✣❸■ ❍➴❈ ✣⑨ ◆➂◆●
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
❑❍❖❆ ❚❖⑩◆

✖✖✖✖✖

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P

Ù◆● ❉Ư◆● P❍❺◆ ▼➋▼ ▼❆❚❍❊▼❆❚■❈❆
❈❍❖ P❍×❒◆● P❍⑩P ❘❯◆●❊✲❑❯❚❚❆

●✐→♦ ✈✐➯♥ ữợ
tỹ P ❚❍➚ ❍■➋◆

✣➔ ◆➤♥❣✱ ✵✺✴✷✵✶✺


▼ö❝ ❧ö❝
▲❮■ ❈❷▼ ❒◆
▼Ð ✣❺❯
✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✸
✹
✻

✶✳✶

❈→❝ ❦❤→✐ ♥✐➺♠

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

ỵ tỗ t t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② ✳

✼

✷ P❍×❒◆● P❍⑩P ❘❯◆●❊ ✲ ❑❯❚❚❆ ✣➮■ ❱❰■ ❈➷◆●
❚❍Ù❈ ❳❻P ❳➓ ❇❾❈ ❇➮◆ ❘❑
✶✵
✷✳✶

P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✷✳✷

P❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡✲❑✉tt❛ ✲ ❈ỉ♥❣ t❤ù❝ ①➜♣ ①➾ ❜➟❝ ❜è♥ ❘❑

✶✶

✷✳✸

P❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ tt ố ợ PP

ì ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛

✳ ✳ ✳ ✳

✶✽

✷✳✺

Ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾

✸ Ù◆● ❉Ư◆● ▼❆❚❍❊▼❆❚■❈❆ ❈❍❖ P❍×❒◆● P❍⑩P
❘❯◆●❊ ✲ ❑❯❚❚❆
✷✵
✸✳✶

❙ü r❛ ✤í✐ ✈➔ ♣❤→t tr✐➸♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✵

✸✳✷

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈ỵ✐ ▼❛t❤❡♠❛t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✶

❱⑨■ ◆➆❚ ❱➋ ❈❆❘▲ ❘❯◆●❊ ❱⑨ ❲■▲❍❊▲▼ ❑❯❚❚❆
❑➌❚ ▲❯❾◆
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦
✷

✸✷
✸✸
✸✹


▲❮■ ❈❷▼ ❒◆
❚❙✳ ▲➯ ❍↔✐ ❚r✉♥❣

❙❛✉ ♠ët t❤í✐ ❣✐❛♥ ❤å❝ t ự ữợ sỹ ữợ sỹ ❝❤➾
❜↔♦ t➟♥ t➻♥❤ ❝õ❛ t❤➛② ❣✐→♦

✱ ✤➳♥ ♥❛② ❦❤â❛ ❧✉➟♥ tèt

♥❣❤✐➺♣ ❝õ❛ ❡♠ ✤➣ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤✳
❊♠ ①✐♥ ❜➔② tä sü ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❇❛♥ ●✐→♠ ❍✐➺✉ tr÷í♥❣
✣↕✐ ❤å❝ ❙÷ P❤↕♠ ✲ ✣↕✐ ❍å❝ ✣➔ ◆➤♥❣✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥✱ ✤➣ t↕♦
❝ì ❤ë✐ ❝❤♦ ❡♠ ✤÷đ❝ ❧➔♠ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ❈❤ó♥❣ ❡♠ ①✐♥ ❣û✐ ❧í✐ ❝↔♠
ì♥ s➙✉ s➢❝ ✤➳♥ t➜t ❝↔ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr♦♥❣ tr÷í♥❣✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ t❤➛② ❝ỉ
❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✤➣ t➟♥ t➻♥❤ ❝❤➾ ❞↕②✱ tr✉②➲♥ ✤↕t ❝❤♦ ❡♠ ♥❤ú♥❣
tự ờ qỵ tr sốt tớ ✈ø❛ q✉❛✳ ❳✐♥ ❝↔♠ ì♥ sü ❣✐ó♣
✤ï✱ ❝❤✐❛ s➫ ❝õ❛ t➜t ❝↔ ❝→❝ ❜↕♥ tr♦♥❣ ❧ỵ♣ tr♦♥❣ t❤í✐ ❣✐❛♥ ❡♠ ❧➔♠ ❦❤â❛ ❧✉➟♥✳
❈✉è✐ ❝ị♥❣✱ ❡♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ t r ữớ trỹ
t ữợ ❡♠✱ ❧✉æ♥ q✉❛♥ t➙♠✱ ✤ë♥❣ ✈✐➯♥ ❝❤➾ ❞➝♥ t➟♥ t➻♥❤ ✤➸ ❝❤ó♥❣
❡♠ ❤♦➔♥ t❤➔♥❤ tèt ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ♥➔②✳

❚✉② ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ s♦♥❣ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✈➝♥ ❦❤æ♥❣ tr→♥❤
❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât ✈➲ ♥ë✐ ❞✉♥❣ ❧➝♥ ❤➻♥❤ t❤ù❝ tr➻♥❤ ❜➔②✱ ❝❤ó♥❣ ❡♠ r➜t

❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ỡ

ữủ sỹ õ õ ừ qỵ t ❝æ ✈➔ ❜↕♥ ✤å❝✳

✣➔ ◆➤♥❣✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✺
❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥

P❤↕♠ ❚❤à ❍✐➲♥

✸


é
ỵ ỹ ồ t
Pữỡ tr ♣❤➙♥ ❧➔ ♠æ ❤➻♥❤ ♠æ t↔ ❦❤→ tèt q✉→ tr➻♥❤ ❝❤✉②➸♥ ✤ë♥❣
tr♦♥❣ tü ♥❤✐➯♥ ✈➔ ❦➽ t❤✉➟t✳ ❚r♦♥❣ ❧➽♥❤ ✈ü❝ t♦→♥ ù♥❣ ❞ư♥❣ t❤÷í♥❣ ❣➦♣ r➜t
♥❤✐➲✉ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ tợ ữỡ tr tữớ
ự ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✤â♥❣ ♠ët ✈❛✐ trá q✉❛♥ trå♥❣
tr♦♥❣ ❧➼ t❤✉②➳t t♦→♥ ❤å❝✳
❚r♦♥❣ ✤↕✐ ✤❛ sè tr÷í♥❣ ❤đ♣ ♥❤÷✿ ❝→❝ ❜➔✐ t♦→♥ ❝â ❤➺ sè ❜✐➳♥ t❤✐➯♥✱ ❝→❝
❜➔✐ t♦→♥ ♣❤✐ t✉②➳♥✱ ❝→❝ ❜➔✐ t♦→♥ tr➯♥ ♠✐➲♥ ❜➜t ❦➻✱ ✳✳✳ t❤➻ ♥❣❤✐➺♠ t÷í♥❣
♠✐♥❤ ❤♦➦❝ ❦❤ỉ♥❣ ❝â ❤♦➦❝ ♥➳✉ ❝â t❤➻ r➜t ♣❤ù❝ t↕♣ ✈➔ ✈✐➺❝ ❦❤↔♦ s→t t➼♥❤
❝❤➜t ♥❣❤✐➺♠ ❣➦♣ r➜t ♥❤✐➲✉ ❦❤â ❦❤➠♥✱ ❞♦ ✤â ♣❤↔✐ t➻♠ ✤➳♥ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣
①➜♣ ①➾ ✤➸ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣✳
❉♦ ♥❤✉ ❝➛✉ t❤ü❝ t✐➵♥ ❝ị♥❣ ✈ỵ✐ sü ♣❤→t tr✐➸♥ ❝õ❛ ♠➻♥❤✱ tr♦♥❣ t♦→♥ ❤å❝
①✉➜t ❤✐➺♥ ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤➸ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
t❤÷í♥❣✿ P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ t➼❝❤✱ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✉é✐ ❚❛②❧♦r✱ ♣❤÷ì♥❣ ♣❤→♣

