TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI 2
KHOA TOÁN
*************
ĐOÀN THỊ PHƯƠNG NHUNG
PHƯƠNG PHÁP WKB ĐỂ GIẢI 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
*************
ĐOÀN THỊ PHƯƠNG NHUNG
PHƯƠNG PHÁP WKB ĐỂ GIẢI 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
▲❮■ ❈❷▼ ❒◆
❚r÷î❝ ❦❤✐ tr➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✱ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣
❜✐➳t ì♥ s➙✉ s➢❝ tî✐
P●❙✳❚❙ ❑❤✉➜t ❱➠♥ ◆✐♥❤
♥❣÷í✐ ✤➣ t➟♥ t➻♥❤ ❤÷î♥❣ ❞➝♥ ✤➸
❡♠ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳
❊♠ ❝ô♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tî✐ t♦➔♥ t❤➸ ❝→❝ t❤➛② ❝æ tr♦♥❣ ❑❤♦❛
❚♦→♥✱ ❚r÷í♥❣ ✤↕✐ ❤å❝ s÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷ ✤➣ ❞↕② ❜↔♦ ❡♠ t➟♥ t➻♥❤ tr♦♥❣ s✉èt q✉→
tr➻♥❤ ❤å❝ t➟♣ t↕✐ ❦❤♦❛✳
◆❤➙♥ ❞à♣ ♥➔② ❡♠ ❝ô♥❣ ①✐♥ ✤÷ñ❝ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tî✐ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧
✤➣ ❧✉æ♥ ❜➯♥ ❡♠✱ ✤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❝ ✈➔ ❝❤÷❛
❤➲ ✤÷ñ❝ sû ❞ö♥❣ ✤➸ ❜↔♦ ✈➺ ♠ët ❤å❝ ✈à ♥➔♦✱ ❝→❝ t❤æ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❦❤â❛ ❧✉➟♥
✤➣ ✤÷ñ❝ ❝❤➾ rã ♥❣✉ç♥ ❣è❝ ♠ët ❝→❝❤ rã r➔♥❣✳ ❊♠ ❤♦➔♥ t♦➔♥ ❝❤à✉ tr→❝❤ ♥❤✐➺♠ tr÷î❝
♥❤➔ tr÷í♥❣ ✈➲ sü ❝❛♠ ✤♦❛♥ ♥➔②✳
❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽
❙✐♥❤ ✈✐➯♥
✣♦➔♥ ❚❤à P❤÷ì♥❣ ◆❤✉♥❣
ử ử
é
ộ ụ tứ
ộ ụ tứ
ở tử ừ ộ ụ tứ
tr t ộ ụ tứ
Pữỡ tr
ữỡ tr
t
ừ ữỡ tr
tỗ t t ừ t
Pì PP ể
t ổ
ử ử
Pì PP ể
ởt số trữ r ừ ợ
t t t t t
ú t ụ
ỹ rở tự ú
ủ ỵ ừ 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❖
✹✷
✐✐
ớ
ỵ ồ t
t ồ ữủ t s qt t õ ỗ
ố tỹ t ũ ợ sỹ t tr ừ ở t ồ ồ
ồ t ỹ ồ tt ồ ự ử
r ỹ ự ử tữớ rt ỳ t q
ữỡ tr tữớ ự ữỡ tr
tữớ õ trỏ q trồ tr ỵ tt ồ ú t t r
ởt số t ữỡ tr tữớ õ t t ữủ tr
ợ ữỡ tr s tứ t tỹ t ổ t ữủ
ởt t r t
ú ừ ữỡ tr
t t tứ õ ồ t r ữỡ
ú ữỡ tr tữớ
r õ tr ởt ữỡ t t
ữủ ồ ữỡ t rr t
r rrs r t tr ởt tr
ỳ ữỡ t ữủ t ử ừ
ừ ữỡ tr
ữợ sỹ ữợ t t ừ P t ồ t õ
tốt Pữỡ P ữỡ tr tữớ t
s ỡ ữỡ
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆●
✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉
◆❣❤✐➯♥ ❝ù✉ ù♥❣ ❞ö♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❲❑❇ ✈➔♦ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ♣❤÷ì♥❣ tr➻♥❤
✈✐ ♣❤➙♥ t❤÷í♥❣✳
✸✳ ✣è✐ t÷ñ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉
✲ ✣è✐ t÷ñ♥❣ ♥❣❤✐➯♥ ❝ù✉✿ P❤÷ì♥❣ ♣❤→♣ ❲❑❇✳
✲ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ P❤÷ì♥❣ ♣❤→♣ ❲❑❇ ✈➔ ♠ët sè ù♥❣ ❞ö♥❣ ❝õ❛ ❝❤ó♥❣ tr♦♥❣
❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✳
✹✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉
•
P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ t➔✐ ❧✐➺✉
•
P❤÷ì♥❣ ♣❤→♣ ✤→♥❤ ❣✐→
•
P❤÷ì♥❣ ♣❤→♣ ❤➺ t❤è♥❣ ❤â❛
✺✳ ❈➜✉ tró❝ ❦❤â❛ ❧✉➟♥
◆ë✐ ❞✉♥❣ ❦❤â❛ ❧✉➟♥ ❣ç♠ ✸ ❝❤÷ì♥❣✿
❈❤÷ì♥❣ ✶ ✏❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳✑ ❈❤÷ì♥❣ ♥➔② ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ❝❤✉é✐ ❧ô②
t❤ø❛✱ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣
n,
❜➔✐ t♦→♥ ❈❛✉❝❤②✳
❈❤÷ì♥❣ ✷ ✏P❤÷ì♥❣ ♣❤→♣ ❲❑❇ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❜➔✐ t♦→♥ ❦❤æ♥❣ ♥❤✐➵✉✳✑ ▼ö❝ ✤➼❝❤ ❝õ❛
❝❤÷ì♥❣ ♥➔② ❧➔ ❣✐î✐ t❤✐➺✉ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❲❑❇ ✤➸ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝ ❜➔✐ t♦→♥ ❦❤æ♥❣
♥❤✐➵✉ ✈➔ ♠ët sè ✈➼ ❞ö →♣ ❞ö♥❣✳
❈❤÷ì♥❣ ✸ ✏P❤÷ì♥❣ ♣❤→♣ ❲❑❇ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❜➔✐ t♦→♥ ❜à ♥❤✐➵✉✳✑ ▼ö❝ ✤➼❝❤ ❝õ❛
