Phương trình sai phân hệ số hằng và một vài ứng dụng

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

✖✖✖✖✖

❱➹ ❚❍➚ ❍×❮◆●

P❍×❒◆● ❚❘➐◆❍ ❙❆■ P❍❹◆ ❍➏ ❙➮ ❍➀◆●
❱⑨ ▼❐❚ ❱⑨■ Ù◆● ❉Ö◆●

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆ ❍➴❈

✣➔ ◆➤♥❣✱ ✷✵✷✶


✷

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

✖✖✖✖✖

❱➹ ❚❍➚ ❍×❮◆●

P❍×❒◆● ❚❘➐◆❍ ❙❆■ P❍❹◆ ❍➏ ❙➮ ❍➀◆●
❱⑨ ▼❐❚ ❱⑨■ Ù◆● ❉Ö◆●
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤
▼➣ sè✿ ✽✹✻✵✶✵✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆
ở ữợ ồ

✷✵✷✶


▲❮■ ❈❆▼ ✣❖❆◆
❚♦➔♥ ❜ë ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥
❝ù✉ tê♥❣ q✉❛♥ ❝õ❛ tæ✐✱ ữủ t ữợ sỹ ữợ ừ
❚r✉♥❣✳ ◆❤ú♥❣ ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tê♥❣ ❤đ♣ tø
❝→❝ t➔✐ ❧✐➺✉ ❦❤♦❛ ❤å❝ ✤→♥❣ t✐♥ ❝➟②✳ ❚ỉ✐ ①✐♥ ❝❤à✉ tr→❝❤ ♥❤✐➺♠ ✈ỵ✐ ♥❤ú♥❣ ❧í✐
❝❛♠ ✤♦❛♥ ❝õ❛ ♠➻♥❤✳

❚→❝ ❣✐↔
❱ã ❚❤à ❍÷í♥❣


▲❮■ ❈❷▼ ❒◆
▲í✐ ✤➛✉ t✐➯♥ ❝õ❛ ❧✉➟♥ ✈➠♥ t→❝ ❣✐↔ ỷ ớ ỡ s s tợ t
ữợ r t t ữợ t→❝ ❣✐↔ tr♦♥❣
s✉èt q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ✤➸ t→❝ ❣✐↔ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♥➔②✳
❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ♥❤➜t ✤➳♥ t➜t ❝↔ ❝→❝ t❤➛② ❝æ
❣✐→♦ ✤➣ t➟♥ t➻♥❤ ❞↕② ❜↔♦ t→❝ tr sốt tớ ồ t ừ õ ồ
ỗ t❤í✐✱ t→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❝→❝ ❛♥❤ ❝❤à ❡♠ tr♦♥❣
❧ỵ♣ ❚♦→♥ ●✐↔✐ t➼❝❤ ❑✸✾ ✲ ✣➔ ◆➤♥❣ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣
q✉→ tr ồ t t ợ


ó ữớ

▼Ư❈ ▲Ư❈
❈❍×❒◆● ✶✳ ❑✐➳♥ t❤ù❝ ❝ì sð ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✶✳ ❑❤→✐ ♥✐➺♠ ✈➲ s❛✐ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✷✳ P❤➙♥ ❧♦↕✐ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ♠ët ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✶✳✸✳ P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ❝❛♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✶✳✸✳✶✳ ❍➔♠ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈➔ ♣❤ö t❤✉ë❝ t✉②➳♥ t➼♥❤✳ ✣à♥❤
t❤ù❝ ❑❛③♦r❛t✐✳ ❉➜✉ ❤✐➺✉ ♥❤➟♥ ❜✐➳t ♣❤ö t❤✉ë❝ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✶✳✸✳✷✳ P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ n t❤✉➛♥ ♥❤➜t ✳ ✶✻
✶✳✸✳✸✳ P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t ❤➺ sè ❤➡♥❣
✷✷
✶✳✸✳✹✳ P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ✳ ✷✼
✶✳✸✳✺✳ P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ❤➺
sè ❤➡♥❣ ✈ỵ✐ ✈➳ ♣❤↔✐ ✤➦❝ t❤ò ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

❈❍×❒◆● ✷✳ ▼❐❚ ❱⑨■ Ù◆● ❉Ư◆● ❈Õ❆ P❍×❒◆● ❚❘➐◆❍
❙❆■ P❍❹◆ ❍➏ ❙➮ ❍➀◆● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✷✳✶✳ Ù♥❣ ❞ö♥❣ t➼♥❤ tê♥❣ ừ số tr ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
ử tr t ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✷✳✸✳ Ù♥❣ ❞ö♥❣ tr♦♥❣ s✐♥❤ ❤å❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✷✳✹✳ ▼ët sè ù♥❣ ❞ö♥❣ ❦❤→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽


✶

❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
◗❯❨➌❚ ✣➚◆❍ ●■❆❖ ✣➋ ❚⑨■ ▲❯❾◆ ❱❿◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

é
ỵ ỹ ồ t
õ t÷đ♥❣ ❦❤♦❛ ❤å❝ ❦ÿ t❤✉➟t tr♦♥❣ t❤ü❝ t✐➵♥ ♠➔ ✈✐➺❝ t➻♠
❤✐➸✉ ♥â t❤÷í♥❣ ❞➝♥ ✤➳♥ ❜➔✐ t♦→♥ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥✱ ❝â t❤➸ ❦➸ ✤➳♥
♥❤÷ ♠ỉ ❤➻♥❤ ❞➙♥ sè✱ ♠ỉ ❤➻♥❤ ❧➣✐ s✉➜t✱✳ ✳ ✳ ◆❣♦➔✐ r❛✱ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥
❝á♥ ❧➔ ♠ët ❝ỉ♥❣ ❝ư ❣✐ó♣ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ✈✐ ♣❤➙♥✱ ✤↕♦ ❤➔♠ ✈➔ ❝→❝ ♣❤÷ì♥❣
tr➻♥❤ ✤↕✐ sè ❝➜♣ ❝❛♦✳
❙ü r❛ ✤í✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ❝ơ♥❣ ①✉➜t ♣❤→t tø ✈✐➺❝ ①→❝ ✤à♥❤
♠è✐ q✉❛♥ ❤➺ t❤✐➳t ❧➟♣ ❜ð✐ ♠ët ❜➯♥ ❧➔ ♠ët ✤↕✐ ❧÷đ♥❣ ❜✐➳♥ t❤✐➯♥ rí✐ r
ữủ f (n) ợ n Z+
ố ợ ữỡ tr tổ tữớ t ❝â t❤➸ ❧➔ ♠ët ❣✐→ trà
sè ✭sè t❤ü❝✱ sè ♣❤ù❝✱✳ ✳ ✳ ✮✳ ❈á♥ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ♠ư❝ t✐➯✉ ❧➔
t➻♠ r❛ ❝ỉ♥❣ t❤ù❝ ❝õ❛ ❤➔♠ ❝❤÷❛ ✤÷đ❝ ❜✐➳t ♥❤➡♠ t❤ä❛ ♠➣♥ ♠è✐ q✉❛♥ ❤➺ ✤➲
r❛✳ ❚❤ỉ♥❣ t❤÷í♥❣ ✤â s➩ ❧➔ ♠ët ❤å ❝→❝ ❤➔♠ sè ✈ỵ✐ s❛✐ ❧➺❝❤ ❜ð✐ ♠ët ❤➡♥❣ sè

c ♥➔♦ ✤â✳ ❍➔♠ ♥➔② s➩ ✤÷đ❝ ①→❝ ✤à♥❤ ❝❤➼♥❤ ①→❝ ❦❤✐ ❝â t❤➯♠ ✤✐➲✉ ❦✐➺♥ ①→❝
✤à♥❤ ❜❛♥ ✤➛✉✱ ❝á♥ ❣å✐ ❧➔ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥✳
❱ỵ✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ ✈➲ ♠ët ❧➽♥❤ ✈ü❝ tr♦♥❣ ❚♦→♥ ❤å❝ ❝â ù♥❣ ❞ö♥❣
rë♥❣ r➣✐ tr♦♥❣ ❝→❝ ✈➜♥ ✤➲ t❤ü❝ t➳ ✈➔ ữủ sỹ ủ ỵ ừ t ữợ
❍↔✐ ❚r✉♥❣ tỉ✐ ❧ü❛ ❝❤å♥ ✤➲ t➔✐ ✏P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ❤➺ sè
❤➡♥❣ ✈➔ ♠ët ✈➔✐ ù♥❣ ❞ö♥❣✑ ❝❤♦ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛ ♠➻♥❤✳

✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉
▼ö❝ t✐➯✉ ❝õ❛ ✤➲ t➔✐ ❧➔ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ❤➺ sè ❤➡♥❣
✈➔ ♠ët ✈➔✐ ù♥❣ ❞ö♥❣✳ ✣➸ ✤↕t ✤÷đ❝ ♠ư❝ t✐➯✉ tr➯♥ ✤➲ t➔✐ s➩ ♥❣❤✐➯♥ ❝ù✉
♥❤ú♥❣ ♥ë✐ ❞✉♥❣ s❛✉✿
✲ ❚r➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ s❛✐ ♣❤➙♥✳

✲ P❤➙♥ ❧♦↕✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥✳


✸

✲ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ❤➺ sè ❤➡♥❣✳
✲ ◆ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐ ✤÷đ❝ ❞ü ✤à♥❤ ❝❤✐❛ t❤➔♥❤ ✷ ❝❤÷ì♥❣✿
❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝ì sð
❈❤÷ì♥❣ ✷✿ ▼ët ✈➔✐ ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ❤➺ sè ❤➡♥❣

✸✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉
P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ❤➺ sè ❤➡♥❣✳

✹✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉
P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ❤➺ sè ❤➡♥❣ ✈➔ ♠ët ✈➔✐ ù♥❣ ❞ư♥❣✳

✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉
❚❤✉ t❤➟♣ ❝→❝ t➔✐ ❧✐➺✉ s÷✉ t➛♠ ✤÷đ❝✱ ❝→❝ s→❝❤ ✈ð ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ✤➲ t➔✐
❧✉➟♥ ✈➠♥✳ ❈→❝ ❦✐➳♥ t❤ù❝ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧✐➯♥ q✉❛♥
ỹ ỵ tt ữỡ tr s t➼❝❤✱ ✣↕✐ sè t✉②➳♥ t➼♥❤✱✳ ✳ ✳

✻✳ Þ ♥❣❤➽❛ ❦❤♦❛ ❤å❝ ✈➔ t❤ü❝ t✐➵♥ ❝õ❛ ✤➲ t➔✐
✣➲ t➔✐ ❝â ❣✐→ tr t ỵ tt õ t sỷ ử ✈➠♥ ❧➔♠ t➔✐ ❧✐➺✉
t❤❛♠ ❦❤↔♦ ❞➔♥❤ ❝❤♦ s✐♥❤ ✈✐➯♥ ♥❣➔♥❤ ❚♦→♥ ✈➔ ♥❤ú♥❣ ✤è✐ t÷đ♥❣ ❝â ❝❤✉②➯♥
♥❣➔♥❤ ❧✐➯♥ q✉❛♥✳

✼✳ ❈➜✉ tró❝ ❧✉➟♥ ✈➠♥
▲❮■ ◆➶■ ✣❺❯
❈❤÷ì♥❣ ■✳ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð
✶✳✶✳ ❑❤→✐ ♥✐➺♠ ✈➲ s❛✐ ♣❤➙♥

✶✳✷✳ P❤➙♥ ❧♦↕✐ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ♠ët
✶✳✸✳ P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ❝❛♦
✶✳✸✳✶✳ ❍➔♠ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈➔ ♣❤ö t❤✉ë❝ t✉②➳♥ t➼♥✳ ✣à♥❤ t❤ù❝
❑❛③♦r❛t✐✳ ❉➜✉ ❤✐➺✉ ♥❤➟♥ ❜✐➳t ♣❤ö t❤✉ë❝ t✉②➳♥ t➼♥❤
✶✳✸✳✷✳ P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ n t❤✉➛♥ ♥❤➜t
✶✳✸✳✸✳ P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t ❤➺ sè ❤➡♥❣
✶✳✸✳✹✳ P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t


✹

✶✳✸✳✺✳ P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t số
ợ tũ
ữỡ Ù◆● ❉Ư◆● ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ❙❆■ P❍❹◆
❍➏ ❙➮ ❍➀◆●
✷✳✶✳ Ù♥❣ ử t tờ ừ số tr ỗ k
ử tr t ỵ
ử tr s ❤å❝✳
✷✳✹✳ ▼ët sè ù♥❣ ❞ö♥❣ ❦❤→❝✳


✺

✶
❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð
❈❍×❒◆●

✶✳✶✳ ❑❤→✐ ♥✐➺♠ ✈➲ s❛✐ ♣❤➙♥
❳➨t ❤➔♠ sè ♠ët ❜✐➳♥ t❤ü❝ y(t) ✈➔ h > 0.

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❇✐➸✉ t❤ù❝✿
∆y(t) = y(t + h) − y(t)

✭✶✳✶✮

✤÷đ❝ ❣å✐ ❧➔ s❛✐ ♣❤➙♥ ❤ú✉ ❤↕♥ t❤ù ♥❤➜t ❤❛② s❛✐ ♣❤➙♥ ❤ú✉ ❤↕♥ ❝➜♣ ♠ët ❝õ❛

y(t)✳
▼ët ❝→❝❤ tü ♥❤✐➯♥ t❛ s➩ ♠➦❝ ✤à♥❤ ❤➔♠ y(t) ❧➔ ①→❝ ✤à♥❤ t↕✐ ❝→❝ ✤✐➸♠
t t t ú ỵ r tr ỵ t❤✉②➳t ✈✐ ♣❤➙♥ t❤➻ h ❝❤➼♥❤ ❧➔
sè ❣✐❛ ❝õ❛ ✤è✐ sè✱ ❝á♥ ∆y(t) ❝❤➼♥❤ ❧➔ sè ❣✐❛ ❝õ❛ ❤➔♠ sè t↕✐ ✤✐➸♠ t✳ ❚r♦♥❣
t➔✐ ❧✐➺✉ ❝õ❛ ❝❤ó♥❣ t❛ sè h ỏ õ t ồ ữợ ỳ ❝➜♣
❝❛♦ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❜✐➸✉ t❤ù❝✿

∆n y(t) = ∆(∆n−1 y(t)).

✭✶✳✷✮

❱➼ ❞ư ✶✳✶✳✷✳ ✣è✐ ✈ỵ✐ n = 2 t❛ ❝â✿
∆2 y(t) = ∆(∆y(t)) = ∆(y(t + h) − y(t))
= (y(t + 2h) − y(t + h) − (y(t + h) − y(t)) = y(t + 2h) − 2y(t + h) + y(t).
✣➸ t✐➺♥ ❧ñ✐ ✈➔ ♥❤➜t q✉→♥ ✈➲ ♠➦t ❧♦❣✐❝ t❛ s➩ ❦➼ ❤✐➺✉ ∆0 y(t) = y(t)✳ ❇➡♥❣
♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣ t♦→♥ ❤å❝ ❦❤ỉ♥❣ ❦❤â ✤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ s❛✐ ♣❤➙♥
❤ú✉ ❤↕♥ ❜➟❝ n ❧➔ t✉②➳♥ t➼♥❤✱ tù❝ ❧➔✿

∆n (f (t) + g(t)) = ∆n (f (t)) + ∆n (g(t))
∆n (Cf (t)) = C∆n (f (t)).
●✐→ trà ∆n y(t) ❞➵ ❞➔♥❣ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ q✉❛ ❣✐→ trà ❝õ❛ ❤➔♠ y(t) t↕✐ ❝→❝
✤✐➸♠ t, t + h, ..., t + nh✳ ❚❛ ❝â ✤÷đ❝ ❝ỉ♥❣ t❤ù❝ s❛✉ ✤➙②✿
n

n

(−1)n−k Cnk y(t + kh).