①➜♣ ①➾ ❧✐➯♥ t✐➳♣ P✐❝❛r❞✱ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ sè ♥❤÷✿ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r✱ ❊✉❧❡r
❝↔✐ t✐➳♥✱ ❆❞❛♠✱ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡✲❑✉tt❛✱ ✳✳✳ ✳ ▼❛t❤❡♠❛t✐❝❛ ❧➔ ❝ỉ♥❣ ❝ư
❧➟♣ tr➻♥❤ ♠↕♥❤ ✈ỵ✐ ❤ì♥ ✼✵✵ ❤➔♠ ❝â s➤♥ tr♦♥❣ t❤ü ✈✐➺♥ ❤➔♠✳❱✐➺❝ sû ❞ư♥❣
▼❛t❤❡♠❛t✐❝❛ ♥❤❛♥❤ ❣➜♣ ♥❤✐➲✉ ❧➛♥ s♦ ✈ỵ✐ ❣✐↔✐ ❜➡♥❣ t❛② t❤ỉ♥❣ t❤÷í♥❣ ❜➡♥❣
❝→❝❤ ù♥❣ ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣✳
❚r♦♥❣ ♣❤↕♠ ✈✐ ✈➔ ②➯✉ ❝➛✉ ❝õ❛ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ❝❤ó♥❣ ❡♠
s➩ tr➻♥❤ ❜➔② ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡✲

✷✳ ▼ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉

❑✉tt❛ ✤➸ ú ữỡ tr

ợ t ữỡ ❘✉♥❣❡ ✲ ❑✉tt❛✱ ♠ët tr♦♥❣ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣
r➜t ❤❛② ✤➸ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✈ỵ✐ ✤ë
❝❤➼♥❤ ①→❝ ❝❛♦✱ ✈➔ t✐➳t ❦✐➺♠ t❤í✐ ❣✐❛♥✳
✹


●✐ỵ✐ t❤✐➺✉ ♠ët ♥❣ỉ♥ ♥❣ú ❧➟♣ tr➻♥❤ ♠↕♥❤✱ ▼❛t❤❡♠❛t✐❝❛✱ ❤é trñ ❝❤♦ ❣✐↔♥❣
✈✐➯♥ ✈➔ s✐♥❤ ✈✐➯♥ tr♦♥❣ ✈✐➺❝ ❣✐↔♥❣ ❞↕② ✈➔ ❤å❝ t➟♣ ●✐↔✐ t➼❝❤ sè ♥â✐ ❝❤✉♥❣ ✈➔

✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉

✈✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♥â✐ r✐➯♥❣✳

❚❤❛♠ ❦❤↔♦ ✈➔ ❞à❝❤ ❝→❝ t➔✐ ❧✐➺✉ t✐➳♥❣ ❆♥❤✳
❚ê♥❣ ❤ñ♣ ✈➔ tr➻♥❤ ❜➔②✳

✺

❈❤÷ì♥❣ ✶

❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
✶✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠
P❤÷ì♥❣ tr➻♥❤ ✈✐ ữỡ tr õ
F (x, y, y,
yă, . . . , y (n) ) = 0,
tr♦♥❣ ✤â

✭✶✳✶✮

②❂②✭①✮ ❧➔ ➞♥ ❤➔♠ ❝➛♥ t➻♠ ✈➔ ♥❤➜t t❤✐➳t ♣❤↔✐ ❝â sü t❤❛♠ ❣✐❛ ❝õ❛

✤↕♦ ❤➔♠ ✭✤➳♥ ❝➜♣ ♥➔♦ ✤â✮ ❝õ❛ ➞♥✳
❚r♦♥❣ tr÷í♥❣ ❤đ♣ ➞♥ ❤➔♠ ❝➛♥ t➻♠ ❧➔ ❤➔♠ ♥❤✐➲✉ ❜✐➳♥ ✭①✉➜t ❤✐➺♥ ❝→❝
✤↕♦ ❤➔♠ r✐➯♥❣✮ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝á♥ ❣å✐ ❧➔

r✐➯♥❣✳

♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠

❚❤ỉ♥❣ t❤÷í♥❣ t❛ ①➨t ữỡ tr ợ số ởt ❜✐➳♥

②❂②✭①✮ ①→❝ ✤à♥❤ tr➯♥ ❦❤♦↔♥❣ ♠ð I ⊂ R ✱ ❦❤✐ ✤â ❤➔♠ ❋ tr♦♥❣ ✤➥♥❣
n+1
t❤ù❝ tr➯♥ ①→❝ ✤à♥❤ tr♦♥❣ ởt t ừ R ì R
r trữớ ❤đ♣

t❤ü❝

➞♥ ❤➔♠ ❝➛♥ t➻♠ ❧➔ ✈❡❝t♦r ✲ ❤➔♠ ✭❤➔♠ ✈ỵ✐ ❣✐→ trà ✈❡❝t♦r✮

❋ ❧➔ ♠ët →♥❤ ①↕ ♥❤➟♥ ❣✐→ trà tr♦♥❣
Rm ✱ ❦❤✐ ✤â ✭✶✳✶✮ ✤÷đ❝ ❤✐➸✉ ❧➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳
P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❝➜♣ ■ ❝â ❞↕♥❣ tê♥❣ q✉→t✿
y(x) = (y1 (x), . . . , ym (x))T ∈ Rm ✱

F (x, y, y)
˙ = 0,
tr♦♥❣ ✤â

❋

①→❝ ✤à♥❤ tr♦♥❣ ♠✐➲♥

G ∈ R✱

✭✶✳✷✮

✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ õ

t t ữợ ồ ữủ t

y = f (x, y),
ợ



tử tr ♠✐➲♥

D ⊂ R2 ✳
✻

✭✶✳✸✮


ỵ tỗ t t ừ t
t ố ợ ữỡ tr ✈✐ ♣❤➙♥ ❝➜♣ ♠ët
❚➻♠ ♥❣❤✐➺♠

②✭①✮ t❤ä❛✿

y˙ = f (x, y),
y(x0 ) = y0
tr♦♥❣ ✤â

(x0 , y0 ) ∈ D

✈➔

y(x0 ) = y0

✭✶✳✹✮

✤÷đ❝ ❣å✐ ❧➔ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉✳ ❚❛ ①➨t

❜➔✐ t ố ợ ữỡ tr ữủ ố ợ
tr õ
sỷ

✤à♥❤ ✈➔ ❧✐➯♥ tö❝ tr➯♥ ♠✐➲♥ ♠ð

D ⊂ R2 ✳

❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✹✮✱ t➼❝❤ ♣❤➙♥ ✷ ✈➳ ❝õ❛ ♣❤÷ì♥❣

tr➻♥❤ tr♦♥❣ ✭✶✳✹✮ t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✤è✐ ✈ỵ✐

②✭①✮ ❧➔✿

x

f (t, y(t))dt,

y(x) = y0 +

✭✶✳✺✮

x0
❘ã r➔♥❣ ♠é✐ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✹✮ ❝ơ♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✺✮ ✈➔ ♥❣÷đ❝ ❧↕✐✱ ♠é✐
♥❣❤✐➺♠ ❝õ❛ ✭✶✳✺✮ ✤➲✉ ❦❤↔ ✈✐ ✈➔ ❧✐➯♥ tö❝ tr➯♥ ♠ët ❦❤♦↔♥❣

■

♥➔♦ ✤â ✈➔ t❤ä❛

✭✶✳✹✮✳

❇ê ✤➲ ✶✳✶✳ ●✐↔ sû ❢✭①✱②✮ ❧✐➯♥ tö❝ tr➯♥ ❤➻♥❤ ❝❤ú ♥❤➟t
D = {(x, y) ∈ R2 , |x − x0 | ≤ a, |y − y0 | ≤ b},

✣➦t M = ♠❛① ⑤❢✭①✱②✮⑤✱ ❤❂ ♠✐♥ {a, b/M }✱ ✭t ❛ ❝â t❤➸ ❣✐↔ sû M = 0✮✳
❚❛ ✤➦t✿ y0(x) = y0, ∀x ∈ [x0 − h, x0 + h],
x

y1 (x) = y0 +

f (t, y0 (t))dt,
x0

x

f (t, yn−1 (t))dt, n ≥ 1.

yn (x) = y0 +
x0

❉➣② yn(x) ✤÷đ❝ ❣å✐ ❧➔ ❞➣② ①➜♣ ①➾ P✐❝❛r❞✳

✶✳✷✳✷✳ ❙ü tỗ t t
▲✐♣s❝❤✐t③✮

❢✭①✱②✮ ①→❝ ✤à♥❤ tr➯♥
♠✐➲♥ D ⊂ R✳ ❚❛ ♥â✐ ❢ t❤ä❛ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③ t❤❡♦ ❜✐➳♥ ② tr➯♥ ❉ ♥➳✉ tỗ
t số ữỡ ồ số st s❛♦ ❝❤♦✿

❈❤♦ ❤➔♠

|f (x, y1 ) − f (x, y2 )| ≤ L|y1 − y2 |✱

✈ỵ✐ ♠å✐

✼

(x, y1 ), (x, y2 ) ∈ D.