❝❤÷ì♥❣ ♥➔② ❧➔ ❣✐î✐ t❤✐➺✉ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❲❑❇ ✤➸ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ❜à ♥❤✐➵✉ ✈➔ ♠ët
sè ✈➼ ❞ö →♣ ❞ö♥❣✳
❑❤â❛ ❧✉➟♥ ♥➔② ✤÷ñ❝ tr➻♥❤ ❜➔② tr➯♥ ❝ì sð ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✤÷ñ❝ ❧✐➺t ❦➯ tr♦♥❣
♣❤➛♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ✣â♥❣ ❣â♣ ❝õ❛ ❡♠ t❤➸ ❤✐➺♥ ð ❝❤é✱ →♣ ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣
❲❑❇ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤✱ t➻♠ ✤÷ñ❝ ♠ët sè ✈➼ ❞ö ♠✐♥❤ ❤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✳ ❊♠ ♠♦♥❣ ♥❤➟♥ ✤÷ñ❝ sü ✤â♥❣ ❣â♣ þ ❦✐➳♥ ♣❤↔♥
❜✐➺♥ ❝õ❛ q✉þ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥✳
❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦
✸
❈❤÷ì♥❣ ✶
▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆
❇➚
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② t→❝ ❣✐↔ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ❝❤✉é✐ ❧ô② t❤ø❛✱ ♣❤÷ì♥❣
tr➻♥❤ ✈✐ ♣❤➙♥✱ ❜➔✐ t♦→♥ ❈❛✉❝❤②✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷ñ❝ 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
✤÷ñ❝ ❣å✐ ❧➔ t➙♠ ❝õ❛ ❝❤✉é✐ ❧ô② t❤ø❛✳ ✣➸ þ r➡♥❣ ❝❤✉é✐ ❧ô② t❤ø❛ ❧✉æ♥ ❤ë✐
+∞
tö t↕✐ ✤✐➸♠ x = x0 . ◆➳✉ ✤➦t y = x − x0 t❤➻ ❝â t❤➸ ✤÷❛ ❝❤✉é✐ ✈➲ ❞↕♥❣
an y n ,
n=0
❝❤✉é✐ ❝â t➙♠ t↕✐
y = 0✳
✶✳✶✳✷ ❇→♥ ❦➼♥❤ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ❧ô② t❤ø❛
✣à♥❤ ❧➼ ❆❜❡❧
❈❤♦ ❝❤✉é✐ ❧ô② t❤ø❛
+∞
an xn = a0 + a1 x + a2 x2 + ...
n=0
✹
✭✶✳✶✮
õ tốt ồ
õ tỗ t ởt số
ợ
0 R +
ộ ở tử tr
ợ
R
Pì
s
( R, R) ở tử tr ộ [r, r]
0 < r < R.
ồ
x
|x| > R
t ộ
tr t ộ ụ tứ
ởt õ t tr t ộ ụ tứ
ỵ
a (x x )
+
sỷ ộ ụ tứ
n
0
n
õ ở tử
R>0
n=0
+
an (x x0 )n , x (x0 R, x0 + R).
f (x) =
n=0
õ
ổ tr
(x0 R, x0 + R).
f (n) (x0 )
f (x) =
(x x0 )n x (x0 R, x0 + R).
n!
n=0
f (n) (x0 )
an =
n = 0, 1...
n!
+
ỵ
x0 .
sỷ õ ồ tr ởt õ ừ
Rn (x)
ữ r ừ ổ tự r
Rn (x) =
f (n+1) (x0 + (x x0 )
(x x0 )n+1
(n + 1)!
õ tr ừ
x0 : limRn (x) = 0
tr ữủ t ộ r t
ỵ
tr ởt
ồ
f (n) (n = 1, 2, ...)
f (x)
õ t
x0 .
(x0 , x0 + )
tỗ t ởt số
x0 .
ừ
M >0
M (n = 1, 2...) ợ ồ x (x0 , x0 + ) t
ộ r t
t
x0
õ
s
f (n) (x)
õ t tr ữủ t
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆●
✶✳✷ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♥
✶✳✷✳✶ ❑❤→✐ ♥✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♥
P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♥ ❝â ❞↕♥❣ tê♥❣ q✉→t ✿
F (x, y, y , ..., y (n) ) = 0
✭✶✳✷✮
tr♦♥❣ ✤â ❤➔♠ ❋ ①→❝ ✤à♥❤ tr♦♥❣ ♠ët ♠✐➲♥ ● ♥➔♦ ✤➜② ❝õ❛ ❦❤æ♥❣ ❣✐❛♥
♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ ❝â t❤➸ ✈➢♥❣ ♠➦t ♠ët sè tr♦♥❣ ❝→❝ ❜✐➳♥
y (n)
Rn+2
✳ ❚r♦♥❣
x, y, y , ..., y (n−1)
♥❤÷♥❣
♥❤➜t t❤✐➳t ♣❤↔✐ ❝â ♠➦t✳ ◆➳✉ tø ✭✶✳✷✮ t❛ ❣✐↔✐ r❛ ✤÷ñ❝ ✤↕♦ ❤➔♠ ❝➜♣ ❝❛♦ ♥❤➜t✱ tù❝
❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ ❝â ❞↕♥❣
y (n) = f (x, y, y , . . . , y (n−1) )
t❤➻ t❛ ✤÷ñ❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♥ ✤➣ ❣✐↔✐ r❛ ✤è✐ ✈î✐ ✤↕♦ ❤➔♠ ❝➜♣ ❝❛♦ ♥❤➜t✳
❈➜♣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ ✤↕♦ ❤➔♠ ❝➜♣ ❝❛♦ ♥❤➜t ❝â ♠➦t tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤✳
❍➔♠ sè
y = ϕ(x), x ∈ (a, b)
✤÷ñ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ ♥➳✉ t❤❛②
y = ϕ(x), y = ϕ (x), . . . , y (n) = ϕ(n) (x)
✭✶✳✷✮ trð t❤➔♥❤ ✤ç♥❣ ♥❤➜t tr➯♥ ❦❤♦↔♥❣
✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ t❤➻ ♣❤÷ì♥❣ tr➻♥❤
(a, b).