∆ y(t) =
k=0

✭✶✳✸✮


✻

❚❛ ✤✐ ❝❤ù♥❣ tä ❝ỉ♥❣ t❤ù❝ tr➯♥ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣ t♦→♥ ❤å❝✳
❍✐➸♥ ♥❤✐➯♥ ✈ỵ✐ n = 1 ❝ỉ♥❣ t❤ù❝ ✭✶✳✸✮ ❝â ❞↕♥❣ ∆y(t) = −y(t) + y(t + h)✱
❝❤➼♥❤ ❧➔ ✭✶✳✶✮✳ ●✐↔ sû ✭✶✳✸✮t❤ä❛ ♠➣♥ ❦❤✐ ✤è✐ ✈ỵ✐ s❛✐ ♣❤➙♥ ❤ú✉ ❤↕♥ ❝➜♣

n − 1✱ t❛ ❝â✿
n−1
n

∆ y(t) = ∆(∆

n−1

k
(−1)n−k−1 Cn−1
y(t + kh))

y(t)) = ∆(

k=0

n−1

n−1
n−k−1

(−1)

=

k
Cn−1
y(t

k
(−1)n−k−1 Cn−1
y(t + kh).

+ (k + 1)h) −
k=0

k=0

Ð sè ❤↕♥❣ t❤ù ♥❤➜t t❛ ✤➦t k + 1 = m✱ ❦❤✐ ✤â✿
n−1

n

(−1)

n−k−1

k
Cn−1
y(t

m−1
(−1)n−m Cn−1
y(t + mh),

+ (k + 1)h) =

m=1
n

k=0

✈➔ ❧↕✐ ✈✐➳t m := k, t❛ ♥❤➟♥ ✤÷đ❝ ❜✐➸✉ t❤ù❝

k−1
(−1)n−k Cn−1
y(t + kh)✳ ❑❤✐

k=1

✤â t❛ ❝â✿

n

n−1

n

n−k

∆ y(t) =

(−1)

k−1
Cn−1
y(t

k=1

k
(−1)n−k−1 Cn−1
y(t + kh)

+ kh) −
k=0

n−1

=

n−1

(−1)

n−k

k−1
n−1
Cn−1
y(t+kh)+(−1)0 Cn−1
y(t+nh)+

k=1

k
(−1)n−k Cn−1
y(t+kh)
k=0

n−1
k−1
n−1
(−1)n−k Cn−1
y(t + kh) + Cn−1
y(t + nh)

=
k=1
n−1

k
0
(−1)n−k Cn−1

y(t + kh) + (−1)n Cn−1
y(t).

+
k=1

k−1
n−1
k
0
▼➦t ❦❤→❝ t❛ ❧↕✐ ❝â Cn−1
+ Cn−1
= Cnk ✈➔ Cn−1
= Cn−1
= 1. ❉♦ ✤â ❝ỉ♥❣

t❤ù❝ ❝✉è✐ t❛ ✈✐➳t ✤÷đ❝ ữợ
n1
n

n
nk

y(t) = y(t+nh)+

(1)

Cnk y(t+kh)+(1)n y(t)

k=1

(1)nk Cnk y(t+kh).

=
k=0



ữ t ổ tự ữủ ự
ỵ r ữ tr♦♥❣ ❝æ♥❣ t❤ù❝ ✭✶✳✸✮ t❛ t❤ü❝ ❤✐➺♥ ♣❤➨♣ ✤ê✐ ❜✐➳♥
❝õ❛ ❝❤➾ sè m = n − k ✈➔ sû ❞ö♥❣ ❝æ♥❣ t❤ù❝ Cnk = Cnn−k ✱ ❦❤✐ ✤â t❛ ♥❤➟♥


✼

✤÷đ❝✿

n
n

∆ y(t) =

n
(−1)m Cm
y(t + (n − m)h).
m=0

▼ët ❝→❝❤ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü✱ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣ t♦→♥ ❤å❝✱ t❛
❝ơ♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❝ỉ♥❣ t❤ù❝✿

n

Cnk ∆k y(t)).

y(t + nh) =

✭✶✳✺✮

k=0

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ P❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣✿
F (t, y(t), y(t + h), ..., y(t + nh)) = 0

✭✶✳✻✮

✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥✳
◆➳✉ tr♦♥❣ ✭✶✳✻✮ t❛ ❜✐➸✉ ❞✐➵♥ ❝→❝ s❛✐ ♣❤➙♥ ❤ú✉ ❤↕♥ ❜ỵ✐ ❝ỉ♥❣ t❤ù❝ ✭✶✳✸✮
t❤➻ t❛ ♥❤➟♥ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤✿

G(t, y(t), y(t + h), ..., y(t + nh)) = 0

✭✶✳✼✮

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ P❤÷ì♥❣ tr➻♥❤ ✭✶✳✼✮ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥
❝➜♣ n✳

❱➼ ❞ư ✶✳✶✳✺✳ ❳→❝ ✤à♥❤ ❜➟❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ✤➙②✿
∆3 y(t) + ∆2 y(t) − ∆y(t) − y(t) = 0
❚❛ ❝â✿ ∆y(t) = y(t + h) − y(t)

∆2 y(t) = y(t + 2h) − 2y(t + h) + y(t)
∆3 y(t) = y(t + 3h) − 3y(t + 2h) + 3y(t + h) − y(t).
❉♦ ✤â✿ ∆3 y(t) + ∆2 y(t) − ∆y(t) − y(t) = y(t + 3h) − 2y(t + 2h).
✣➦t τ = t + h✱ ❦❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❝✉è✐ ✈✐➳t ✤÷đ❝ ữợ

y( + h) 2y( ) = 0,
ữỡ tr➻♥❤ ✭s❛✐ ♣❤➙♥✮ ❝➜♣ ♠ët✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻✳ ▼ët ❤➔♠ ❧✐➯♥ tư❝ y(t) ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ ✭✶✳✼✮ tr➯♥ t➟♣ Ω✱ ♥➳✉ t❤❛② ♥â ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ t❤➻ t❛ ♥❤➟♥ ✤÷đ❝
✤➥♥❣ t❤ù❝ ✤ó♥❣ tr➯♥ Ω.

❱➼ ❞ư ✶✳✶✳✼✳ ❍➔♠ sè y = 3t ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ y(t+2)−9y(t) =


✽

0 tr➯♥ R.
❘ã r➔♥❣ ♠å✐ ❤➔♠ sè ❝â ❞↕♥❣ y(t) = C(t)3t ✱ ✈ỵ✐ C(t) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥
✈ỵ✐ ❝❤✉ ❦ý T = 2 ❝ơ♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✳
❚✐➳♣ t❤❡♦ t❛ s➩ ❧✉æ♥ ❣✐↔ sû h = 1✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✼✮ ❝â ❞↕♥❣✿

G(t, y(t), y(t + 1), ..., y(t + n)) = 0.

✭✶✳✽✮

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✽✳ ◆❣❤✐➺♠ ✭rí✐ r↕❝✮ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✽✮ t÷ì♥❣ ù♥❣
t↕✐ ✤✐➸♠ t0 ∈ Z+ ❧➔ ❝❤✉é✐ sè y0 , y1 , ..., yk , ... s❛♦ ❝❤♦✿

G(t0 + k, yk , ..., yk+n ) = 0.

✭✶✳✾✮

✈ỵ✐ k = 0, 1, 2, ..., ❝á♥ Z+ ❧➔ t➟♣ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✳
❇➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ✈✐➺❝ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✼✮ ♥➡♠
y(t) ừ ữỡ tr ỗ t❤í✐ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉
❦✐➺♥ ✭✤➛✉✮ s❛✉ ✤➙②✿

y(t0 ) = y0 , y(t0 + 1) = y1 , ..., y(t0 + n − 1) = yn−1 .
❈→❝ sè y0 , y1 , ..., yn−1 ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ❣✐→ trà ✤➛✉ ❝õ❛ ♥❣❤✐➺♠ y(t)✱ t0 ✤÷đ❝
❣å✐ ❧➔ ✤✐➸♠ ✤➛✉✳
◆➳✉ y(t) ❧➔ ♥❣❤✐➺♠ ❧✐➯♥ tư❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✼✮ tr➯♥ [t0 , +∞)✱ t❤➻
❦❤✐ ✤â ❞➣② y(t0 ), y(t0 + 1), ..., y(t0 + k), ... s➩ ❧➔ ♥❣❤✐➺♠ rí✐ r↕❝ ❝õ❛ ✭✶✳✼✮✳
❚✐➳♣ t❤❡♦ t❛ s➩ ❧➜② t0 = 0✳ ▲ó❝ õ rớ r t s t ữợ

y(t) ✤÷đ❝ ♥❣➛♠ ❤✐➸✉ ❧➔ ❤➔♠ ♥➔② ❝❤➾ ①→❝ ✤à♥❤ t↕✐ ❝→❝ ✤✐➸♠ ❝õ❛ t➟♣
Ω0 = t0 , t0 + 1, ..., t0 + k, ... ✈➔ y(t0 + k) = yk .
❈❤ó♥❣ t❛ ❣✐û sû ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✼✮ ❝❤➾ ❝â t ữủ tữỡ ự ố
ợ y(t + n) y(t) tự ữủ ữợ

y(t + n) = Φ1 (t, y(t), y(t + 1), ..., y(t + n − 1))

✭✶✳✶✵✮

y(t) = Φ2 (t, y(t + 1), ..., y(t + n)).