ỵ

sỷ
tr tử t❤ä❛ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③ t❤❡♦ ❜✐➳♥ ② tr➯♥ ❤➻♥❤ ❝❤ú
♥❤➟t
✭✤à♥❤ ỵ tỗ t t

D = {(x, y) ∈ R2 , |x − x0 | ≤ a, |y − y0 | ≤ b},

❑❤✐ ✤â ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ tỗ t t tr
I := [x0 − h, x0 + h], ✈ỵ✐ ❤❂ ♠✐♥ {a, b/M } ✈➔ ▼❂ ♠❛① ⑤❢✭①✱②✮ ⑤✳

❈❤ù♥❣ ♠✐♥❤

❛✮✳ ❙ü tỗ t ự r Pr ở tö ✤➲✉ tr➯♥

■

✤➳♥ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤②✳ ✣➛✉ t✐➯♥ ❜➡♥❣ q✉② ♥↕♣ t❛ ❝❤ù♥❣

♠✐♥❤ r➡♥❣✿

|yk+1 (x) − yk (x)| ≤ M Lk
✈ỵ✐ ♠å✐

|x − x0 |k+1
,
(k + 1)!

x ∈ I✳

❦ ❂✵ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ trð t❤➔♥❤ | xx f (t, y0(t))dt| ≤ M |x − x0|,
✈ỵ✐ x0 ≤ x✱ ✈➻ ▼❂ ♠❛① ⑤❢✭①✱②✮ ⑤ s✉② r❛ f (x, y) ≤ M ✱ t❛ ❝â
❱ỵ✐

0

x

x

f (t, y0 (t))dt ≤

|f (t, y0 (t))|dt,

x0

x0

x

⇒|

f (t, y0 (t))dt| ≤ M (x − x0 ) ≤ M |x − x0 |,
x0

✈➟② ❜➜t ✤➥♥❣ t❤ù❝ ✤ó♥❣✳

k − 1✱ ❦❤✐ ✤â ✈ỵ✐ x0 ≤ x ≤ x0 + h
x
|yk+1 (x) − yk (x)| = | x0 [f (t, yk (t)) − f (t, yk−1 (t))]dt| ≤
●✐↔ sû t❛ ❝â ✤✐➲✉ ❦✐➺♥ ✤â ✈ỵ✐

≤|

x
x0

[f (t, yk (t)) − f (t, yk−1 (t))]dt| ≤ L

≤ M Lk

x
x0

x
x0

|yk (t) − yk−1 (t)|dt ≤

(x − x0 )k /k!dt = M Lk (x − x0 )(k+1) /(k + 1)!.

x0 − h ≤ x ≤ x0 t❛ ❝❤ù♥❣ ♠✐♥❤
❳➨t ❞➣② yk (x) tr t õ

ợ

tữỡ tỹ

|yk+p (x) yk (x)| ≤ |yk+p (x) − yk+p−1 (x)| + |yk+p−1 (x) − yk+p−2 (x)|+
. . . + |yk+1 (x) − yk (x)| ≤
≤

M
L

j≥k+1

M (L|x−x0 |)k+p
L{
(k+p)!

(Lh)j
j! ✳

✽

+ ... +

(L|x−x0 |)k+1

}
(k+1)!

≤

t❛ ❝â


❈❤✉é✐ sè

∞ (Lh)j
❧➔ ❤ë✐ tư✱ ♥➯♥ ♣❤➛♥ ❞÷ ❝õ❛ ♥â ❝â t❤➸ ❧➔♠ ❝❤♦ ❜➨
j=0 j!

❦ ✤õ ❧ỵ♥✳ ❚❤❡♦ t✐➯✉ ❝❤✉➞♥ ❈❛✉❝❤②✱ ❞➣② {yk (x)} ❤ë✐ tö ✤➲✉ tr➯♥ ■
✤➳♥ ❤➔♠ ②✭①✮✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ②✭①✮ ❧➔ ♥❣❤✐➺♠ t❛ ❝❤➾ q ợ tr

tũ ỵ

tự

x

yk+1 (x) = y0 +
❱➻ ❞➣② ❤➔♠

yk (x) ❤ë✐ tö ✤➲✉✱ ❢

❧✐➯♥ tö❝ tr➯♥ ❤➻♥❤ ❝❤ú ♥❤➟t

❉ ♥➯♥ ❞➣② ❤➔♠

❢✭t✱②✭t✮✮ ❞♦ ✤â ❝â t ợ
q t ữủ t❤ù❝ ✭✶✳✺✮✳ ❱➟② ②✭①✮ ❧➔ ♥❣❤✐➺♠ ❝õ❛
{f (t, yk (t))}

■

f (t, yk (t))dt,
x0

❤ë✐ tö ✤➲✉ tr➯♥

✤➳♥ ❤➔♠

❜➔✐ t♦→♥ ❈❛✉❝❤② ✭✶✳✹✮✳
❜✮✳ ❚➼♥❤ ❞✉② ♥❤➜t✳ ●✐↔ sû ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝á♥ ❝â ♥❣❤✐➺♠
t❛ ❝â

③✭①✮ ❦❤✐ ✤â

x

y(x) − z(x) =

[f (t, y(t)) − f (t, z(t))]dt.
x0

❙✉② r❛

x

|y(x) − z(x)| = |

[f (t, y(t)) − f (t, z(t))]dt| ≤ 2M |x − x0 |.
x0

❚ø ✤â

x

|y(x) − z(x)| = |

x

[f (t, y(t)) − f (t, z(t))]dt| ≤ L
x0

|y(t) − z(t)|dt
x0

2

0)
.
≤ 2M L (x−x
2

▲➦♣ ❧↕✐ q✉→ tr tr t ự ữủ r ợ ồ số tü ♥❤✐➯♥

❑❤✐

k→∞

t❤➻

|y(x) − z(x)| ≤ 2M Lk

|x − x0 |k+1
, x ∈ I.
(k + 1)!

|y(x) − z(x)| = 0

■

②✭①✮ ❧➔ ❞✉② ♥❤➜t✳

tr➯♥

✾

⇒ y(x) ≡ z(x).