✶✳✷✳✷ ❇➔✐ t♦→♥ ❈❛✉❝❤②
❛✮ ❇➔✐ t♦→♥ ❈❛✉❝❤②
◆➳✉ tø ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ ❣✐↔✐ r❛ ✤÷ñ❝ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ✈î✐ ✤↕♦ ❤➔♠ ❝➜♣ ❝❛♦ ♥❤➜t
y (n) = f x, y, y , . . . , y (n−1)
tr♦♥❣ ✤â ❢ ❧➔ ❤➔♠ ①→❝ ✤à♥❤ tr➯♥ ♠✐➲♥
●✐↔ sû ✤✐➸♠ tr♦♥❣
✭✶✳✸✮
D ⊂ Rn+1 .
(n−1)
∈ D ⊂ Rn .
x0 , y0 , y0 , . . . , y0
❇➔✐ t♦→♥✿
y (n) = f (x, y, y , . . . , y (n−1) ),
(x, y, y , . . . , y (n−1) ) ∈ D
(n−1)
y(x0 ) = y0 , y (x0 ) = y0 , . . . , y (n−1) (x0 ) = y0
✻
✭✶✳✹❛✮
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆●
●å✐ ❧➔ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❜❛♥ ✤➛✉✳
✣✐➲✉ ❦✐➺♥ ✭✶✳✹❛✮ ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉✳
❜✮ ❇➔✐ t♦→♥ ❣✐→ trà ❜✐➯♥ ✤è✐ ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❝➜♣ ❤❛✐
❈❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❝➜♣ ❤❛✐
F (x, y, y , y ) = 0
✭✶✳✺✮
❇➔✐ t♦→♥ ❜✐➯♥ ❤❛✐ ✤✐➸♠ ❝➜♣ ❤❛✐ ✤è✐ ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ ✤÷ñ❝ ✤➦t r❛ ♥❤÷ s❛✉✿
❚➻♠ ❤➔♠
y = y(x)
s❛♦ ❝❤♦ ♥â t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ tr♦♥❣ ❦❤♦↔♥❣
(a, b)
✈➔
t↕✐ ❤❛✐ ✤✐➸♠ ❛✱❜ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥
ϕ [y (a) , y (a)] = 0
1
ϕ2 [y (b) , y (b)] = 0
✭✶✳✻✮
✶✳✷✳✸ ❈→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
❝➜♣ ♥
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳
◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♥ ✭✶✳✷✮ ❧➔ ❤➔♠
✈✐ ♥ ❧➛♥ tr➯♥ ❦❤♦↔♥❣
❛✮
(a, b)
y = ϕ(x)❦❤↔
s❛♦ ❝❤♦
(x, ϕ(x), ϕ (x), . . . , ϕn (x)) ∈ G, ∀x ∈ (a, b).
❜✮ ◆â ♥❣❤✐➺♠ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ tr➯♥
◆❣❤✐➺♠ tê♥❣ q✉→t✳
❚❛ ❣✐↔ t❤✐➳t r➡♥❣
(a, b).
G
❧➔ ♠✐➲♥ tç♥ t↕✐ ✈➔ ❞✉② ♥❤➜t ♥❣❤✐➺♠
❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮✱ tù❝ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② tç♥ t↕✐ ✈➔ ❞✉② ♥❤➜t ✤è✐
✈î✐ ♠é✐ ✤✐➸♠
❍➔♠
(n−1)
x0 , y0 , y0 , . . . , y0
∈ D ⊂ Rn .
φ(x, y, C1 , C2 , . . . , Cn ) ①→❝ ✤à♥❤ tr♦♥❣ ♠✐➲♥ ❜✐➳♥ t❤✐➯♥ ❝õ❛ ❝→❝ ❜✐➳♥ x, C1 , C2 , . . . , Cn
❝â t➜t ❝↔ ❝→❝ ✤↕♦ ❤➔♠ r✐➯♥❣ t❤❡♦
x
❧✐➯♥ tö❝ ✤➳♥ ❝➜♣
✼
n
✤÷ñ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ tê♥❣ q✉→t
õ tốt ồ
Pì
ừ ữỡ tr tr
g
tr
g
tứ ữỡ tr
y0 = (x0 , C1 , C2 , . . . , Cn )
y = (x0 , C1 , C2 , . . . , Cn )
x
0
...
y (n1) = (n1)
(x0 , C1 , C2 , . . . , Cn )
x
0
õ t ữủ
)
(n1)
0
C2 = 2 (x0 , y0 , y0 , . . . , y0
)
...