✭✶✳✶✶✮

✈➔

◆➳✉ ❤➔♠ Φ1 (t, u1 , u2 , ..., un ) ①→❝ ✤à♥❤ ❜ð✐ ✈➳ ♣❤↔✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
✭✶✳✶✵✮ ①→❝ ✤à♥❤ t↕✐ ♠å✐ ✤✐➸♠ t ∈ Z+ ✈➔ ♠å✐ ❣✐→ trà ❝õ❛ u1 , u2 , ..., un
t❤➻ ♠ët ♥❣❤✐➺♠ rí✐ r↕❝ ❞✉② ♥❤➜t ✤÷đ❝ ①→❝ ✤à♥❤ ✱ ♥➳✉ ✈ỵ✐ ♠å✐ sè t0 ∈

Z+ , y0 , y1 , ..., yn1 ữủ trữợ ú õ tự yn+k = Φ1 (t0 +


✾

k, yk , ...yn+k−1 ) ❜✐➸✉ ❞✐➵♥ ❝æ♥❣ t❤ù❝ tr✉② ỗ tổ q õ r
ữủ yn , yn+1 , ...
❚✐➳♣ t❤❡♦✱ ✤➸ ✤✐ ✤➳♥ ❦❤→✐ ♥✐➺♠ ✤✐➸♠ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❈❛✉❝❤② ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✼✮✱ t❛ ①❡♠ ①➨t ♠ët ✈➼ ❞ư ✤ì♥ ❣✐↔♥ s❛✉ ✤➙②✿

❱➼ ❞ư ✶✳✶✳✾✳ ✣è✐ ợ ữỡ tr y(t + 1) = y 2 (t) t❤➻ ❞➣② 1, 1, ..., 1, ...
❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✤➛✉ y(0) = 1✱ ❝á♥ ♥❣❤✐➺♠
❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✤➛✉ y(0) = −1 ❧➔ ❞➣② ✲✶✱✶✱✳✳✳✱✶✱✳✳✳✳ ❍✐➸♥
♥❤✐➯♥ tø k ≥ 1 t❤➻ ợ t t ữủ ♥❣❤✐➺♠
♥❤÷ ♥❤❛✉✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✵✳ ✣✐➸♠ (t0 , y0 , y1 , ..., yn1 ) Z+ ì Rn ữủ ồ ❧➔
✤✐➸♠ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② ✭✶✳✼✮✱ ♥➳✉ ✈ỵ✐ ❜➜t ❦ý ♥❣❤✐➺♠

ϕ(t) ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤②✱ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✤➛✉✿
ϕ(t0 ) = ϕ0 , ϕ(t0 + 1) = ϕ1 , ..., ϕ(t0 + n − 1) = ϕn−1 .
(y0 , y1 , ..., yn−1 ) = (ϕ0 , ϕ1 , ..., ϕn−1 )
s✉② r❛ ✈ỵ✐ ♠å✐ k ≥ 1

(yk , yk+1 , ..., yk+n−1 ) = (ϕ1 , ϕk+1 , ..., ϕk+n−1 ),
tù❝ ❧➔ ✈ỵ✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❦❤→❝ ♥❤❛✉ t❤➻ s✐♥❤ r❛ ❝→❝ ♥❣❤✐➺♠ ❦❤→❝ ♥❤❛✉✳

◆❤➻♥ ❝❤✉♥❣ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ❧➔ ❝â ✈ỉ sè ♥❣❤✐➺♠✱ ❝è ♥❤✐➯♥ ❝ơ♥❣
❝â t❤➸ ❧➜② ✈➼ ❞ư ❝❤♦ tr÷í♥❣ ❤đ♣ ✈ỉ ♥❣❤✐➺♠ ✭t❤ü❝✮✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷ ♣❤÷ì♥❣
tr➻♥❤ y 2 (t + 1) + y 2 (t) + 1 = 0.
◆➳✉ ♥❤÷ ✤á✐ ❤ä✐ ❤➔♠ Φ2 (t, u1 , u2 , ..., u2 ) ✈➳ ♣❤↔✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✶✮
t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ t÷ì♥❣ tü ♥❤÷ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤è✐ ✈ỵ✐ Φ1 (t, u1 , u2 , ..., u2 )
t❤➻ ộ ừ t 0 ì Rn tỗ t↕✐ ✈➔ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐
t♦→♥ ❈❛✉❝❤②✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✶✳ ●✐↔ sû D ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ n + 1 ❝❤✐➲✉
Rn+1 ✈➔ ♠è✐ ✤✐➸♠ ❝õ❛ D ✤➲✉ tỗ t t ừ
t ❈❛✉❝❤② ✭✶✳✼✮✳ ❍➔♠ y(t) = y(t, C1 , ..., Cn ) ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ tê♥❣
q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✼✮✱ ♥➳✉ t❤ä❛ ♠➣♥ ❤❛✐ ✤✐➲✉ ❦✐➺♥✿
✶✳ ❱ỵ✐ ♠å✐ ❣✐→ trà trữợ C1 , ..., Cn ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣


✶✵

tr➻♥❤ ✭✶✳✼✮✱
✷✳ ▼å✐ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② ✭✶✳✼✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✤➛✉ ✤÷đ❝ ❧➜② tø

D ❝â t❤➸ ♥❤➟♥ ✤÷đ❝ tø ♥❣❤✐➺♠ tê♥❣ q✉→t ♠ët ❝→❝❤ ❞✉② ♥❤➜t✳

✶✳✷✳ P❤➙♥ ❧♦↕✐ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ♠ët
❳➨t ♣❤÷ì♥❣ tr➻♥❤✿
✭✶✳✶✷✮

∆y(t) = f (t), t ∈ Ω0

❤❛② y(t + 1) = y(t) + f (t)✳ ✣➦t ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❝✉è✐ ❧➛♥ ❧÷đt ❝→❝ ❣✐→ trà

t = t0 , t = t0 + 1, . . . , t = n − 1 rỗ ở ỗ t ờ
n := t t❛ ♥❤➟♥ ✤÷đ❝✿
t−1

f (k), C = y (t0 )

y(t) = C +



k=t0

ỵ r ữỡ tr ởt y (x) = f (x) tữỡ ự ợ ✭✶✳✶✸✮
❝â ❞↕♥❣✿

x

y(x) = C +

f (x)dx.
x0

❇➙② ❣✐í t❛ t✐➳♥ ❤➔♥❤ ①❡♠ ①➨t ♣❤÷ì♥❣ tr➻♥❤✿

y(t + 1) = y(t)p(t), p(t) = 0, t ∈ Ω0 .

✭✶✳✶✹✮

▲➛♥ ❧÷đt ✤➦t ✈➔♦ ✭✶✳✶✹✮ ❝→❝ ❣✐→ trà t = t0 , t = t0 + 1, . . . , t = n − 1 rỗ
ợ s õ tỹ ờ n := t✱ t❛ ♥❤➟♥ ✤÷đ❝✿

t

t−1

y(k) =
k=t0 +1

t−1

p(k)
k=t0

y(k).

