp − i ε −P (t)dt x ✣➸ ❜✐➸✉ t❤ù❝ ♥➔② ♣❤ò ❤đ♣ ợ số F G yII (x) tr ỗ ❝❤➨♦ ❝õ❛ II ✈➔ III, ❝→❝ ❤➡♥❣ ♣❤↔✐ ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦  yIII (x) = 2C[−P (x)]−1/4 sin   ε π −P (t)dt + ✭✸✳✹✾✮ x ❑➳t q✉↔ tr♦♥❣ ✭✸✳✹✾✮ ✤÷đ❝ t❤✐➳t ❧➟♣ ❜➡♥❣ ❝→❝❤ s♦ s→♥❤ ①➜♣ ①➾ t✐➺♠ ❝➟♥ ❝❤♦ ✈➔ yII (x) tr ỗ II r ỗ õ t ữợ t 2Ca1/4 (x)1/4 sin III ❧➔ yIII (x) ε2/3 ε2/5 (ε → 0+) tr♦♥❣ ✭✸✳✹✾✮ ❜ð✐ 1/2 π a (−x)3/2 + 3ε ❱➔ sû ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ s❛✉ ✤è✐ ✈ỵ✐ t➼♥❤ ❝❤➜t t✐➺♠ ❝➟♥ ❝õ❛ Ai(t) = √ (−t)−1/4 sin φ(t), π (−x) yIII (x) Ai(t) π φ(t) ∼ (−t)3/2 + , ✸✾ ✈ỵ✐ ✤è✐ sè ➙♠ ✈➔ ❧ỵ♥ t → −∞ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ õ õ t ữợ ữủ Pì ◆❍❯◆● yIII (x) ✈ỵ✐ E=0 ❜ð✐ −1/2 π a (−x)3/2 + 3ε Dπ −1/2 a−1/12 ε1/6 (−x)−1/4 sin ◆❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ♠➔ t❛ t➻♠ ✤÷đ❝ ①→❝ ❜ð✐ ✈➻ ❝õ❛ ♠✐➲♥ D II ✈➔ C ✈➔ III yII (x) yIII (x) tr ỗ ❝â sü ❧✐➯♥ ❤➺ ✈ỵ✐ ♥❤❛✉ ❜ð✐ ✭✸✳✹✼✮✳ ✣✐➲✉ ♥➔② ❝❤➼♥❤ ❧➔ sü ♣❤➙♥ t➼❝❤ ❚â♠ ❧↕✐✱ t❛ ✤➣ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ y(x) tr♦♥❣ ♠é✐ ♠✐➲♥ I, II ✈➔ III ❈→❝ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ✤â ❧➔   x yI (x) = C[P (x)]−1/4 exp − ε ε2/3 , ε → 0+ P (t)dt , x > 0, x ✭✸✳✺✵✮ √ yII (x) = π(aε)−1/6 CAi(ε−2/3 a1/3 x), |x|  yIII (x) = 2C[−P (x)]−1/4 sin  1, ε → 0+  ε ✭✸✳✺✶✮ −P (t)dt + π , ✭✸✳✺✷✮ x ε2/3 , ε → 0+ x < 0, (−x) ❈æ♥❣ t❤ù❝ t❤ù ♥❤➜t ✈➔ t❤ù ❜❛ ❝á♥ ✤÷đ❝ ❣å✐ ❧➔ ❝ỉ♥❣ t❤ù❝ ❧✐➯♥ ❤➺ ✈➻ ❝❤ó♥❣ t❤➸ ❤✐➺♥ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ t➼♥❤ ❝❤➜t ❞❛♦ ✤ë♥❣ ✈➔ t➼♥❤ ❝❤➜t ❣✐↔♠ ❞➛♥ t❤❡♦ ❤➔♠ ♠ô ❝õ❛ y(x) ✈➲ ❝→❝ ♣❤➼❛ ❝õ❛ ✤✐➸♠ ♥❣♦➦t✳ ❍➡♥❣ sè C ❝❤÷❛ ①→❝ ✤à♥❤ ✈➻ ❝❤➾ ❝â ♠ët ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ✣✐➲✉ ❦✐➺♥ ❜✐➯♥ t❤ù ❤❛✐ ❧➔ ❝➛♥ t❤✐➳t ✤➸ ①→❝ ✤à♥❤ Ai(0) = 3−2/3 / Γ 0.3550280539, C = C y(+∞) = ❱➼ ❞ö✱ ♥➳✉ t❤➻ (aε)1/6 Γ ✹✵ 32/3 π −1/2 y(0) = t❤➻ tø ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆● ❑➳t ❧✉➟♥ ❈❤÷ì♥❣ ✸ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❈❤÷ì♥❣ ✸ tr➻♥❤ ❜➔② ♠ët sè ♥ë✐ ❞✉♥❣ s❛✉ ✶✳ ▼ët sè ✤➦❝ trữ r ợ ❲❑❇ t❤❡♦ ❞↕♥❣ ❤➔♠ ♠ô ✸✳ ❙ü ♠ð rë♥❣ ❞↕♥❣ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❲❑❇ ✹✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ ❤đ♣ ❧➼ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ ❲❑❇ ✺✳ ❙ü ❦➳t ❤đ♣ ❝→❝ ♣❤➨♣ ❣➛♥ ✤ó♥❣ t✐➺♠ ❝➟♥✿ ◆❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝â ♠ët ✤✐➸♠ ♥❣♦➦t ✹✶ ❑➌❚ ▲❯❾◆ ❉ü❛ tr➯♥ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❦❤â❛ ❧✉➟♥ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ✈➜♥ ✤➲ s❛✉ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❝❤✉é✐ ❧ơ② t❤ø❛✱ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ n, ❜➔✐ t♦→♥ ❈❛✉❝❤②✳ ✷✳ P❤÷ì♥❣ ♣❤→♣ ❲❑❇ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❜➔✐ t♦→♥ ❦❤ỉ♥❣ ♥❤✐➵✉✳ ✸✳ P❤÷ì♥❣ ♣❤→♣ ❲❑❇ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❜➔✐ t♦→♥ ❜à ♥❤✐➵✉✳ Ù♥❣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❲❑❇ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❝á♥ ♥❤✐➲✉ ✤✐➲✉ ợ ọ ữ ữủ tr ❜➔② tr♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❈✉è✐ ❝ò♥❣✱ t✉② ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ❞♦ t❤í✐ ❣✐❛♥ ✈➔ ❦❤↔ ♥➠♥❣ ❝â ❤↕♥ ♥➯♥ ❝→❝ ✈➜♥ ✤➲ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ✈➝♥ ❝❤÷❛ ✤÷đ❝ tr➻♥❤ ❜➔② s➙✉ s➢❝ ✈➔ ❦❤ỉ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ❝â ♥❤ú♥❣ s❛✐ sât tr♦♥❣ ❝→❝❤ tr➻♥❤ ❜➔②✳ ❊♠ ữủ sỹ õ ỵ t ổ ❜↕♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✹✷ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬❆✳❪ [1] ✲ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❱✐➺t ◆❣✉②➵♥ ❚❤➳ ❍♦➔♥✱ P❤↕♠ P❤✉ ✭✷✵✶✵✮✱ ❈ì sð ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈➔ ❧➼ t❤✉②➳t ê♥ ✤à♥❤✱ ◆❳❇ ●✐→♦ ❞ö❝ ❱✐➺t ◆❛♠✳ [2] ❚r➛♥ ✣ù❝ ▲♦♥❣✱ ◆❣✉②➵♥ ✣➻♥❤ ❙❛♥❣✱ ❍♦➔♥❣ ◗✉è❝ ❚♦➔♥ ✭✷✵✵✺✮✱ ●➼❛♦ tr➻♥❤ ❣✐↔✐ t➼❝❤ t➟♣ ✷✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ●✐❛ ❍➔ ◆ë✐✳ [❇] [3] ✲ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❆♥❤ ❖✳▼✳ ❇❡♥❞❡r ❛♥❞ ❙✳❆✳ ❖rs③❛❣ ✭✶✾✾✾✮✱ ❆❞✈❛♥t❡❞ ▼❛t❤❡♠t✐❝❛❧ ▼❡t❤♦❞s ❢♦r ❙❝✐❡♥✲ t✐sts ❛♥❞ ❊♥❣✐♥❡❡rs✱ ◆❡✇ ❨♦r❦✿ ▼❝●r❛✇ ✕ ❍✐❧❧✳ [4] ❚✳❙✳▲✳ ❘❛❞✐❤✐❦❛✱ ❚✳❑✳❱✳ ■②❡♥❣❛r✱ ❚✳❘❛❥❛ ❘❛♥✐ ✭✷✵✶✺✮✱ ❆♣♣r♦①✐♠❛t❡ ❛♥❛❧②t✐❝❛❧ ♠❡t❤♦❞s ❢♦r s♦❧✈✐♥❣ ♦r❞✐♥❛r② ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❈❘❈ Pr❡ss✱ ❚❛②❧♦r ●r♦✉♣✱ ❆ ❈❤❛♣♠❛♥ & ❍❛❧❧ ❜♦♦❦✳ ✹✸ & ❋r❛♥❝✐s ... PHẠM HÀ NỘI KHOA TOÁN ************* ĐOÀN THỊ PHƯƠNG NHUNG PHƯƠNG PHÁP WKB ĐỂ GIẢI PHƯƠNG TRÌNH VI PHÂN THƯỜNG KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Giải tích Người hướng dẫn khoa học PSG.TS