(n1)
0
Cn = n (x0 , y0 , y0 , . . . , y0
)
(n1)
C10 = 1 (x0 , y0 , y0 , . . . , y0
y = (x, C10 , C20 , . . . , Cn0 )
C10 , C20 , . . . , Cn0
ừ ữỡ tr ự ợ ộ
ữủ tứ
(n1)
(x0 , y0 , y0 , . . . , y0
)
t tr
G.
r
ừ ữỡ tr t õ ộ ừ õ t
t ừ t ữủ ữủ ồ r ừ
ữỡ tr ữủ tứ tờ qt ợ tr
ừ sốC1 , C2 , ..., Cn r
ý
ừ ữỡ tr t ộ ừ õ t
t ừ t ù ữủ ồ
ừ ữỡ tr õ t ởt ồ ử tở ởt số số tũ
ỵ ữ số số tũ ỵ ổ ữủ q
n 1
tỗ t t ừ t
sỷ tr
st t
G Rn+1
u1 , u2 , . . . , un
tỗ t t
f (x, u1 , u2 , . . . , un ) tử tọ
õ ợ t ý tr
y = y(x)
(n1)
(x0 , y0 , y0 , . . . , y0
)G
ừ ữỡ tr tọ
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆●
✤➛✉
(n−1)
y(x0 ) = y0 , y (x0 ) = y0 , . . . , y (n−1) (x0 ) = y0
◆❣❤✐➺♠ ♥➔② ①→❝ ✤à♥❤ t↕✐ ❧➙♥ ❝➟♥✱ ♥â✐ ❝❤✉♥❣✱ ❦❤→ ❜➨ ❝õ❛
❑➳t ▲✉➟♥ ❈❤÷ì♥❣ ✶
x0 .
◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❈❤÷ì♥❣ ✶ ❧➔ ♥➯✉ ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲
✶✳ ❈❤✉é✐ ❧ô② t❤ø❛
✷✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♥
✸✳ ❇➔✐ t♦→♥ ❈❛✉❝❤②
✾
❈❤÷ì♥❣ ✷
P❍×❒◆● P❍⑩P ❲❑❇ ●■❷■
●❺◆ ✣Ó◆● ❇⑨■ ❚❖⑩◆
❑❍➷◆● ◆❍■➍❯
✷✳✶ ❇➔✐ t♦→♥ ❦❤æ♥❣ ♥❤✐➵✉
❚r÷î❝ t✐➯♥ ❡♠ s➩ tr➻♥❤ ❜➔② ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❲❑❇ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝❤♦ ❜➔✐ t♦→♥ ❦❤æ♥❣
❜à ♥❤✐➵✉ ♠➔ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣
d2 y
+ f (x)y = 0
dx2
tr♦♥❣ ✤â
f (x)
❧➔ ♠ët ❤➔♠ sè ❜✐➳♥ t❤✐➯♥ ❝❤➟♠ ❝õ❛ ①✳
❑❤→✐ ♥✐➺♠ ❜✐➳♥ t❤✐➯♥ ❝❤➟♠ ✤÷ñ❝ ❤✐➸✉ ❧➔ ❤➔♠ ✤â ❝â
tr➻♥❤ ✭✷✳✶✮ ✈î✐
✭✷✳✶✮
f (x)
|f (x)| ♥❤ä✳ ◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣
✤÷ñ❝ ❝♦✐ ❧➔ ♠ët ❤➡♥❣ sè ❣✐ó♣ ❡♠ t❤➜② r➡♥❣ ♥❣❤✐➺♠ ❝â t❤➸ ✤÷ñ❝
✈✐➳t t❤❡♦ ❞↕♥❣
y(x) = eiψ(x) .
✭✷✳✷✮
▲➜② ✤↕♦ ❤➔♠ ❤❛✐ ✈➳ ❜✐➸✉ t❤ù❝ ✤è✐ ✈î✐ ①✱ t❛ ❝â
y (x) = ieiψ(x) ψ (x).
✶✵
✭✷✳✸✮
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆●
▲➜② ✤↕♦ ❤➔♠ ❧➛♥ ♥ú❛✱ t❛ ✤÷ñ❝
2
y (x) = i2 eiψ(x) (ψ (x)) + ieiψ(x) ψ (x) .
✭✷✳✹✮
❚❤➳ ❝→❝ ❜✐➸✉ t❤ù❝ ✭✷✳✹✮ ✈➔ ✭✷✳✷✮ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ t❛ t❤✉ ✤÷ñ❝
−ψ 2 (x) + iψ (x) + f (x) = 0.
●✐↔ sû r➡♥❣
ψ (x)
♥❤ä✱ ❦❤✐ ✤â ❜ä q✉❛ ✤↕✐ ❧÷ñ♥❣
ψ (x)
✭✷✳✺✮
tø ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✺✮ ❝â t❤➸
t❤✉ ✤÷ñ❝
ψ (x) = ±
f (x),
ψ (x) = ±
✣✐➲✉ ❦✐➺♥ ❤ñ♣ ❧➼ ❝õ❛ ❞↕♥❣ tr➯♥ ❧➔ ✭
|ψ (x)| ≈
f (x)dx.
ψ (x)
1
2
✭✷✳✻✮
✭✷✳✼✮
♥❤ä✮
f (x)
f (x)
≤ |f (x)| .
✭✷✳✽✮
❉♦ ✤â✱ ✤✐➲✉ ❦✐➺♥ ❤ñ♣ ❧➼ ❝õ❛ ♣❤➨♣ ❣➛♥ ✤ó♥❣ ❧➔ ❦❤↔ q✉❛♥ ❦❤✐ sü t❤❛② ✤ê✐ ❝õ❛
tr♦♥❣ ♠ët ❜÷î❝ sâ♥❣ ♥➯♥ ❝â ❣✐→ trà ♥❤ä ❦❤✐ s♦ s→♥❤ ✈î✐
|f (x)|
f (x)
✳
✣➸ t➻♠ ♣❤➨♣ ❣➛♥ ✤ó♥❣ t❤ù ❤❛✐✱ ❡♠ t✐➳♥ ❤➔♥❤ ♥❤÷ s❛✉✿
❚❤➳ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✽✮ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✺✮✱ t❛ ✤÷ñ❝
2
i f (x)
,
2 f (x)
i f (x)
f (x) +
,
4 f (x)
i
f (x)dx + ln f (x) ≈ ±
4
(ψ (x)) ≈ f (x) ±
ψ (x) ≈
ψ(x) ≈
✶✶
✭✷✳✾✮
✭✷✳✶✵✮
i
f (x)dx + ln (f (x)) 4 .