❦

◆❤÷ ✈➟② ♥❣❤✐➺♠


ữỡ

Pì PP

P ❇❾❈ ❇➮◆ ❘❑
✷✳✶ P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣
✣➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ t❛ ❝â t❤➸ ❣✐↔✐ t❤❡♦ ❤❛✐ ♣❤÷ì♥❣ ♣❤→♣✿
P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝✿ ❇➡♥❣ ❝→❝❤ ❞ü❛ ✈➔♦ ❝→❝❤ t➼♥❤ t➼❝❤
♣❤➙♥ trü❝ t✐➳♣✱ ①→❝ ✤à♥❤ ❝→❝ tờ qt ừ rỗ ỹ
✤➛✉ ✤➸ ①→❝ ✤à♥❤ ♥❣❤✐➺♠ r✐➯♥❣ ❝➛♥ t➻♠✳
P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣✿ ❳✉➜t ♣❤→t tø ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉✱
♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❝â t❤➸ →♣ ❞ư♥❣ ❝❤♦ ♠ët ❧ỵ♣ ❝→❝ ữỡ tr
rở ỡ rt s ợ ữỡ ♣❤→♣ trü❝ t✐➳♣✱ ❞♦ ✤â ♣❤÷ì♥❣ ♣❤→♣ ♥➔②
✤÷đ❝ ❞ị♥❣ ♥❤✐➲✉ tr♦♥❣ t❤ü❝ t➳✳
❚r♦♥❣ t❤ü❝ t➳ ❝â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ♥❤÷✿ P❤÷ì♥❣
♣❤→♣ ❊✉❧❡r✱ ❊✉❧❡r ❝↔✐ t✐➳♥✱ ❘✉♥❣❡ ✲ ❑✉tt❛✳✳✳ ✳ ❚r♦♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤â
t❤➻ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛ ✤÷đ❝ ❞ị♥❣ ♥❤✐➲✉ tr♦♥❣ t❤ü❝ t➳ ✈➻ t➼♥❤ ❤✐➺✉
q✉↔ ♠➔ ♥â ♠❛♥❣ ❧↕✐ ✈ỵ✐ ✤ë ❝❤➼♥❤ ①→❝ ❝❛♦✳ P❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲❑✉tt❛ ❞♦
❤❛✐ ♥❤➔ ❜→❝ ❤å❝ ♥❣÷í✐ ✣ù❝ ❧➔ ❈❛r❧ ❘✉♥❣❡ ✈➔ ❲✐❧❤❡❧♠ ❑✉tt❛ ✤÷❛ r❛✳
❚❛ ❧✉ỉ♥ ❣✐↔ t❤✐➳t ❜➔✐ t♦→♥ ✤➦t r❛ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ✈➔ ♥❣❤✐➺♠ ✤â ✤õ
trì♥✱ ♥❣❤➽❛ ❧➔ ♥â ❝â ✤↕♦ ❤➔♠ ✤➳♥ ❝➜♣ ✤õ ❝❛♦✳

✶✵


✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡✲❑✉tt❛ ✲ ❈ỉ♥❣ t❤ù❝ ①➜♣ ①➾ ❜➟❝ ❜è♥ ❘❑
❚rð ❧↕✐ ❜➔✐ t♦→♥ ✭✶✳✹✮
trà ❣➛♥ ✤ó♥❣

y(xi )

x ∈ [a, b]

t↕✐ ❝→❝ ✤✐➸♠

✳ ❈→❝❤ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ✭✶✳✹✮ ❧➔ t➻♠ ❝→❝ ❣✐→

xi = 0, 1, 2 . . . n✱

tr♦♥❣ ✤â✿

a = x0 < x1 < . . . < xn = b✱
xi = x0 + ih; i = 0, 1, 2, . . . n − 1,
h=
y0 = α0 ✱ (α0 =)✱ t❛ s➩ ❧➛♥
❣✐→ trà ❣➛♥ ✤ó♥❣ yi t↕✐ xi t❛

❚❛ ❝â
❚ø ❝→❝

b−a
n ✳

y1 t↕✐ x1 ✱ y2
t↕✐ xi+1 ✳

❧÷đt ①→❝ ✤à♥❤✿
s➩ t➼♥❤

yi+1

t↕✐

x2 ✱ . . .

✣➸ t❤➔♥❤ ❧➟♣ ♥❤ú♥❣ ❝æ♥❣ t❤ù❝ ❘✉♥❣❡ ✲ ❑✉tt❛ ❝â ✤ë ❝❤➼♥❤ ①→❝ ❝❛♦ ❤ì♥
❝ỉ♥❣ t❤ù❝ ❊✉❧❡r ✈➔ ❊✉❧❡r ❝↔✐ t✐➳♥ t❛ ❞ị♥❣ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r ♥❣❤✐➺♠
t↕✐

xi

②✭①✮

✈ỵ✐ ♥❤✐➲✉ sè ❤↕♥❣ ❤ì♥✳

Ð ✤➙② t❛ ❝❤➾ tr➻♥❤ ❜➔② ❝❤ù♥❣ ♠✐♥❤ ❝æ♥❣ t❤ù❝ ❘✉♥❣❡ ✲ ❑✉tt❛ ❝â ✤ë ❝❤➼♥❤
①→❝ ❝➜♣ ❜è♥✳ ❈æ♥❣ t❤ù❝ ❘✉♥❣❡ ✲ ❑✉tt❛ ✤ë ❝❤➼♥❤ ①→❝ ❝➜♣ ❤❛✐ ✈➔ ❝➜♣ ❜❛
✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü✳
❳➨t ❜➔✐ t♦→♥ ✭✶✳✹✮✱ ✤➸ ❝â ✤÷đ❝ ❝æ♥❣ t❤ù❝ ①➜♣ ①➾ ❜➟❝ ❜è♥ ❘❑ t❛ t❤ü❝
❤✐➺♥ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r ❝õ❛ ♥❣❤✐➺♠ ✤ó♥❣

y(xi+1 ) = y(xi ) +

x−xi
˙ i)
1! y(x

+

4

(xxi )2
ă(xi )
2! y

i)
(4)
+ (xx
4! y (xi ) +

ợ

✈ỵ✐ ✻ sè ❤↕♥❣✿
+

(x−xi )3 ✳✳✳
y (xi )+
3!

(x−xi )5 (5)
5! y (ci ),

ci ∈ (xi , x).

❚❤❛②

x = xi+1 = xi + h,

t❛ ❝â✿

h2

h3 ✳✳✳
h4 (4)
h5 (5)
y
y (ci ),
(xi ) + y (xi ) +
y(xi+1 ) = y(xi ) + hy(x
i ) + yă(xi ) +
2
6
24
120



tr õ t õ

y i = f (xi , yi ),
yăi = fx (xi , yi ) + f˙y (xi , yi )y˙ i (xi ),
✳✳✳

y i = f˙x (xi , yi )f˙y (xi , yi ) + (f˙y )2 (xi , yi )y˙ i (xi ),

(4)
yi = f˙x (xi , yi )(f˙y )2 (xi , yi ) + (f˙y )3 (xi , yi )y˙ i (xi ).
✶✶


❚❤❛② ❝→❝ ❜✐➸✉ t❤ù❝ ♥❤➟♥ ✤÷đ❝ ✈➔♦ ✭✷✳✶✮ ✈➔ ✤➦t✿

y(xi+1 ) = yi+1
y(x
˙ i ) = y˙i
t❛ ✤÷đ❝✿

yi+1 = yi + hy˙ i +

h2 ˙
2 (fx

+ f˙y y˙ i ) +

h3 ˙ ˙
6 [fx fy

+ (f˙y )2 y˙ i ] +

h4 ˙ ˙ 2
24 [fx (fy ) +

h5 (5)
3
˙
+ (fy ) y˙ i ] +
y (ci ).
120

✭✷✳✷✮

yi+1 = yi + r1 k1 + r2 k2 + r3 k3 + r4 k4 ,

✭✷✳✸✮

❚❛ ✤➦t✿

tr♦♥❣ ✤â✿













❈→❝ ❣✐→ trà ❝õ❛

k1 = hf (xi , yi ),
k2 = hf (xi + a1 h, yi + b1 k1 ),
k3 = hf (xi + a2 h, yi + b2 k1 + b3 k2 ),
k4 = hf (xi + a3 h, yi + b4 k1 + b5 k2 + b6 k3 ).

k1 ✱ k2 ✱ k3

✤➣ ✤÷đ❝ ①→❝ ✤à♥❤ ð tr➯♥✳ ✣➸ ①→❝ ✤à♥❤

✭✷✳✹✮

k4

t❛ t✐➳♥

❤➔♥❤ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r ❝õ❛ ❤➔♠ ❤❛✐ ❜✐➳♥ s❛✉✿

f (xi + a3 h, yi + b4 k1 + b5 k2 + b6 k3 ) =
= y˙ i + a3 hf˙x + (b4 k1 + b5 k2 + b6 k3 )f˙y + O(h4 ) =
= y˙ i + a3 hf˙x + b4 k1 f˙y + b5 k2 f˙y + b6 k3 f˙y + O(h4 ),
❤❛②✿

f (xi + a3 h, yi + b4 k1 + b5 k2 + b6 k3 ) = y˙i + a3 hf˙x + b4 hy˙ i f˙y + b5 hy˙ i f˙y +
+b5 a1 h2 f˙x f˙y + b5 b1 h2 y˙ i (f˙y )2 + b6 hy˙ i f˙y + b6 a2 h2 f˙x f˙y + b6 b2 h2 y˙ i (f˙y )2 +
+b6 b3 h2 y˙ i (f˙y )2 + h3 b6 b3 a1 f˙x (f˙y )2 + b6 b3 b1 h3 y˙ i (f˙y )3 + O(h4 ).
❙✉② r❛✿

