Phương pháp WKB để giải phương trình vi phân thường

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✐ ❲❑❇✳

✶✾