✭✷✳✶✶✮
õ tốt ồ
Pì
(x) tứ ữỡ tr ữỡ tr t ữủ tờ qt
ừ t
y(x) e
i
e
i
e
i
e
i
f (x)dx
f (x)dx
c1
f (x)dx
1
c2
i2
eln (f (x)) 4
(f (x)) 4
tr õ
i
ei ln (f (x)) 4
1
i
f (x)dx + ln (f (x)) 4
eln (f (x))
1
4
f (x)dx + c2 exp i
c1 exp i
f (x)dx .
số tũ ỵ ứ õ t t ữủ ởt ừ
tờ qt ừ ữỡ tr tr ởt ừ ữỡ tr
ú ỵ Pữỡ ổ t ổ
f (x)
f (x)
t ờ q
t tr q t tự
q trồ ỡ t t ừ t t r ũ
f (x) > 0
ở tr ũ
ồ
f (x) > 0
f (x) = x = 4
ợ t
f (x)
x
x,
(f (x))
x
1
(f (x))
õ
x0
f (x)dx + exp
x
x0 , f (x) > 0,
4
õ ở tỹ
õ ụ ổ ũ ủ
x0
exp
y (x)
cos2x
t trữớ ủ s t s
x0 , f (x) < 0
1
4
t õ t tợ ổ ữủ ồ
y (x)
sin2x
x = 0
tọ ừ ữỡ tr tr
tr ừ
x = 0
õ t t ụ t
õ ữớ t tợ ổ t
f (x) = 4
tr ố ợ
t sỷ
f (x) = x
f (x) < 0
f (x)dx .
x
õ
x
x
f (x) dx + exp i
exp i
x0
f (x) dx .
x0
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆●
✷✳✷ ❱➼ ❞ö →♣ ❞ö♥❣
❱➼ ❞ö ✷✳✶✳
❚➻♠ ❤➔♠ ❲❑❇ ❧✐➯♥ ❦➳t ✈î✐ ♣❤÷ì♥❣ tr➻♥❤
y + xy = 0
tr♦♥❣ ✤â
x
0.
▲í✐ ❣✐↔✐✿
❙♦ s→♥❤ ❜➔✐ t♦→♥ ✤➣ ❝❤♦ ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ t❛ t❤➜② r➡♥❣
f (x) = x
❱➻
x
0,
t❤✉ ✤÷ñ❝ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❲❑❇ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✹✮ ❧➔
x
1
exp i
y(x) = √
4
x
√
x
xdx + exp −i
0
√
xdx
0
x
1
1
=(x)− 4 exp ±i
(x) 2 dx
0
=(x)
✯❈❤ó þ✿ ◆➳✉f (x)
= −x,
− 14
3
2
exp ±i (x) 2 .
3
t❤➻ t❛ sû ❞ö♥❣ ❝æ♥❣ t❤ù❝ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✸✮ ✤➸
❝â ✤÷ñ❝ ♣❤➨♣ ❣➛♥ ✤ó♥❣ ❲❑❇✳ ▲÷✉ þ r➡♥❣ ♥❣❤✐➺♠ t❤✉ ✤÷ñ❝ ❧➔ ♥❤÷ ♥❤❛✉ ❜➡♥❣ ❝→❝❤
sû ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ t✐➺♠ ❝➟♥ ✭tø ✈➼ ❞ö ♥➔②✱ rã r➔♥❣ t❤➜② r➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❲❑❇
❝â t→❝ ✤ë♥❣ ♠↕♥❤ ♠➩ ❧➔ ♥â ❝✉♥❣ ❝➜♣ ♥❣❤✐➺♠ tr♦♥❣ ♠ët ❜÷î❝ ❞✉② ♥❤➜t✳✮
P❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ❱➼ ❞ö ✷✳✶ ✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❆✐r② ✈➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛
♥â ✤÷ñ❝ ❣å✐ ❧➔ ♥❤ú♥❣ ❤➔♠ sè ❆✐r②✳ ◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈î✐ ♠ët ✤✐➸♠ ♥❣♦➦t
t❤÷í♥❣ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ❤➔♠ ❆✐r②✳
❱➼ ❞ö ✷✳✷✳
❚➻♠ ❤➔♠ ❲❑❇ ❧✐➯♥ ❦➳t ✈î✐ ♣❤÷ì♥❣ tr➻♥❤
y + 3xy = 0
tr♦♥❣ ✤â
x
0.
▲í✐ ❣✐↔✐✿
❙♦ s→♥❤ ❜➔✐ t♦→♥ ✤➣ ❝❤♦ ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮✱ t❛ t❤➜② r➡♥❣
❱➻
x
0,
f (x) = 3x.
t❤✉ ✤÷ñ❝ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❲❑❇ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✸✮
✶✸
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆●
❧➔
y(x) ≈ √
4
0
0
√
√
1
exp
−3xdx + exp
−3xdx
−3x
x
x
0
0
≈ (−3x)
−1
4
1
exp
x
≈ (−3x)
−1
4
1
i(3x) 2 dx + exp
3
2√
exp ±i
3(x) 2 .