k4 = hy˙ i + h2 a3 f˙x + h2 b4 y˙ i f˙y + h3 b5 a1 f˙x f˙y + h3 b5 b1 y˙ i (f˙y )2 +
+h2 b6 y˙ i f˙y + h3 b6 a2 f˙x f˙y + h3 b6 b2 y˙ i (f˙y )2 + h3 b6 b3 y˙ i (f˙y )2 +
+h4 + b6 b3 a1 f˙x (f˙y )2 + h4 b6 b1 b3 y˙ i (f˙y )3 + O(h5 ).
❚❤❛②

k1 , k2 , k3 , k4

✈➔♦ ✭✷✳✹✮ t❛ ✤÷đ❝✿
✶✷


yi+1 = yi + h(r1 + r2 + r3 + r4 )y˙ i + h2 (r2 a1 + r3 a2 + r4 a3 )f˙x +

+h2 (r2 b1 + r3 b2 + r3 b3 + r4 b4 + r4 b5 + r4 b6 )y˙ i f˙y +
+h3 (r3 b3 a1 + r4 b5 a1 + r4 b6 a2 )f˙x f˙y +
+h3 (r3 b1 b3 + r4 b5 b1 + r4 b6 b2 + r4 b6 b3 )y˙ i (f˙y )2 +
+ h4 (r4 b6 b3 a1 )f˙x (f˙y )2 + h4 r4 b6 b1 b3 y˙ i (f˙y )3 .

✭✷✳✺✮

❙♦ s→♥❤ ❝→❝ ❤➺ sè ❧ô② t❤ø❛ ❝õ❛ ❤ tr♦♥❣✭ ✷✳✷✮ ✈➔ ✭✷✳✺✮ t❛ ❝â✿


r1 + r2 + r3 + r4 = 1,






r2 a1 + r3 a2 + r4 a3 = 1/2,





 r2 b1 + r3 b2 + r3 b3 + r4 b4 + r4b 5 + r4 b6 = 1/2,














r3 b3 a1 + r4 b5 a1 + r4 b6 a2 = 1/6,
r3 b1 b3 + r4 b5 b1 + r4 b6 b2 + r4 b6 b3 = 1/6,
r4 b6 b3 a1 = 1/24,
r4 b6 b1 b3 = 1/24.

r1 = r4 = 1/6, r2 = r3 = 2/6, a1 = a2 =
1/2, a3 = 1, b1 = b3 = 1/2, b2 = b4 = b5 = 0, b6 = 1.

●✐↔✐ ❤➺ ♥❤➟♥ ữủ t

õ tr t

yi+1
ợ

♥ ✲✶✳

k1 = hf (xi , yi ),
h
k1
k2 = hf (xi + , yi + ),
2
2

k2
h
k3 = hf (xi + , yi + ),
2
2
k4 = hf (xi + h, yi + k3 ),
1
= yi + (k1 + 2k2 + 2k3 + k4 ).
6

◆❤÷ ✈➟② tr♦♥❣ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r ❝õ❛

✭✷✳✻✮

②✭①✮ t❛ ❜ä q✉❛ sè ❤↕♥❣ O(h5)✱ t❤➻ t❛

♥❤➟♥ ✤÷đ❝ ❝ỉ♥❣ t❤ù❝ ❘✉♥❣❡ ✲ ❑✉tt❛ ❝â ✤ë ❝❤➼♥❤ ①→❝ ❝➜♣ ❜è♥✳ ◆❣❤➽❛ ❧➔✿

|yi − y(xi )| ≤ M h4 ,
tr♦♥❣ ✤â

▼

❧➔ ❤➡♥❣ sè ❞÷ì♥❣ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦

✶✸

❤✳

❚r♦♥❣ ❝→❝ ❝æ♥❣ t❤ù❝ tr➯♥ t❤➻ ❝æ♥❣ t❤ù❝ ❘✉♥❣❡ ✲ ❑✉tt❛ ❜➟❝ ❜è♥ ✭✷✳✻✮
✤÷đ❝ sû ❞ư♥❣ ♥❤✐➲✉ ♥❤➜t ✈➻ ❝ỉ♥❣ t❤ù❝ ❝❤♦ ✤ë ❝❤➼♥❤ ①→❝ ❝❛♦ ♠➔ ❦❤æ♥❣ q✉→
♣❤ù❝ t↕♣✳ ❈ỉ♥❣ t❤ù❝ ✭✷✳✻✮ ✤÷đ❝ ❣å✐ ❧➔ ❝ỉ♥❣ t❤ù❝ ①➜♣ ①➾ ❜➟❝ ❜è♥ ❘❑✳
❱✐➺❝ →♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ✭✷✳✻✮ ✤➸ t➼♥❤ ❝→❝ ❣✐→ trà ①➜♣ ①➾✿

y1 , y2 , . . . , yn+1

✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛✳
❚ø ✤→♥❤ ❣✐→ s❛✐ sè ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛✿

|yi − y(xi | ≤ M h4 ,

✭✷✳✼✮

t❛ t❤➜②✿ ❚rà t✉②➺t ố ừ s số s rt ữợ
❦❤✐ t❛ ❣✐↔♠

❤ ✤✐ 21 t❤➻ trà t✉②➺t ✤è✐ s➩ ❣✐↔♠ ✤✐ ✶✻ ❧➛♥✿
[( 21 )4 =

❤

❣✐↔♠✳ ❚ø

1
16 ]

❱➻ ✈➟② t t ữớ t tữớ ồ ữợ

s 12 ữợ

trữợ õ

ử
t t tr s

dy
= y 2,
dx

ợ

ũ ữỡ ❘✉♥❣❡ ✲ ❑✉tt❛ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ tr➯♥✳

❇↔♥❣ ✭2.1✮ ❝❤♦ ❝→❝ ❦➳t q✉↔ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛ tr➯♥
ợ ữợ





ợ






ợ

❂✵✱✷

●✐→ trà ❝❤➼♥❤ ①→❝ ❝õ❛ ♥❣❤✐➺♠

✵✱✷

✵✱✼✼✽✻

✵✱✼✼✽✻

✵✱✼✼✽✻

✵✱✹

✵✱✺✵✽✶✾

✵✱✺✶✷✼

✵✱✺✵✽✶✽

✵✱✻

✵✱✶✼✼✾✶

✵✱✶✽✽✾✶

✵✱✶✼✼✽✽

✵✱✽

✲✵✱✷✷✺✺

✲✵✱✷✶✷✵✼

✲✵✱✷✷✺✺✹

✶

✲✵✱✼✶✽✷✸

✲✵✱✼✵✶✽✷

✲✵✱✼✶✽✷✽

❇↔♥❣ ✷✳✶✿

❚ø ❜↔♥❣ ❣✐→ trà tr t t ợ ữợ

ọ t ở ❝❤➼♥❤ ①→❝

❝➔♥❣ ❝❛♦✱ ❝â ❣✐→ trà ❣➛♥ ✈ỵ✐ ❣✐→ trà ❝❤➼♥❤ ①→❝ ❤ì♥✳
✶✹


Pữỡ tt ố ợ PP ♥ ✭♥❃✶✮
P❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛ ❝â t❤➸ →♣ ❞ư♥❣ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
❝➜♣