✭✶✳✶✺✮

k=t0

◆➳✉ y (t0 ) = 0 t❤➻ tø ✤✐➲✉ ❦✐➺♥ p(t) = 0, t ∈ Ω0 s✉② r❛ y(t) = 0, t ∈ 0
ữợ ừ ữỡ tr ❝❤♦

t−1
k=t0

y(k) = 0✱ t❛ ♥❤➟♥
✤÷đ❝ ❝→❝ ♥❣❤✐➺♠ ❦❤ỉ♥❣ s✉② ❜✐➳♥ ❝✠
✉❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✹✮✳
t−1

y(t) = y (t0 )

p(k), y (t0 ) = 0.

✭✶✳✶✻✮

k=t0

✣➦t y (t0 ) := C t❛ ♥❤➟♥ ✤÷đ❝ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ✭✶✳✶✺✮ ❝â ❞↕♥❣✿
t

y(t) = C

p(k).
k=t0

✭✶✳✶✼✮




ú ỵ r tr tỹ t t ổ tự ❝â ❝❤ù❛ ❝↔ ♥❣❤✐➺♠ t➛♠
t❤÷í♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✺✮✱ ❦❤✐ C = 0✳ ❚÷ì♥❣ tü✱ t❛ ❣➦♣ ❧↕✐ ❝ỉ♥❣
t❤ù❝ ♠ỉ t ừ ữỡ tr ởt ợ ❜✐➳♥ sè ♣❤➙♥
❧② y = p(x)y ✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❝â ✤÷đ❝ ❝ỉ♥❣ t❤ù❝ t÷ì♥❣ tü ♥❤÷
✭✶✳✶✼✮✿

y(x) = Ce

x
x0

p(x)dx

✭✶✳✶✽✮

.

P❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✹✮ ❝á♥ ✤÷đ❝ ❝♦✐ ❧➔ tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
s❛✉ ✤➙②✿
✭✶✳✶✾✮

y(t + 1) = p(t)y(t) + f (t), p(t) = 0, t ∈ Z+ .

❇➔✐ t♦→♥ ①➙② ❞ü♥❣ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ✤÷đ❝ ❣✐↔✐
q✉②➳t ❜ð✐ ▲❛❣r❛♥❣❡r✳ ❚✐➳♥ ❤➔♥❤ ①❡♠ ①➨t ♣❤÷ì♥❣ ♣❤→♣ ①➙② ❞ü♥❣ ♥❣❤✐➺♠
tê♥❣ q✉→t ❝õ❛ ♣❤÷♦♥❣ tr➻♥❤ tr➯♥✱ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ t❤✐➯♥ ❤➡♥❣
sè ❤❛② ❝á♥ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ▲❛❣r❛♥❣❡r✳ ❚❛ ❝♦✐ C tr♦♥❣ ✭✶✳✶✾✮ ♥❤÷ ❧➔
♠ët ❤➔♠ ♣❤ư t❤✉ë❝ ✈➔♦ t✱ ✤➸ ❝❤♦ ❝æ♥❣ t❤ù❝✿
t−1

y(t) = C(t)

✭✶✳✷✵✮

p(k)
k=t0

❝❤♦ t❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✾✮✳ ✣➦t ✭✶✳✷✵✮ ✈➔♦ ✭✶✳✶✾✮ t❛ ♥❤➟♥ ✤÷đ❝✿
t−1

t

p(k) + f (t)

p(k) = p(t)C(t)

C(t + 1)

k=t0

k=t0



t

−1

t

p(k) · ∆C(t) = f (t), ∆C(t) = f (t) 
k=t0

p(k)

.

k=t0

P❤÷ì♥❣ tr➻♥❤ ❝✉è✐ ❝â ❞↕♥❣ ✭✶✳✶✷✮✱ ❞♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ữỡ

tr tr õ t t ữủ ữợ
t1

1

k

f (k) ·

C(t) = C +

p(m)

.

m=t0

k=t0

✣➦t ❜✐➸✉ t❤ù❝ ♥❤➟♥ ✤÷đ❝ ✤è✐ ợ C(t) t t ữủ


1
t1

t1

y(t) =

f (k) Ã

p(k) C +
k=t0

k

k=t0

p(m)
m=t0

.

✭✶✳✷✶✮


✶✷

❚❛ t✐➳♥ ❤➔♥❤ ①❡♠ ①➨t t❤➯♠ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ♥ú❛ ✤➸ ①→❝ ✤à♥❤ ♥❣❤✐➺♠
tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✾✮✱ ❝â t➯♥ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❇❡❝♥✉❧❧✐✳ ❚❛
s➩ ✤✐ t➻♠ y(t) ữợ y(t) = u(t)v(t) õ

y(t + 1) = u(t + 1)v(t + 1) = p(t)u(t)v(t) + f (t),
u(t + 1)∆v(t) = p(t)u(t)v(t) − u(t + 1)v(t) + f (t).
❚❛ ❝❤å♥ ❤➔♠ u(t) s❛♦ ❝❤♦ u(t+1) = u(t)p(t)✱ ✈➼ ❞ư ♥❤÷ u(t) = t−1
k=t0 p(k)✳
❈á♥ ❤➔♠ v(t) ✤÷đ❝ ①→❝ ✤à♥❤ tø ♣❤÷ì♥❣ tr➻♥❤ u(t + 1)∆v(t) = f (t)✳ ❍❛②
−1
∆v(t) = f (t) tk=t0 p(k) ✱ ❞♦ ❞â t❤❡♦ ✭✶✳✶✸✮ t❛ ♥❤➟♥ ✤÷đ❝✿
t−1

v(t) = C +

f (k)

t−1

y(t) =



t−1

,
−1

k

f (k) ·

p(k) C +
k=t0

p(m)
m=t0

k=t0

tø ✤â

−1

t

p(m)


,

m=t0

k=t0

✈➔ ❝ỉ♥❣ t❤ù❝ ❝✉è✐ ♥❤➟♥ ✤÷đ❝ ❝ơ♥❣ ữ
t ụ ữ ỵ ố ợ ữỡ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣
♠ët ❞↕♥❣ y = p(x)y + f (x) t❤➻ ❝æ♥❣ t❤ù❝ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝â ❞↕♥❣✿
x

y(x) = exp

x

p(t)dt
x0

t

f (t) exp −

C+

x0

p(τ )dτ dt .
x0

❚r♦♥❣ ♠ö❝ ♥➔② ✈➔ trð ✈➲ s❛✉ t❛ t❤è♥❣ ♥❤➜t r➡♥❣✱ ♥➳✉ ♥❤÷ tr♦♥❣ ♣❤➨♣ t➼♥❤
tê♥❣ ♠➔ ❣✐→ trà ❝õ❛ ❝❤➾ sè ❜➯♥ tr➯♥ ♥❤ä ❤ì♥ ❣✐→ trà ❝õ❛ ❝❤➾ sè ❜➯♥ ữợ t
tờ õ ỏ tr t ♥❤➙♥ ✤✐➲✉ t÷ì♥❣ tü ①↔② r❛ t❤➻ t➼❝❤
✤â ♥❤➟♥ ❣✐→ trà ❜➡♥❣ 1. ❚✐➳♣ t❤❡♦ t❛ t✐➳♥ ❤➔♥❤ ①❡♠ ①➨t ♣❤÷ì♥❣ tr➻♥❤ s❛✐
♣❤➙♥ ❝➜♣ ♠ët ❘✐❝❝❛t✐ ❝â ❞↕♥❣ s❛✉ ✤➙②✿
✭✶✳✷✷✮

y(t + 1)y(t) + ay(t + 1) + by(t) + c = 0,

✈ỵ✐ a, b, c ❧➔ ❝→❝ ❤➡♥❣ sè ✭t❤ü❝✮✳ ❚❤ü❝ ❤✐➺♥ ♣❤➨♣ ✤ê✐ ❜✐➳♥ y(t) = u(t) + δ ✱
✈ỵ✐ δ ❧➔ ♠ët ❤➡♥❣ sè ✭♥➔♦ ✤➜②✮✱ ✤➦t ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ t❛ ✤÷đ❝✿

u(t + 1)u(t) + (a + δ)u(t + 1) + (b + δ)u(t) + δ 2 + (a + b)δ + c = 0.
▲➜② δ ❝❤➼♥❤ ❧➔ ❣✐→ trà ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ δ 2 + (a + b)δ + c = 0✳
◆❣❤✐➺♠ t➛♠ t❤÷í♥❣ ừ ữỡ tr u(t) = 0 tữỡ ự ợ

y(t) = δ ✳ ◆➳✉ ❝❤♦ y(t) = δ ✱ t❤ü❝ ❤✐➺♥ ♣❤➨♣ ✤ê✐ ❜✐➳♥ v(t) =

1
u(t)

=

1
y(t)−6 .