3
i(3x) 2 dx
x
❑➳t ❧✉➟♥ ❈❤÷ì♥❣ ✷
◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ♠ët sè ♥ë✐ ❞✉♥❣ s❛✉
✶✳ P❤÷ì♥❣ ♣❤→♣ ❲❑❇ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝ ❜➔✐ t♦→♥ ❦❤æ♥❣ ♥❤✐➵✉
✷✳ ❱➼ ❞ö →♣ ❞ö♥❣
✶✹
❈❤÷ì♥❣ ✸
P❍×❒◆● P❍⑩P ❲❑❇ ●■❷■
●❺◆ ✣Ó◆● ❇⑨■ ❚❖⑩◆ ❇➚
◆❍■➍❯
◆❤÷ ✤➣ t➻♠ ❤✐➸✉ tr♦♥❣ ♠ö❝ tr÷î❝✱ t❛ t❤➜② r➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❲❑❇ ❝❤♦ ❧í✐ ❣✐↔✐
❝õ❛ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤❡♦ ♠ët ❝→❝❤ ✤ì♥ ❣✐↔♥✳ ▼ët ❦❤➼❛ ❝↕♥❤ q✉❛♥ trå♥❣
❧➼ t❤✉②➳t ❲❑❇ ♥➔② ❝â t❤➸ ✤÷ñ❝ ❝♦✐ ♥❤÷ ❧➔ ♠ët ❧þ t❤✉②➳t tê♥❣ q✉→t ✤➸ t➻♠ r❛ ❝→❝
♥❣❤✐➺♠ ❝õ❛ ♥❤ú♥❣ ❜➔✐ t♦→♥ ❜à ♥❤✐➵✉ ✈➻ ♥â ❝❤ù❛ ❧➼ t❤✉②➳t ❧î♣ ❜✐➯♥ ♥❤÷ ♠ët tr÷í♥❣
❤ñ♣ ✤➦❝ ❜✐➺t✳
✸✳✶ ▼ët sè ✤➦❝ tr÷♥❣ r✐➯♥❣ ❝õ❛ ❝→❝ ♥❣❤✐➺♠ ❣➛♥
❧î♣ ❜✐➯♥
✸✳✶✳✶ ❚➼♥❤ ❝❤➜t t→♥ ①↕ ✈➔ t➼♥❤ ❝❤➜t ♣❤➙♥ t→♥
❛✮ ❚➼♥❤ ❝❤➜t t→♥ ①↕
❚r♦♥❣ ❧þ t❤✉②➳t ❧î♣ ❜✐➯♥✱ ♥❤÷ ✤➣ t➻♠ ❤✐➸✉ t❤➻ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✤➣ ❝❤➾ r❛ ❝→❝❤ ①➙②
❞ü♥❣ ♠ët ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝❤♦ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜à ♥❤✐➵✉✳ ❱✐➺❝ ①➙② ❞ü♥❣
♥➔② ✤á✐ ❤ä✐ sü ❦➳t ❤ñ♣ ❝→❝ ♥❣❤✐➺♠ ♥❣♦➔✐ t❤❛② ✤ê✐ tø tø ✈î✐ ❝→❝ ♥❣❤✐➺♠ tr♦♥❣ t❤❛②
✤ê✐ ♥❤❛♥❤✳ ❈❤ó þ r➡♥❣ ♥❣❤✐➺♠ ♥❣♦➔✐ ✈➝♥ ❣✐ú ♥❣✉②➯♥ ♥➳✉ ❝❤♦ ♣❤➨♣ t❤❛♠ sè ♥❤✐➵✉
ε
t✐➳♥ tî✐
0+.
❚✉② ♥❤✐➯♥✱ ♥❣❤✐➺♠ tr♦♥❣ trð ♥➯♥ ❣✐→♥ ✤♦↕♥ tr➯♥ ❧î♣ ❜✐➯♥ ✈➻ ❜➲ ❞➔②
✶✺
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
❝õ❛ ❧î♣ ❜✐➯♥ ❝â ①✉ ❤÷î♥❣
0.
✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆●
❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔②✱ ❝â t❤➸ ♥â✐ r➡♥❣ ♥❣❤✐➺♠ ♥➔② ❜à
♠ët sü ❝è ❝ö❝ ❜ë ð ❧î♣ ❜✐➯♥ ✈î✐
ε → 0+.
❙ü ❝è ❝ö❝ ❜ë ①↔② r❛ ❦❤✐ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣
t➠♥❣ ❤♦➦❝ ❣✐↔♠ t❤❡♦ ❝➜♣ sè ♥❤➙♥✳ ❚➼♥❤ ❝❤➜t ♥➔② ✤÷ñ❝ ❣å✐ ❧➔ t➼♥❤ ❝❤➜t t→♥ ①↕ ✈➻ ❝→❝
②➳✉ tè t❤❛② ✤ê✐ ♥❤❛♥❤ ❝❤â♥❣ ❝õ❛ ♥❣❤✐➺♠ ♣❤➙♥ r➣ t❤❡♦ ❝➜♣ sè ♥❤➙♥✭ t→♥ ①↕ ✮ r❛ ❦❤ä✐
✤✐➸♠ ♣❤➙♥ t➼❝❤ ❝õ❛ sü ❝è ❝ö❝ ❜ë✳
❜✮ ❚➼♥❤ ❝❤➜t ♣❤➙♥ t→♥
▼ët sè ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈î✐ ❝→❝ t❤❛♠ sè ♥❤ä ❝â ♥❣❤✐➺♠ t❤➸ ❤✐➺♥ sü ♣❤➙♥ r➣
t♦➔♥ ❝➛✉✳ ❱➼ ❞ö ❜➔✐ t♦→♥ ❣✐→ trà ❜✐➯♥
εy + y = 0,
y(0) = 0, y(1) = 1.
✭✸✳✶✮
◆❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❧➔
√
sin(x/ ε)
√ ,
y(x) =
sin(1/ ε)
◆❣❤✐➺♠ ♥➔② trð ♥➯♥ ❞❛♦ ✤ë♥❣ ♥❤❛♥❤ ✈î✐
ε
ε = (nπ)−2 .