♥ ✭♥ ❃✶✮ ✈➔ ❤➺ t❤è♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët✳

❉↕♥❣ ✈❡❝t♦r ❝õ❛ ổ tự tt ữủ ổ t ữợ

Xn+1 = Xn + 16 (k1 + 2k2 + 2k3 + k4 ),
tr♦♥❣ ✤â

k1 ✱ k2 ✱ k3 ✱ k4

✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿

k1 = f (tn , Xn ),
k2 = hf (tn + h2 , Xn +

k1
2 ),

k3 = hf (tn + h2 , Xn +

k2
2 ),

k4 = hf (tn + h, Xn + k3 )✳

✷✳✸✳✶ ❇➔✐ t♦→♥ ❈❛✉❝❤② ✤è✐ ợ ữỡ tr ởt
t t

x = f (t, x, y),
y˙ = g(t, x, y).
✈ỵ✐

x(t0 ) = x0 ,

y(t0 ) = y0 .
❚❛ ✤➦t✿

Y =

❢

x
y

❢
❣

; =

; k✐ =

❋✐
●✐

❈ỉ♥❣ t❤ù❝ ❧➦♣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛ ố ợ ữợ tứ

(xn , yn )

s ữợ t t❤❡♦✿

xn+1

❂

(xn+1 , yn+1 )❂(x(tn+1 ), y(tn+1 ))✱

xn + 61 (F1 + 2F2 + 2F3 + F4 )✱

yn+1 ❂yn + 16 (G1 + 2G2 + 2G3 + G4 )✳
tr♦♥❣ ✤â ❝→❝ ❣✐→ trà

F 1 ✱ F 2 ✱ F3 ✱ F4
F1

❂

❝õ❛ ❤➔♠

❢

hf (tn , xn , yn ),
✶✺

❧➔✿

❧➔✿


❈→❝ ❣✐→ trà

F2

❂

hf (tn +

h
2 ✱xn

+

F1
2 ✱yn

+

G1
2 ),

F3

❂

hf (tn +

h
2 ✱xn

+

F2
2 ✱yn

+

G2
2 )✱

F4

❂

hf (tn + h✱xn + F3 ✱yn + G3 )✳

G1 ✱ G2 ✱ G3 ✱ G4

❝õ❛ ❤➔♠

G1

❂

❣ ❧➔✿

hg(tn , xn , yn )✱

G2

❂

hg(tn +

h
2 ✱xn

+

F1
2 ✱yn

+

G1
2 )✱

G3

❂

hg(tn +

h
2 ✱xn

+

F2
2 ✱yn

+

G2
2 )✱

G4

❂

hg(tn + h✱xn + F3 ✱yn + G3 )

t ố ợ ữỡ tr
t t s
xă = g(t, x, x)
˙ ✱
Ð ✤➙②

x(t0 ) = x0 ❀ x(t
˙ 0 ) = y0 ❀

❜➡♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐

② ❂x˙ t❛ ❝❤✉②➸♥ ❜➔✐

t♦→♥ ✈➲ ❜➔✐ t♦→♥ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët✿

x˙ = y,
y˙ = g(t, x, y).
❱ỵ✐✿

x(t0 ) = x(0),
y(t0 ) = y(0).
❱✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ ♥➔② t÷ì♥❣ tü ♥❤÷ ❝→❝❤ ❣✐↔✐ ❜➔✐ t♦→♥ ð ✷✳✸✳✶✳

❇✳ ⑩♣ ❞ư♥❣✿

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ s

t [0, 2]





xă = x

ừ ữỡ tr

xă = x
xă + x = 0


ợ

x(0)

= 1


P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❧➔✿

λ2 + 1 = 0
⇔ λ2 = −1 = i2
⇒ λ = i ∨ λ = −i
x = C1 cost + C2 sint

❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❧➔✿
❚ø

x(0) = 0,
x(0)
˙
= 1.
C1 = 0,
C2 = 1.

⇒
◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔✿

①✭t✮ ❂ s✐♥t

✯✮ ●✐↔✐ ❣➛♥ ✤ó♥❣ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ tt
ứ ữỡ tr

xă = x t x = y ✱ tr♦♥❣ ✤â ② ❂ ❝♦♥st✱ s✉② r❛ y˙ = x

ợ

x(0) = 0,
y(0) = 1.
ử ữỡ ❘✉♥❣❡ ✲ ❑✉tt❛ t❛ ✤➦t✿

❢✭t✱①✱②✮ ❂ ②✱
❣✭t✱①✱②✮ ❂ ✲①✱
❤ ❂ ✵✱✺✳

✰✮ t ❂✵❀ s✉② r❛ ① ❂✵✱ ② ❂✶✱
✰✮ t ❂ ✵✱✺❀ t❛ ❝â✿

✈ỵ✐

t ∈ [0, 2]❀

F1 = hf (t0 , x0 , y0 ) = 0, 5 ∗ 1 = 0, 5✳
G1 = hg(t0 , x0 , y0 ) = 0, 5 ∗ 0 = 0✳
F2 ❂hf (t0 +
G2 ❂hg(t0 +
F3 ❂hf (t0 +

h
2 ✱x0

h
2 ✱x0

h
2 ✱x0

+

+

+

F1
2 ✱y0

F1
2 ✱y0

F1
2 ✱y0

+

+

+

G1
2 )

G1
2 )

G1
2 )

= 0, 5(1 + 12 0) = 0, 5.

= −0, 5(0 + 21 0, 5) = −0, 125.

= 0, 5(1 + 21 − 0, 125) = 0, 46875.
✶✼

G3 ❂hg(t0 +

h
2 ✱x0

+

F1
2 ✱y0

+

G1
2 )

= −0, 5(0 + 21 0, 5) = −0, 125.

F4 ❂hf (t0 + h✱x0 + F3 ✱y0 + G3 ) = 0, 5(1 − 0, 125) = 0, 4375.
G4 ❂hg(t0 + h✱x0 + F3 ✱y0 + G3 ) = −0, 5(0 + 0, 46875) = −0, 23438.
❙✉② r❛

x1 = x0 + 16 (F1 + 2F2 + 2F3 + F4 )
= 0 + 16 (0, 5 + 2(0, 5) + 2(0, 46875) + 0, 4375) = 0, 47917✱
y1 = y0 + 61 (G1 + 2G2 + 2G3 + G4 )
= 0 + 61 [0 + 2(−0, 125) + 2(−0, 125) − 0, 23438] = 0, 8776✳
❚÷ì♥❣ tü ✈ỵ✐

t

t2 = 1; t3 = 1, 5; t4 = 2

P❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛

t❛ ❝â ❜↔♥❣ ❣✐→ trà s❛✉ ✭❇↔♥❣ ✷✳✷✮ ✿

❙❛✐ sè ✭%✮

●✐→ trà ❝❤➼♥❤ ①→❝

①

②

①

②

①❂s✐♥✭t✮

②❂❝♦s✭t✮

0

1

0

0

0

1

0, 5

0, 47917

0, 8776

0, 054

0, 0023

0, 47943

0, 87758

1

0, 84104

0, 54058

0, 0511

0, 0519

0, 84147

0, 54030

1, 5

0, 99715

0, 07142

0, 0341

0, 9613

0, 99749

0, 07074

2

0, 90932

−0, 41512

0, 0022

0, 2475

0, 90930

−0, 41615

0

❇↔♥❣ ✷✳✷✿

✷✳✹ ×✉ ✤✐➸♠✱ ❤↕♥ ❝❤➳ ❝õ❛ ữỡ tt
ì
Pữỡ ❑✉tt❛ ✈ø❛ ❝â ✤ë ❝❤➼♥❤ ①→❝ ❝❛♦ ❤ì♥ ❝→❝ ♣❤÷ì♥❣
♣❤→♣ ❦❤→❝ ♥❤÷ ❊✉❧❡r✱ ❊✉❧❡r ❝↔✐ t✐➳♥✱ ❧↕✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ t
t ợ ố ữủ t ữ ở õ t t ờ
ữợ tứ ✤✐➸♠ ♥➔② ✤➳♥ ✤✐➸♠ ❦❤→❝✳
P❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛ ❝â ữ ữủt trở t ộ

ã

ố ữủ t➼♥❤ t♦→♥ t❤➜♣

•

❑❤ỉ♥❣ q✉→ ♣❤ù❝ t↕♣

•

❈❤♦ ✤ë ❝❤➼♥❤ ①→❝ ❝❛♦ ❤ì♥ ✭s❛✐ sè t❤➜♣✮

✶✽


❱➻ ✈➟②✱ tr♦♥❣ t❤ü❝ t➳ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛ ữủ ũ ỡ



s ợ ữỡ r ✈➔ ❊✉❧❡r ❝↔✐ t✐➳♥ ✈➻ t➼♥❤ ❤✐➺✉ q✉↔ ❝õ❛ ♥â✳

❇➯♥ ❝↕♥❤ ♥❤ú♥❣ ÷✉ ✤✐➸♠ ♠➔ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛ tr
q tr t t t tỗ t ♠ët sè ❤↕♥ ❝❤➳ s❛✉✿