õ ố ợ v(t) t ữủ ữỡ tr

(b + δ)v(t + 1) + (a + δ)v(t) + 1 = 0
♥➳✉ b + δ = 0 ✈➔ a + δ = 0 t❤➻ ✤➣② ❝❤➼♥❤ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥
t➼♥❤ ❝➜♣ ♠ët✳ ◆➳✉ b + δ = 0 t❤➻ t❛ ❝â ✤÷đ❝ c = ab✱ t÷ a + δ = 0 t❛ ❝ơ♥❣
♥❤➟♥ ✤÷đ❝ c = ab✳
P❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✷✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② (a + δ = 0) ❝â ❞↕♥❣ y(t+ ✶✮

y(t) + ay(t + 1) + by(t) + ab = 0✱ ❤❛② [y(t + 1) + b][y(t) + a] = 0. P❤÷ì♥❣
tr➻♥❤ ♥❤➟♥ ✤÷đ❝ ❝❤➾ ❝â ❤❛✐ ♥❣❤✐➺♠ y(t) = −b ✈➔ y(t) = −a.

❱➼ ❞ö ✶✳✷✳✶✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✿
y(t + 1)y(t) + 2y(t + 1) + y(t) − 4 = 0
✣➸ tr✐➺t t✐➯✉ ✤÷đ❝ sè ❤↕♥❣ tü ❞♦ (−4) t❛ t❤ü❝ ❤✐➺♥ ♣❤➨♣ ✤ê✐ ❜✐➳♥ y(t) =

u(t)+δ. ❑❤✐ ✤â δ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ δ 2 +3δ −4 = 0. ●✐↔ sû t❛ ❧➜②
δ = 1✱ t❤✉ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ u(t + 1)u(t) + 3u(t + 1) + 2u(t) = 0✳ ◆❣❤✐➺♠
u(t) = 0 ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❝✉è✐ t÷ì♥❣ ù♥❣ y(t) = 1 ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜❛♥
1
✤➛✉✳ ❚✐➳♣ t❤❡♦ t❤ü❝ ❤✐➺♥ ♣❤➨♣ ✤ê✐ ❜✐➳♥ u(t) = e(t)
✭✤✐ t➻♠ ♥❣❤✐➺♠ y(t) = 1
✮✱ ♥❤➟♥ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤
2v(t + 1) + 3v(t) + 1 = 0
❤❛②

3

1
v(t + 1) = − v(t) − .
2
2
◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② t➻♠ ✤÷đ❝ t❤❡♦ ❝ỉ♥❣ t❤ù❝ ✭✶✳✷✶✮✱
✈➔ ✤➦t ✈➔♦ t0 = 0. ❚❛ ❝â ✤÷đ❝✿
t−1
3 t
1
3 −k−1
3 t 1
v(t) = −
C+
−
−
=C −
−
2
2
2
2
5
k=0
◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜❛♥ ✤➛✉ ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝✿

y(t) =

1

+1

t
C − 32 − 15
✣➸ þ r➡♥❣ ♥❣❤✐➺♠ y(t) = 1 ❝â t❤➸ ♥❤➟♥ ✤÷đ❝ ♠ët ❝→❝❤ ❤➻♥❤ t❤ù❝ ❦❤✐ tr♦♥❣
❝æ♥❣ t❤ù❝ ♥❣❤✐➺♠ tê♥❣ q✉→t t❛ ❝❤♦ C = ∞✳

❱➼ ❞ư ✶✳✷✳✷✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✿
p(t)y(t + 1) + q(t)y(t + 1)y(t) − y(t) = 0, p(t) = 0, t ∈ Z+


✶✹

✈➔ ①❡♠ ①➨t tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❦❤✐ p(t) = 2, q(t) = 3
ú ỵ r y(t) = 0 ởt ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✳ ❚✐➳♣
t❤❡♦ t❛ ✤➦t v(t) =

1
y(t)

✭tù❝ ❧➔ ✤✐ t➻♠ ♥❤ú♥❣ ♥❣❤✐➺♠ y(t) = 0 ✮✱ ♥❤➟♥ ✤÷đ❝

♣❤÷ì♥❣ tr➻♥❤ v(t + 1) = p(t)v(t) + q(t) ✈ỵ✐ ♥❣❤✐➺♠ tê♥❣ q✉→t ❧➔


−1

q(k)

p(k) C +

v(t) =

k

t−1

t−1

❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔


t−1

t−1

y(t) =

.

m=0

k=0

k=0

p(m)

1
q(k)
=  p(k) C +
v(t)

k=0
k=0

k

−1

p(m)

−1


m=0

✈➔ ♥❣❤✐➺♠ y(t) = 0✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ p(t) = 2, q(t) = 3 t❛ ❝â ✤÷đ❝✿

y(t) = C2t − 3

−1

✶✳✸✳ P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ❝❛♦

✶✳✸✳✶✳ ❍➔♠ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈➔ ♣❤ö t❤✉ë❝ t✉②➳♥ t➼♥❤✳
✣à♥❤ t❤ù❝ ❑❛③♦r❛t✐✳ ❉➜✉ ❤✐➺✉ ♥❤➟♥ ❜✐➳t ♣❤ö t❤✉ë❝ t✉②➳♥ t➼♥❤
✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ ❈→❝ ❤➔♠ sè ϕ1 (t), ϕ2 (t), ..., ϕn (t) ✤÷đ❝ ❣å✐ ❧➔ ♣❤ư
t❤✉ë❝ t✉②➳♥ t➼♥❤ tr➯♥ t tỗ t ở số C1 , C2 , ..., Cn ổ ỗ
tớ ổ t❤ù❝ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥✿

C1 ϕ1 (t) + C2 ϕ2 (t) + ... + Cn ϕn (t) = 0, t ∈ Ω.

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✷✳ ❈→❝ ❤➔♠ sè ϕ1 (t), ϕ2 (t), ..., ϕn (t) ✤÷đ❝ ❣å✐ ❧➔ ✤ë❝ ❧➟♣
t✉②➳♥ t➼♥❤ tr➯♥ t➟♣ Ω✱ ♥➳✉ tø ✤➥♥❣ t❤ù❝

C1 ϕ1 (t) + C2 ϕ2 (t) + ... + Cn ϕn (t) = 0, t ∈ Ω
t❛ ♥❤➟♥ ✤÷đ❝ C1 = C2 = ... = Cn = 0
ữ ỵ ởt t ❝❤➜t ❤✐➸♥ ♥❤✐➯♥ s❛✉ ✤➙② ❝õ❛ ❝→❝ ❤➔♠ ♣❤ö t❤✉ë❝
t✉②➳♥ t➼♥❤✿
✶✳ ◆➳✉ tr♦♥❣ ❝→❝ ❤➔♠ ϕ1 (t), ϕ2 (t), ..., ϕn (t) ❝â ❤➔♠ ❧➔ ❦❤æ♥❣ t❤➻ ❝→❝
❤➔♠ ✤➣ ❝❤♦ ♣❤ö t❤✉ë❝ t✉②➳♥ t➼♥❤✳
✷✳ ❈→❝ ❤➔♠ ϕ1 (t), ϕ2 (t), ..., ϕn (t) ❧➔ ♣❤ö t❤✉ë❝ t✉②➳♥ t➼♥❤ ❦❤✐ ✈➔
õ ởt t ữợ tờ ủ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ❤➔♠ ❝á♥ ❧↕✐✳