♥❤ä ✈➔ ❣✐→♥ ✤♦↕♥ ❦❤✐
t♦➔♥ ❝ö❝ ✈➻ ♥â ①↔② r❛ tr♦♥❣ ❦❤♦↔♥❣ ❤ú✉ ❤↕♥
0 < x < 1.
✭✸✳✷✮
ε→0+.
❙ü ❝è ❧➔
❇↔♥ ❝❤➜t ❝õ❛ ❝→❝ ♥❣❤✐➺♠
♥➔② ✤÷ñ❝ ❣å✐ ❧➔ t➼♥❤ ❝❤➜t ♣❤➙♥ t→♥✳
❚â♠ ❧↕✐✱ ❧➼ t❤✉②➳t ❲❑❇ ❝✉♥❣ ❝➜♣ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ ✤ì♥ ❣✐↔♥ ✈➔ tê♥❣ q✉→t
✤è✐ ✈î✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ✈➔ ①û ❧➼ ✤÷ñ❝ ❝→❝ t➼♥❤ ❝❤➜t t→♥ ①↕ ✈➔
t➼♥❤ ❝❤➜t ♣❤➙♥ t→♥✳
✸✳✷ ◆❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❲❑❇ t❤❡♦ ❞↕♥❣ ❤➔♠ ♠ô
❈→❝ t➼♥❤ ❝❤➜t t→♥ ①↕ ✈➔ ♣❤➙♥ t→♥ ✤➲✉ ✤÷ñ❝ ♠æ t↔ ❜ð✐ t➼♥❤ ❝❤➜t ❤➔♠ ♠ô✱ tr♦♥❣ ✤â
sè ♠ô ❧➔ sè t❤ü❝ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♠ët✱ ❧➔ sè ↔♦ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ❤❛✐✳ ❉♦ ✤â✱ ✤è✐
✈î✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤➸ ❤✐➺♥ ♠ët tr♦♥❣ ❤❛✐ ❧♦↕✐✱ t❛ ❝â t❤➸ ①➙② ❞ü♥❣ ♥❣❤✐➺♠
❲❑❇ ❞÷î✐ ❞↕♥❣ ❤➔♠ ♠ô ♥❤÷ s❛✉
y(x) ∼ A (x) e
S(x)
δ
✶✻
, δ → 0+.
✭✸✳✸✮
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
❍➔♠
S(x)
✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆●
✤↕✐ ❞✐➺♥ ❝❤♦ ✏♣❤❛✑ ❝õ❛ ♠ët sâ♥❣ ✈➔ ✤÷ñ❝ ❣✐↔ ✤à♥❤ ❧➔ ♠ët ❤➔♠ ❦❤æ♥❣ ✤ê✐
✈➔ ❤➔♠ t❤❛② ✤ê✐ tø tø tr♦♥❣ ✈ò♥❣ sü ❝è✳
❈→❝ tr÷í♥❣ ❤ñ♣ s❛✉ ♣❤→t s✐♥❤
✶✮
S(x)
✷✮
S(x) ❧➔ ↔♦✱ t❤➻ ❝â ♠ët ✈ò♥❣ ❞❛♦ ✤ë♥❣ ♥❤❛♥❤ ✤➦❝ tr÷♥❣ ❜ð✐ ❝→❝ sâ♥❣ ❝â ❜÷î❝ sâ♥❣
❜➟❝
✸✮
❧➔ t❤ü❝✱ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ❝â ✤ë ❞➔② ❧î♣ ❜✐➯♥ ❧➔
δ.
δ.
S(x)
❧➔ ❤➡♥❣ sè✿ ❚➼♥❤ ❝❤➜t ❝õ❛
❝õ❛ ❤➔♠ ❜✐➯♥ ✤ë
y(x)
✤÷ñ❝ ✤➦❝ tr÷♥❣ ❜ð✐ ❤➔♠ sü ❜✐➳♥ t❤✐➯♥ ❝❤➟♠
A(x).
✸✳✸ ❙ü ♠ð rë♥❣ ❤➻♥❤ t❤ù❝ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❲❑❇
◆❣❤✐➺♠ ❣➛♥ ✤ó♥❣ t❤❡♦ ❞↕♥❣ ❤➔♠ ♠ô tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✸✮ ❦❤æ♥❣ ❧➔ ❞↕♥❣ ♣❤ò
❤ñ♣ ✤➸ t❤✉ ✤÷ñ❝ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ t✐➺♠ ❝➟♥ ✈➻ ❝→❝ ❤➔♠ ❜✐➯♥ ✤ë ✈➔ ♣❤❛ ❝❤♦ ❜ð✐
✈➔
S(x)
A(x)
✈➔
t÷ì♥❣ ù♥❣✱ ♣❤ö t❤✉ë❝ ❤♦➔♥ t♦➔♥ ✈➔♦
S(x)
♥❤÷ ❝→❝ ❝❤✉é✐ ❧ô② t❤ø❛ ❝õ❛
1
y(x) ∼ exp
δ
δ.
δ.
A(x)
❉♦ ✤â ♥â ❝â t→❝ ❞ö♥❣ ✤➸ ♠ð rë♥❣
❉♦ ✤â ♥❣❤✐➺♠ ❝â ❞↕♥❣ ♥❤÷ s❛✉
∞
δ n Sn (x), δ → 0.
✭✸✳✹✮
n=0
❇➙② ❣✐í✱ t❛ s➩ ①➨t ♠ët sè ✈➼ ❞ö ✈➲ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜à ♥❤✐➵✉ s❛✉
❱➼ ❞ö ✸✳✶✳
❳➨t ♣❤÷ì♥❣ tr➻♥❤
ε2 y = P (x)y, P (x) = 0 .