•

✣➸ t➼♥❤ ♠ët ❣✐→ trà
❱➻ ✈➟②✱ ♥➳✉ ❤➔♠

yi

t❛ ♣❤↔✐ t➼♥❤ ❣✐→ trà

❢

ð ♥❤✐➲✉ ✤✐➸♠ tr✉♥❣ ❣✐❛♥✳

❢✭①✱②✮ q ự t t t t rt ỗ

❣➦♣ ♥❤✐➲✉ ❦❤â ❦❤➠♥✳

•

❈→❝ ❝ỉ♥❣ t❤ù❝ ❘❑ ❜➟❝ ❝❛♦ t✉② ❝â ✤ë ❝❤➼♥❤ ①→❝ ❝❛♦ ♥❤÷♥❣ ❧↕✐ ♣❤ù❝ t↕♣
✈➔ t➼❝❤ ụ s số s số tr ộ ữợ õ t ♥❤ä ♥❤÷♥❣ ❝❤÷❛ ❝❤➢❝
s❛✐ sè ❝✉è✐ ❝ị♥❣ ✤➣ ♥❤ä✳

✷✳✺ Ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛
❈❤➼♥❤ ♥❤í ÷✉ ✤✐➸♠ ✤â ♠➔ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛ ✤÷đ❝ ❞ị♥❣ ♣❤ê
❜✐➳♥ tr t t ỵ ữ t ✤✐➺♥✱ ❜➔✐ t♦→♥ ❞❛♦ ✤ë♥❣

❦ÿ t❤✉➟t ✈➔ ♥❤✐➲✉ ❜➔✐ t♦→♥ ❦❤→❝ ✤÷đ❝ ♠✐➯✉ t↔ ❜➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
♥❤÷✿ ❇➔✐ t♦→♥ ✧❚➻♠ ❝÷í♥❣ ✤ë ❞á♥❣ ✤✐➺♥ tr♦♥❣ ♠↕❝❤ ✤✐➺♥ ❘▲✧ ✈➔ ❜➔✐ t♦→♥
✈➲ ✧❙ü ❝❤→② ❦✐➺t ❞á♥❣ ❤é♥ ❤ñ♣ ❜ët t ổ tr ỗ ỷ ỏ ỡ




ữỡ

ệ
Pì PP

ỹ r ✤í✐ ✈➔ ♣❤→t tr✐➸♥
❚r♦♥❣ ❝→❝ ♠ỉ♥ ❤å❝ ù♥❣ ❞ư♥❣ ❝➛♥ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t➼♥❤ t♦→♥ ❝ư t❤➸ ✈ỵ✐
t❤í✐ ❣✐❛♥ ♥❤❛♥❤ ♥❤➜t ❧➔ ②➯✉ ❝➛✉ ❝➜♣ t❤✐➳t✳ ▼❛t❤❡♠❛t✐❝❛ ❧➔ ♠ët ❝ỉ♥❣ ❝ư ❧➟♣
tr➻♥❤ ♠↕♥❤ ✈ỵ✐ ❤ì♥ ✼✵✵ ❤➔♠ ❝â s➤♥ tr♦♥❣ t❤÷ ✈✐➺♥ ❤➔♠✱ t❤ü❝ ❤✐➺♥ ♥❤✐➲✉
❝❤ù❝ ♥➠♥❣ ❦❤→❝ ♥❤❛✉✳
P❤✐➯♥ ❜↔♥ ✤➛✉ t✐➯♥ ❝õ❛ ▼❛t❤❡♠❛t✐❝❛ ✤÷đ❝ ❤➣♥❣ ❲♦❧❢r❛♠ ❘❡s❡❛r❝❤ ♣❤→t
❤➔♥❤ ✈➔♦ ♥➠♠ ✶✾✽✽✳ ✣➙② ❧➔ ❤➺ t❤è♥❣ ♣❤➛♥ ♠➲♠ ♥❤➡♠ t❤ü❝ ❤✐➺♥ ❝→❝ t➼♥❤
t♦→♥ tr➯♥ ▼→② t➼♥❤ ✤✐➺♥ tû✳ ◆â ❧➔ tê ❤ñ♣ ❝→❝ t➼♥❤ t♦→♥ ❜➡♥❣ ❦➼ ❤✐➺✉✱ t➼♥❤
t♦→♥ số ỗ t ổ ỳ tr ✈✐ t➼♥❤✳ ❈ỉ♥❣ tr➻♥❤ ♥➔② ✤÷đ❝
①❡♠ ❧➔ t❤➔♥❤ tü✉ ❝❤➼♥❤ tr♦♥❣ ❧➽♥❤ ✈ü❝ ❦❤♦❛ ❤å❝ t➼♥❤ t♦→♥✳
▼❛t❤❡♠❛t✐❝❛ ❧➔ ♥❣æ♥ ♥❣ú t➼❝❤ ❤ñ♣ ✤➛② ✤õ ♥❤➜t ❝→❝ t➼♥❤ t♦→♥ ❦ÿ t❤✉➟t✳
▲➔ ổ ỳ ỹ tr ỵ ỷ ỵ ❝→❝ ❞ú ❧✐➺✉ t÷ì♥❣ ù♥❣✳ ❚❤➳
❤➺ ♥❣ỉ♥ ♥❣ú ❣✐↔✐ t➼❝❤ ✤➛✉ t✐➯♥ ✤â ❧➔✿ ▼❛❝s②♠❛✱ ❘❡❞✉❝❡✱ ✳✳✳ r❛ ✤í✐ tø ♥❤ú♥❣
♥➠♠ ✻✵ ❝õ❛ t❤➳ ❦➾ ❳❳✳ ❈→❝ ♥❣æ♥ ♥❣ú ♥➔② ừ ũ t t ỵ
ữủ ữủ ừ ú ừ ữủ ữợ ❝❤↕②
tr➯♥ ♠→② t➼♥❤ ❧ỵ♥✳
❚❤➳ ❤➺ t✐➳♣ t❤❡♦ ❧➔ ❝→❝ ♥❣ỉ♥ ♥❣ú✿ ▼❛♣❧❡✱ ▼❛t❤❧❛❜✱ ▼❛t❤❡♠❛t✐❝❛✳ ❈→❝
♥❣ỉ♥ ♥❣ú ♥➔② ❝â ÷✉ ✤✐➸♠ ❧➔ ❝❤↕② ♥❤❛♥❤ ❤ì♥✱ ✈➔ ❝❤➜♣ ♥❤➟♥ ❜ë ♥❤ỵ ♥❤ä

✷✵


❤ì♥✱ ❝❤↕② ❤♦➔♥ ❤↔♦ tr➯♥ ❝→❝ ♠→② t➼♥❤ ❝→ ♥❤➙♥✳ ❚r♦♥❣ ❝→❝ ♥❣æ♥ ♥❣ú t➼♥❤
t♦→♥ ❧♦↕✐ ♥➔② ♥ê✐ ❜➟t ❧➔ ổ ỳ tt ợ ữ ữủt trở
t t ỗ t s t ỷ ỵ ỳ ổ
t ổ ỳ t t♦→♥ ❦❤→❝✳
◆❤í ✈➔♦ ❦❤↔ ♥➠♥❣ ♠ỉ ❤➻♥❤ ❤â❛ ✈➔ ♠ỉ ♣❤ä♥❣ ❝→❝ ❤➺ ❧ỵ♥ ❦➸ ❝↔ ❝→❝ ❤➺
✤ë♥❣ ♠➔ ▼❛t❤❡♠❛t✐❝❛ ổ ữủ ự ử tr ỹ t ỵ ❦ÿ
t❤✉➟t ✈➔ t♦→♥ ♠➔ ❝á♥ ✤÷đ❝ ♠ð rë♥❣ ù♥❣ ❞ư♥❣ tr♦♥❣ ✈→❝ ❧➽♥❤ ✈ü❝ ❙✐♥❤ ❤å❝
✈➔ ❝→❝ ❦❤♦❛ ❤å❝ ❦❤→❝✳
▼❛t❤❡♠❛t✐❝❛ ❝â ♥❤✐➲✉ ❱❡rs✐♦♥ ❞♦ ❧✉ỉ♥ ❧✐➯♥ tư❝ ✤÷đ❝ ❝↔✐ t✐➳♥ ✈➔ ❤♦➔♥
t❤✐➺♥✿ ✶✳✷✱ ✷✳✵✱ ✷✳✷✱ ✸✳✵✱ ✹✳✵✱ ✹✳✷✱ ✳✳✳ P❤✐➯♥ ❜↔♥ ♠ỵ✐ ♥❤➜t ❤✐➺♥ ♥❛② ❝õ❛
▼❛t❤❡♠❛t✐❝❛ ❧➔ ✼✳✵ ✳ ▼❛t❤❡♠❛t✐❝❛ ❝✉♥❣ ❝➜♣ r➜t ♥❤✐➲✉ ❝❤ù❝ ♥➠♥❣✱ ♠ët
sè ❝❤ù❝ ♥➠♥❣ t❤æ♥❣ ❞ư♥❣ ❧➔✿ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ❣✐↔✐ ♠❛ tr➟♥✱
ữỡ tr ỗ t
tt ♠➲♠ ✤÷đ❝ sû ❞ư♥❣ ✈➔ ❣✐↔♥❣ ❞↕② t↕✐
♥❤✐➲✉ tr÷í♥❣ ❈❛♦ ✤➥♥❣✱ ✣↕✐ ❤å❝✱ ✳✳✳ ✤➙② ❧➔ ❝ỉ♥❣ ❝ư ❤é trđ tr ờ
ợ ữỡ ổ ồ