✶✺

✸✳ ◆➳✉ tr♦♥❣ ❝→❝ ❤➔♠ ϕ1 (t), ϕ2 (t), ..., ϕn (t) ❝â k ❤➔♠ ♣❤ö t❤✉ë❝ t✉②➳♥
t➼♥❤ (k < n)✱ t❤➻ ❝→❝ ❤➔♠ ✭ n ❤➔♠✮ ❧➔ ♣❤ö t❤✉ë❝ t✉②➳♥ t➼♥❤✳
❚❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤✱ ✈➼ ❞ö t➼♥❤ ❝❤➜t ✸✳ ❈❤➥♥❣ ❤↕♥✱ ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣
q✉→t t❛ ❝â t❤➸ ❝♦✐ ♥❤÷ k ❤➔♠ sè ✤➛✉ t✐➯♥ ❝õ❛ ϕ1 (t), ϕ2 (t)..., ϕn (t) ❧➔ ♣❤ư
t❤✉ë❝ t✉②➳♥ t➼♥❤✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥❣÷đ❝ ❧↕✐ t❤➻ t❛ ❝❤➾ ❝➛♥ ✤→♥❤ ❧↕✐ sè
t❤ù tü ❝õ❛ ❝→❝ ❤➔♠ ✤➣ ❝❤♦ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ t❛ ❝â✿

C1 ϕ1 (t) + C2 ϕ2 (t) + ... + Ck ϕk (t) ≡ 0, t ∈ Ω

✭✶✳✷✸✮

✈ỵ✐ C12 + C22 + ... + Ck2 = 0. ❳➨t tê ❤ñ♣ t✉②➳♥ t➼♥❤✿

C1 ϕ1 (t) + C2 ϕ2 (t) + ... + Ck ϕk (t) + 0.Ck+1 ϕk+1 (t) + ... + 0.Cn ϕn (t) ✭✶✳✷✹✮
Ð ✤➙② ❝→❝ ❤➡♥❣ sè C1 , ..., Ck tứ ỗ tớ

k
2
i=1 Ci

=

n
2
i=1 Ci

= 0

1 (t), ϕ2 (t), ..., ϕn (t) ❧➔ ♣❤ö t❤✉ë❝ t✉②➳♥ t➼♥❤✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✸✳ ✣à♥❤ t❤ù❝
ϕ1 (t)ϕ2 (t)
K(t) =

...
ϕ1 (t + n − 1)ϕ2 (t + n − 1)

...
ϕ1 (t + 1)ϕ2 (t + 1)

ϕn (t)
...

ϕn (t + n − 1)
✭✶✳✷✺✮

✤÷đ❝ ❣å✐ ❧➔ ✤à♥❤ t❤ù❝ ❑❛③♦r❛t✐ ❜➟❝ n ❝õ❛ ❝→❝ ❤➔♠ ϕ1 (t), ϕ2 (t), ..., ϕn (t)✳
...

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✹✳ ✭❉➜✉ ❤✐➺✉ ❝➛♥ ✤➸ ❝→❝ ❤➔♠ ♣❤ö t❤✉ë❝ t✉②➳♥ t➼♥❤✮✳ ◆➳✉
❝→❝ ❤➔♠ ϕ1 (t), ϕ2 (t), ..., ϕn (t) ❧➔ ♣❤ö t❤✉ë❝ t✉②➳♥ t➼♥❤ tr➯♥ Ω t❤➻ ✤à♥❤ t❤ù❝
❑❛③♦r❛t✐ ❝õ❛ ❝❤ó♥❣ ❜➡♥❣ ❦❤æ♥❣ tr➯♥ Ω✳

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â
C1 ϕ1 (t) + C2 ϕ2 (t) + ... + Cn ϕn (t) ≡ 0, t


n
2
i=1 Ci



= 0. sỷ tỗ t t0 ∈ Ω ✤➸ ❝❤♦ K (t0 ) = 0. ❑❤✐ ✤â tø ✭✶✳✷✻✮
t❛❝â ✤÷đ❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✿
C1 ϕ1 (t0 ) + C2 ϕ2 (t0 ) + ... + Cn ϕn (t0 ) = 0


C1 ϕ1 (t0 + 1) + C2 ϕ2 (t0 + 1) + ... + Cn ϕn (t0 + 1) = 0
...


C1 ϕ1 (t0 + n − 1) + C2 ϕ2 (t0 + n − 1) + ... + Cn ϕn (t0 + n − 1) = 0
✭✶✳✷✼✮
❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✼✮ ❧➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t ❜➟❝ n
✈ỵ✐ ➞♥ Ci , i = 1, n ✈ỵ✐ ✤à♥❤ t❤ù❝ K (t0 ) = 0✳ ❚❤❡♦ ỵ rr t

õ t tữớ ♥❤➜t Ci = 0✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐

ϕn (t + 1)




tt Ci ổ ỗ tớ ổ ứ t❛ ❝â ✤÷đ❝ ✤✐➲✉ ♣❤↔✐
❝❤ù♥❣ ♠✐♥❤✳

✶✳✸✳✷✳ P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ n t❤✉➛♥ ♥❤➜t

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✺✳ P❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣✿
y(t+n)+p1 (t)y(t+n−1)+p2 (t)y(t+n−2)+. . .+pn (t)y(t) = f (t) ✭✶✳✷✽✮
✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ n✳
❚❛ s➩ ❣✐↔ t❤✐➳t r➡♥❣ ❝→❝ ❤➺ sè pi (t), i = 1, n ✈➔ ✈➳ ♣❤↔✐ f (t) ❝õ❛ ữỡ
tr tr Z+ ỗ tớ pi (t) = 0, t ∈ Z+ ✳ ❚r♦♥❣ ❝→❝
✤✐➲✉ ❦✐➺♥ tr➯♥ t❤➻ ❜➜t ❦ý ✤✐➸♠ ♥➔♦ t❤✉ë❝ Z+ × Rn ụ tỗ
t t ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✽✮✳
◆➳✉ f (t) ≡ 0 t❤➻ ✭✶✳✷✽✮ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣

n t❤✉➛♥ ♥❤➜t ✈➔ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t ✭❚❑❚◆✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥❣÷đ❝ ❧↕✐✳
P❤÷ì♥❣ tr➻♥❤
λ(t + n) + p1 (t)z(t + n − 1) + p2 (t)(t + n − 2) + . . . + pn (t)z(t) = 0 ✭✶✳✷✾✮
✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t n t t tữỡ
ự ợ t✐➺♥ ❧đ✐ t❤➻ tr♦♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✽✮ ✈➔ ✭✶✳✷✾✮
t❛ sû ❞ö♥❣ ❝→❝ ❝❤ú ❝→✐ ❦❤→❝ ♥❤❛✉✳

✣à♥❤ ❧➼ ✶✳✸✳✻✳ ✭❚✐➯✉ ❝❤✉➞♥ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ♥❣❤✐➺♠ ❚❚◆✮❈→❝
♥❣❤✐➺♠ z1 (t), z2 (t), ..., zn (t) ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✾✮ ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤

tr➯♥ Z+ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✤à♥❤ t❤ù❝ ❑❛③♦r❛t✐ ❝õ❛ ❝❤ó♥❣ ❦❤→❝ ❦❤ỉ♥❣ tr➯♥ Z+ ✳

❈❤ù♥❣ ừ ừ ỵ ữủ ♥❣❛② ❝↔ ✤è✐
✈ỵ✐ ♥❤ú♥❣ ❤➔♠ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ữủ s r tứ
ỵ ự tọ ừ ỵ tr÷í♥❣ ❤đ♣
♥❣❤✐➺♠ ❝õ❛ ❚❚◆✳ ●✐↔ sû z1 (t), z2 (t), ..., zn (t) ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ tr➯♥

Z+ ✈➔ tỗ t t Z+ K(t ) = 0.
♥❤➜t n ♣❤÷ì♥❣ tr➻♥❤ n ➞♥ sè✿
❳➨t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ∗t✉②➳♥ t➼♥❤ t❤✉➛♥
∗
C1 z1 (t ) + C2 z2 (t ) + ... + Cn zn (t∗ ) = 0


C1 z1 (t∗ + 1) + C2 z2 (t∗ + 1) + ... + Cn zn (t∗ + 1) = 0
...