✭♣❤÷ì♥❣ tr➻♥❤ ♥➔② ✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤
✭✸✳✺✮
Schr¨
odinger✮
▲í✐ ❣✐↔✐✿
●✐↔ sû ♥❣❤✐➺♠ ❝õ❛ ♥â ❝â ❞↕♥❣ ❝❤♦ tr÷î❝ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✹✮✳ ▲➜② ✤↕♦ ❤➔♠ ❤❛✐
❧➛♥✱ t❛ t❤✉ ✤÷ñ❝
1
y (x) ∼
δ
∞
1
δ Sn (x) exp
δ
n=0
∞
n
✶✼
δ n Sn (x),
n=0
δ→0 ,
✭✸✳✻✮
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
y (x) ∼
1
δ2
∞
✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆●
1
(δ S n (x)) +
δ
n=0
2
n
∞
1
δ S n (x) exp
δ
n=0
∞
n
δ n Sn (x), δ → 0.
✭✸✳✼✮
n=0
❚❤➳ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✹✮ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✼✮ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✺✮ t❤✉ ✤÷ñ❝
ε2
1 0
δ S 0 (x) + δ 1 S 1 (x) + δ 2 S 2 (x) + .....
δ2
2
+
1
S 0 (x) + δ S 1 (x) + δ 2 S 2 (x) . . .
δ
=P (x) exp
1
δ
1
exp
δ
∞
Sn (x)
n=0
∞
Sn (x).
n=0
❚❤✉ ❣å♥ t❛ ✤÷ñ❝
ε2
1
2
(S 0 (x) + δ 1 S 1 (x) + δ 2 S 2 (x) + .....)
δ2
1
+
S 0 (x) + δ S 1 (x) + δ 2 S 2 (x) . . .
δ
=P (x).
✣➦t
δ = ε,
❦❤✐ ✤â t❛ ✤÷ñ❝
2
(S0 (x) + εS1 (x) + ε2 S 2 (x) + .....) + εS 0 (x) + ε2 S 1 (x) + ε3 S 2 (x) + ...
=P (x) .
✣ç♥❣ ♥❤➜t ❤➺ sè ❝õ❛ ❝→❝ ❧ô② t❤ø❛ ❝ò♥❣ ❜➟❝ ❝õ❛
❤➔♠ sè
ε
ð ❤❛✐ ✈➳ t❛ s➩ ①→❝ ✤à♥❤ ✤÷ñ❝ ❝→❝
S0 (x), S1 (x), S2 (x).
P❤÷ì♥❣ tr➻♥❤ ❝❤♦
S0 (x)
✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❊✐❦♦♥❛❧✱ tù❝ ❧➔
ε0 : S02 (x) = P (x).
◆❣❤✐➺♠ ❝õ❛ ♥â ❧➔
x
S0 (x) = ±
P (t) dt.
x0
✶✽
✭✸✳✽✮
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
P❤÷ì♥❣ tr➻♥❤ ✤è✐ ✈î✐
S1 (x)
✣❖⑨◆ ❚❍➚ P❍×❒◆● ◆❍❯◆●
✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝❤✉②➸♥
ε1 : 2S0 (x)S1 (x) + S0 (x) = 0.
◆❣❤✐➺♠ ❝õ❛ ♥â ❧➔
1
S1 (x) = − ln P (x).
4
❈→❝ sè ❤↕♥❣ ❜➟❝ ❝❛♦ ❤ì♥ ❝â t❤➸ t❤✉ ✤÷ñ❝ ❜➡♥❣ t➟♣ ❤ñ♣ ❝→❝ ❤➺ sè ❝õ❛ ❝→❝ ❧ô② t❤ø❛
❝❛♦ ❤ì♥ ❝õ❛
ε
♥❤÷
n−1
O(εn ) : 2S0 (x)Sn (x) + Sn−1 (x) +
Si (x)Sn−i (x) = 0, n ≥ 2.
✭✸✳✾✮
i=1
❉♦ ✤â ♥❣❤✐➺♠ ①➜♣ ①➾ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✹✮ ❝â t❤➸ ✤÷ñ❝ ✈✐➳t ♥❤÷ s❛✉
1
y(x) ≈ exp (S0 (x) + εS1 (x) + ε2 S2 (x) + ...).
ε
S0 (x), S1 (x)
❚❤➳ ❣✐→ trà ❝õ❛
✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② t❛ ❝â
y(x) ≈ exp
x
1
±
ε
P (t) dt + ε ln P
−1
4
(x)
a
x
1
y(x) ≈ exp
ε
P (t) dt . exp ln P
−1
4
(x)
a
x
1
+ exp (−
ε
P (t)dt) exp ln P
−1
4
(x)
a
x
≈ C1 . P
−1
4
1
(x) exp
ε
x
P (t)dt + C2 .P
−1
4
−1
(x) exp
ε
a
tr♦♥❣ ✤â
✈➔
a
C1
✈➔
C2
P (t)dt.
✭✸✳✶✵✮
a
❧➔ ❝→❝ ❤➡♥❣ sè tò② þ ❝â t❤➸ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉
❧➔ ✤✐➸♠ ❝è ✤à♥❤ tr♦♥❣ ✈✐➺❝ ❧➜② t➼❝❤ ♣❤➙♥✳
◆❤÷ ✈➟② t❛ ❝â t❤➸ ①→❝ ✤à♥❤ ♥❤ú♥❣ ❤➔♠ sè ❝❤÷❛ ❜✐➳t
S2 (x), S3 (x), S4 (x), ... tø ♣❤÷ì♥❣
tr➻♥❤ ✭✸✳✾✮ ✤➸ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝❤➼♥❤ ①→❝ ❤ì♥ ❝õ❛ ❝❤✉é✐ ❲❑❇✳
✶✾