ữỡ tr ợ tt
Pữỡ tr ✈✐ ♣❤➙♥ ❝➜♣ ♠ët
❱➼ ❞ö ✸✳✶
✐✮ y˙ = y − x − 1;

y(0) = 1.

✰✮ ❚➻♠ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ✈➔ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾
❜➟❝ ❜è♥ ❘❑✱ ✈ỵ✐

x ∈ [0, 1]; h = 0, 5.

❚❛ ❝â✿

y˙ = y − x − 1,
⇔ y˙ − y = −x − 1.
✣➙② ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ợ

q(x) = x 1;
tờ qt ừ ữỡ tr➻♥❤ ❝â ❞↕♥❣✿

✷✶

p(x) = −1✱


y = e−

p(x)dx

[

q(x)e

p(x)dx = −dx = −x;
q(x)e p(x)dx dx = (−x − 1)e−x dx =
= xe−x + 2e−x .

p(x)dx

dx + C]

❚❛ ❝â✿

−xe−x dx −

e−x dx =

⇒ y = ex [xe−x + 2e−x + C] = x + 2 + Cex .
y0 = 1 ⇒ C = −1.
◆❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔✿

y = x + 2 − ex .

⑩♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❘❑✹ ✤➸ t➻♠ ❝→❝ ❣✐→ trà

x = 0, 5; 1.
ợ

ợ

y1 , y2 ,

tữỡ ự ợ

✤➦t✿ f (x, y) = y − x − 1✳

x1 = 0, 5 :
k1 = hf (x0 , y0 ) = 0, 5(1 − 0 − 1) = 0,

1
1
k2 = 0, 5[1 + 0 − (0 + 0, 5) − 1] = −0, 125,
2
2
1
1
k3 = 0, 5[1 + (−0, 125) − (0 + 0, 5) − 1] = −0, 15625,
2
2
k4 = 0, 5[1 − 0, 15625 − (0 + 0, 5) − 1] = −0, 32813.
1
⇒ y1 = 1 + [0 + 2(−0, 125) + 2(−0, 15625) − 0, 32813] = 0, 85156.
6
✯✮ ❱ỵ✐ x2 = 1 :
k1 = 0, 5(0, 85156 − 0, 5 − 1),
1
1
k2 = 0, 5[0, 85156 + (−0, 32422) − (0, 5 + 0, 5) − 1] = −0, 532028,
2
2
1
1
k3 = 0, 5[0, 85156 + (−0, 532028) − (0, 5 + 0, 5) − 1] = −0, 58179,
2
2
k4 = 0, 5[0, 85156 − 0, 58179 − (0, 5 + 0, 5) − 1] = −0, 86512.
1
⇒ y2 = 0, 85156 + [−0, 32422 + 2(−0, 53028) + 2(−0, 58179) − 0, 86512]

6
= 0, 28231.
✰✮ ❙û ❞ö♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✭❣✐↔✐ trü❝ t✐➳♣✮

■♥[1] := DSolve[{y[x]
˙ == y[x] − x − 1, y[0] == 1}, y, x]
✷✷


Out[1] = {{y → F unction[{x}, 2 − ex + x]}}

■♥[2] := u := 2 − ex + x
■♥[3] := x = 0.5
■♥[6] := x = 1
Out[3] = x = 0.5

Out[6] = 1

■♥[4] := u

■♥[7] := u

Out[4] = 2.5 − e0.5

Out[7] = 3 − e

■♥[5] := N [2.5 − e0.5]

■♥[8] := N [3 − e]

Out[5] = 0.851279

Out[8] = 0.28172

❚ø ✤â t❛ ❝â ❜↔♥❣ ❣✐→ trà s❛✉✿

①

❈ỉ♥❣ t❤ù❝ ❘❑ ✈ỵ✐

❤ ❂✵✱✺

●✐→ trà ❝❤➼♥❤ ①→❝ ❣✐↔✐ ❜➡♥❣ ♠→② t➼♥❤

✵

✶

✶

✵✱✺

✵✱✽✺✶✺✻

✵✱✽✺✶✷✽

✶

✵✱✷✽✷✸✶

✵✱✷✽✶✼✷
❇↔♥❣ ✸✳✶✿

✰✮ ❙û ❞ö♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✭→♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲
❑✉tt❛✮

■♥[1] := f [x❴, y❴] := y − x − 1;
x0 = 1; y0 = 2;
xf = 2;
n = 10;
h = (xf − x0)/n;
rk[x0❴, xf ❴, y0❴, n❴] :=
M odule[{h, x, y, X, Y, i, j, k1, k2, k3, k4, k},
h = (xf − x0)/n;
x = x0; y = y0;
✷✸


X = {x}; Y = {y};
Do[k1 = f [x, y];
k2 = f [x + h/2, y + h ∗ k1/2];
k3 = f [x + h/2, y + h ∗ k2/2];
k4 = f [x + h, y + h ∗ k3];
k = (k1 + 2k2 + 2k3 + k4)/6;
y = y + h ∗ k;
x = x + h;
X = Append[X, x]; Y = Append[Y, y],
{i, 1, n}];
{X, Y }];
X, Y = rk[0., 1., 1, 10];

P rint[T able[T ranspose[{X, Y }]]]
Out[1] = {{0., 1}, {0.5, 0.851563}, {1., 0.282654}}

■♥[2] := f [x❴, y❴] := e−x

2

+

1
1+x ;

x0 = 0; y0 = 1;
xf = 1;
n = 2;
h = (xf − x0)/n;
rk[x0❴, xf ❴, y0❴, n❴] :=
M odule[{h, x, y, X, Y, i, j, k1, k2, k3, k4, k},
h = (xf − x0)/n;
x = x0; y = y0;
X = {x}; Y = {y};
Do[k1 = f [x, y];
k2 = f [x + h/2, y + h ∗ k1/2];
k3 = f [x + h/2, y + h ∗ k2/2];
k4 = f [x + h, y + h ∗ k3];
k = (k1 + 2k2 + 2k3 + k4)/6;
✷✹


y = y + h ∗ k;

x = x + h;
X = Append[X, x]; Y = Append[Y, y],
{i, 1, n}];
{X, Y }];
X, Y = rk[0., 1., 1, 2];
P rint[T able[T ranspose[{X, Y }]]]
Out[2] =
{{0., 1}, {0.1., 1.194998}, {0.2., 1.37969}, {0.3., 1.5536}, {0.4., 1.71613},
0.5., 1.8665}, {0.6., 2.00516}, {0.7., 2.13131}, {0.8., 2.24546}, {0.9., 2.3481},
{1., 2.43991}}
ỷ ử tt ỗ t ❜✐➸✉ ❞✐➵♥ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥

■♥[1] := P lot[x + 2 − ex, {x, 0, 1}];
❖✉t ❬✶❪❂

✷✺