C1 z1 (t∗ + n − 1) + C2 z2 (t∗ + n − 1) + ... + Cn zn (t∗ + n − 1)) = 0.
✭✶✳✸✵✮


✶✼

✣à♥❤ t❤ù❝ ❝õ❛ ❤➺ ♥➔② t❤❡♦ ❣✐↔ t❤✐➳t K (t∗ ) = 0✱ ❞♦ ❞â ✭✶✳✸✵✮ ❝â ♥❣❤✐➺♠
❦❤ỉ♥❣ t➛♠ t❤÷í♥❣✱ ❦➼ ❤✐➺✉ (C1∗ , C2∗ , .., Cn∗ ) ❧➔ ♥❣❤✐➺♠ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣
✤â✳ ❳➨t ❤➔♠✿

z ∗ (t) = C1∗ z1 (t) + C2∗ z2 (t) + ... + Cn∗ zn (t),
✤➙② ❝ơ♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✷✾✮ ✭tê ❤đ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ♥❣❤✐➺♠✮✳ ❚❛ t✐➳♣

tö❝ ❧➔♠ rã ①❡♠ t↕✐ ✤✐➸♠ t∗ t❤➻ ✤✐➲✉ ❦✐➺♥ ✤➛✉ ♥➔♦ s➩ t❤ä❛ ♠➣♥ ❝❤♦ ♥❣❤✐➺♠
✤➣ ❝❤♦✳ ❚ø ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ❤➺ ✭✶✳✸✵✮t❛ ❝â ✤÷đ❝✿

z ∗ (t∗ ) = 0, z ∗ (t∗ + 1) = 0, ..., z ∗ (t∗ + n − 1) = 0.
❚ø t➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② t❛ ♥❤➟♥ ✤÷đ❝ z ∗ (t) = 0, t ∈

Z+ ✳ ◆❤÷ t❤➳✿

C1∗ z1 (t)

n

+

C2∗ z2 (t)

+ ... +

Cn∗ zn (t)

+

Ci2 = 0

= 0, t ∈ Z ,
i=1

✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❤➺ ✤➣ ❝❤♦✦

✣à♥❤ ❧➼ ✶✳✸✳✼✳ ✭◆❡✉♠❛♥♥✮ ✣à♥❤ t❤ù❝ ❑❛③♦r❛t✐ ❝õ❛ ❜➜t ❦ý n ♥❣❤✐➺♠ ❝õ❛

♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✾✮ ✤➲✉ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤✿

K(t + 1) = (−1)n pn (t)K(t).

✭✶✳✸✶✮

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû z1 (t), z2 (t), ..., zn (t) ❧➔ ♥❣❤✐➺♠ ♥➔♦ ✤➜② ❝õ❛ ✭✶✳✷✾✮
✭❦❤æ♥❣
♥❤➜t t❤✐➳t ♣❤↔✐ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✮✳ ❳➨t ❤➺


 p1 (t)z1 (t + n − 1) + p2 (t)z1 (t + n − 2) + ... + pn (t)z1 (t) = −z1 (t + n)

p1 (t)z2 (t + n − 1) + p2 (t)z2 (t + n − 2) + ... + pn (t)z2 (t) = −z2 (t + n)

 .....................................................
p1 (t)zn (t + n − 1) + p2 (t)zn (t + n − 2) + ... + pn (t)zn (t) = −zn (t + n)
✭✶✳✸✷✮
tr♦♥❣ ✤â p1 (t), ..., pn (t) ✤÷đ❝ ❝♦✐ ♥❤÷ ❧➔ ỗ t ợ
s


p1 (t) = ∆1
p2 (t)∆ = ∆2
..........................


pn (t)∆ = ∆n .
❳➨t ♣❤÷ì♥❣ tr➻♥❤✿

pn (t)∆ = ∆n .

✭✶✳✸✸✮

Ð ✤➙② ∆ ❧➔ ✤à♥❤ t❤ù❝ ❤➺ sè ❝õ❛ ❤➺ ✭✶✳✸✷✮✱ ❝á♥ ∆n ♥❤➟♥ ✤÷đ❝ ❦❤✐ t❛ ❜ä ✤✐
❝ët t❤ù n ❝õ❛ ∆ ✈➔ t❤❛② ✈➔♦ ✤â ❧➔ ✭❝ët✮ ✈➳ ♣❤↔✐ ❝õ❛ ✭✶✳✸✷✮✳ ❚❛ ❝❤ù♥❣ tä


✶✽

r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❝✉è✐ ✭✶✳✸✸✮ t❤ü❝ ❝❤➜t ❧➔ ♠ët ❝→❝❤ ✈✐➳t ❦❤→❝ ❝õ❛ ✭✶✳✸✶✮✳
❚❛ ✤✐ ❜✐➸✉ ❞✐➵♥ ∆ ✈➔ ∆n q✉❛ ✤à♥❤ t❤ù❝ ❑❛③♦r❛t✐✳ ✣➸ t❤✉➟♥ t✐➺♥ ❝❤♦ ❝æ♥❣
✈✐➺❝ tr➯♥ t❛ ✤✐ ❤➻♥❤ ❞✉♥❣ ❧↕✐ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ✤à♥❤ t❤ù❝✿
✶✳ ●✐→ trà ❝õ❛ ✤à♥❤ t❤ù❝ ❧➔ ❦❤æ♥❣ ✤ê✐ trữợ (det A =

det AT .
✤ê✐ ❝❤é ❜➜t ❦ý ❤❛✐ ❤➔♥❣ ✭❤♦➦❝ ❤❛✐ ❝ët✮ ❝õ❛ ✤à♥❤ t❤ù❝ t❤➻ ❣✐→ trà
❝õ❛ ✤à♥❤ t❤ù❝ ✤ê✐ ❞➜✉✳
✸✳ ◆➳✉ tr♦♥❣ ♠ët ❤➔♥❣ ✭❤♦➦❝ ❝ët✮ ❝õ❛ ✤à♥❤ t❤ù❝ ❝â ♥❤➙♥ tû ❝❤✉♥❣ l t❤➻
t❛ ❝â t❤➸ ✤÷❛ l r❛ ❦❤ä✐ ❞➜✉ ❝õ❛ ✤à♥❤ t❤ù❝✳
❚❛ ❝â✿

z1 (t + n − 1) z1 (t + n − 2) ... z1 (t + 1) z1 (t)
z2 (t + n − 1) z2 (t + n − 2) ... z2 (t + 1) z2 (t)
∆=
....
zn (t + n − 1) zn (t + n − 2) ... zn (t + 1) zn (t)
z1 (t) z1 (t + n − 1) ... z1 (t + 1)
z (t) z2 (t + n − 1) ... z2 (t + 1)
= (−1)n−1 2

....
zn (t) zn (t + n − 1) ... zn (t + 1)

= (−1)n−1 (−1)n−2

z1 (t) z1 (t + 1) z1 (t + n − 1) ... z1 (t + 2)
z2 (t) z2 (t + 1) z2 (t + n − 1) ... z2 (t + 2)
= ...
...
zn (t) zn (t + 1) zn (t + n − 1) ... zn (t + 2)

= (−1)n−1 (−1)n−2 ...(−1)1

= (−1)

n−1
k=1

z1 (t) z1 (t + 1) z1 (t + 2) ... z1 (t + n − 1)
z2 (t) z2 (t + 1) z2 (t + 2) ... z2 (t + n − 1)
= ...
...
zn (t) zn (t + 1) zn (t + 2) ... zn (t + n − 1)

z1 (t) z2 (t) ... zn (t)
z1 (t + 1) z2 (t + 1) ... zn (t + 1)
k
....
z1 (t + n − 1) z2 (t + n − 1) ... zn (t + n − 1)
= (−1)n (−1)

n(n−1)
2

K(t)

❚÷ì♥❣ tü ♥❤÷ t❤➳ t❛ ❝â ✤÷đ❝✿

z1 (t + n − 1) z1 (t + n − 2) ... z1 (t + 1) − z1 (t + n)
z2 (t + n − 1) z2 (t + n − 2) ... z2 (t + 1) − z2 (t + n)
∆=
....
zn (t + n − 1) zn (t + n − 2) ... zn (t + 1) − zn (t + n)