❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ✣❸■ ❍➴❈ ✣⑨ ◆➂◆● ✖✖✖✖✖ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P P❍➆P ❇■➌◆ ✣✃■ ▲❆P▲❆❈❊ ❱⑨ Ù◆● ❉Ư◆● ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❙✐♥❤ ✈✐➯♥ t❤ü❝ ữợ ▲➯ ❍↔✐ ❚r✉♥❣ ✣➔ ◆➤♥❣✱ ✵✻✴✷✵✶✹ ▼ö❝ ❧ö❝ ▼Ð ✣❺❯ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✶✳✶ ✶✳✷ ✶✳✸ ✸ ✺ P❤÷ì♥❣ ♣❤→♣ ❦❤❛✐ tr✐➸♥ t❤ø❛ sè r✐➯♥❣ ♣❤➛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➼❝❤ ♣❤➙♥ s✉② rë♥❣ ✈➔ sü ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ◆❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✳ ✳ ✷ P❍➆P ❇■➌◆ ✣✃■ ▲❆P▲❆❈❊ ✺ ✻ ✶✸ ✶✺ ✷✳✶ P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ t➼❝❤ ❝❤➟♣ ✷✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ Ù◆● ❉Ö◆● P❍➆P ❇■➌◆ ✣✃■ ▲❆P▲❆❈❊ ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ✷✻ ✸✳✶ Ù♥❣ ❞ư♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✸✳✷ Ù♥❣ ❞ư♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝❤♦ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✸✵ ✸✳✸ ◆❤ú♥❣ ÷✉ ✤✐➸♠ ✈➔ ♥❤÷đ❝ ✤✐➸♠ ❝õ❛ ✈✐➺❝ →♣ ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ tr♦♥❣ ✈✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ❑➌❚ ▲❯❾◆ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✷✻ ✸✸ ✸✹ é ỵ ỹ t➔✐✿ ◆❣➔② ♥❛②✱ ●✐↔✐ t➼❝❤ ❚♦→♥ ❤å❝ ✤➣ ❝â sü ❜✐➳♥ ✤ê✐ ♠↕♥❤ ♠➩✱ tr♦♥❣ ✤â ❧➽♥❤ ✈ü❝ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❦❤ỉ♥❣ ♥❣ø♥❣ ✤÷đ❝ ♣❤→t tr✐➸♥ ✈➻ ♥â ❝â r➜t ♥❤✐➲✉ ù♥❣ ❞ö♥❣ t❤ü❝ t✐➵♥✳ ❱➻ t❤➳✱ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♥❤÷ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r✱ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✉é✐ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱✳✳✳ ❤❛② ù♥❣ ❞ư♥❣ t✐♥ ❤å❝✳ ❚r♦♥❣ sè ✤â✱ ♣❤÷ì♥❣ ♣❤→♣ ✈➟♥ ❞ư♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤÷đ❝ ữỡ õ ỵ ♥❣❤➽❛✳ ❱ỵ✐ ♠♦♥❣ ♠✉è♥ ❝â t❤➸ ❤✐➸✉ ❦➽ ❤ì♥ ✈➲ ❝→❝ ❞↕♥❣ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr ũ ợ sỹ ủ ỵ t t ữợ t r ❡♠ ✤➣ ❝❤å♥ ✤➲ t➔✐ ✏P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➔ ✷✳ ▼ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉✿ ù♥❣ ❞ư♥❣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✑ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣✳ ❚❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ✏P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➔ ù♥❣ ❞ö♥❣ ữỡ tr t ữợ ✤➼❝❤ ❧➔ r➧♥ ❧✉②➺♥ ❦❤↔ ♥➠♥❣ t✐➳♣ ❝➟♥✱ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ♠ët ✈➜♥ ✤➲ ❚♦→♥ ❤å❝ ❝á♥ ❦❤→ ♠ỵ✐ ✤è✐ ✈ỵ✐ ❜↔♥ t❤➙♥✳ ❚ø ✤â✱ ❤➻♥❤ t❤➔♥❤ ❦❤↔ ♥➠♥❣ tr➻♥❤ ❜➔② ♠ët ✈➜♥ ✤➲ t♦→♥ ❤å❝ trø✉ t÷đ♥❣ ♠ët ❝→❝❤ ❧♦❣✐❝ ✈➔ ❝â ❤➺ t❤è♥❣✳ ▲✉➟♥ ✈➠♥ ♥❤➡♠ ự ỳ ợ ữỡ tr õ t ù♥❣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✤➸ ❣✐↔✐ tr➯♥ ❝ì sð tê♥❣ ❤đ♣ ❧↕✐ ❝→❝ ❦❤→✐ ♥✐➺♠✱ ✤à♥❤ ỵ t t ố ữỡ tr➻♥❤ ✈✐ ♣❤➙♥✳ ❚❤ü❝ ❤✐➺♥ ❜➔✐ ❧✉➟♥ ✈➠♥ ♥➔②✱ t→❝ ❣✐↔ ♠✉è♥ ❝õ♥❣ ❝è ✈➔ ❤➺ t❤è♥❣ ❧↕✐ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ✈➲ t➼❝❤ ♣❤➙♥ s✉② rë♥❣✱ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈➔ ❧➔♠ q✉❡♥ ✈ỵ✐ ❝→❝❤ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ♠ët ✈➜♥ ✤➲ ❝õ❛ ✸✳ ❇è ❝ö❝ ❝õ❛ ❧✉➟♥ ✈➠♥✿ t♦→♥ ❤å❝✳ ◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➔ ❦➳t ❧✉➟♥✱ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ✸ ❧➔♠ ❤❛✐ ♣❤➛♥ P❤➛♥ ✶✿ P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ P❤➛♥ ♥➔② s➩ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡✱ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➔ ♥❤ú♥❣ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣ ✤➸ ❧➔♠ ❝ì sð ❝❤♦ ♣❤➛♥ s❛✉ ❧➔ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✳ P❤➛♥ ✷✿ ⑩♣ ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ P❤➛♥ ♥➔② tr➻♥❤ ❜➔② ❝→❝❤ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ rỗ ❞ị♥❣ ♥❤✐➲✉ ✈➼ ❞ư ✤➸ ♠✐♥❤ ❤å❛ rã ❤ì♥ ✈➲ ❝→❝❤ ❣✐↔✐ ♥➔②✳ ✹ ❈❤÷ì♥❣ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝→❝ ❦✐➳♥ t❤ù❝ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✱ ❬✸❪✱ ❬✹❪✳ ✶✳✶ P❤÷ì♥❣ ♣❤→♣ ❦❤❛✐ tr✐➸♥ t❤ø❛ sè r✐➯♥❣ ♣❤➛♥ ❈❤♦ ♣❤➙♥ t❤ù❝ x ❜➟❝ m ✈➔ n t÷ì♥❣ ù♥❣✱ ❑❤❛✐ tr✐➸♥ u(x) ✱ tr♦♥❣ ✤â u(x) ✈➔ v(x) v(x) ✈ỵ✐ m < n✱ t❛ ❧➔♠ ♥❤÷ s❛✉✿ f (x) = v(x) ❧➔ ❝→❝ ✤❛ t❤ù❝ ❝õ❛ t❤➔♥❤ ❝→❝ t❤ø❛ sè ✤ì♥ ❣✐↔♥ ❝â ❞↕♥❣ v(x) = (x − x1 )k1 (x − x2 )k2 (x − xr )kr tr♦♥❣ ✤â k1 + k2 + + kr = n ◆❤÷ ✈➟②✱ ❝â t❤➸ ❦❤❛✐ tr✐➸♥ ❤➔♠ ❝➜♣ ❝â ❞↕♥❣✿ Aij ✱ (x − xi )ki −j+1 f (x) = tr♦♥❣ ✤â i u(x) v(x) t❤➔♥❤ tê♥❣ ❝→❝ ♣❤➙♥ sè ❧➜② t➜t ❝↔ ❝→❝ ❣✐→ trà tø ✤➳♥ r✱ ✤➳♥ ki ✳ Aij ❚❛ ❝â F (x) = ✱ t➜t ❝↔ ❤➺ sè Aij ❝õ❛ ❦❤❛✐ tr✐➸♥ ki −j+1 i=1 j=1 (x − xi ) dj−1 ✤÷đ❝ t➻♠ t❤❡♦ ❝ỉ♥❣ t❤ù❝ Aij = lim { j−1 [(x − x1 )ki F (x)]} (j − 1)! x→x1 dx ❝á♥ ❥ ❧➜② t➜t ❝↔ ❝→❝ ❣✐→ trà sè tø r ♥➔② ki ❚❤❛② ❝❤♦ ❝ỉ♥❣ t❤ù❝ ♥➔② ❝â t❤➸ ❞ị♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❝➜♣ tr♦♥❣ ♣❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ❦❤✐ t➼♥❤ t➼❝❤ ♣❤➙♥ ❝→❝ ♣❤➙♥ sè ❤ú✉ t✛✳ ✣➦❝ ❜✐➺t ✤✐➲✉ ♥➔② r➜t t❤✉➟♥ ❧đ✐ tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ♣❤ù❝ ❝õ❛ ♠➝✉ sè v(x) ✤ì♥ ✈➔ ✤ỉ✐ ♠ët ❧✐➯♥ ❤ñ♣✳ ✺ v(x) ◆➳✉ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ✤ì♥ t❤➻ ❦❤❛✐ tr✐➸♥ ❝â ❞↕♥❣ ✤ì♥ ❣✐↔♥✿ n v(x) = (x − x1 )(x − x2 ) (x − xn )xj = xk j = k ✱ F (x) = tr♦♥❣ ✤â Aj = ❱➼ ❞ö ✶✳✶✳ Aj ✱ j=1 x − xj u(xj ) ✳ v (xj ) P❤➙♥ t➼❝❤ ♣❤➙♥ t❤ù❝ F (x) = − (x + 2)(4x + 10) (x + 1)(x + 2)2 t❤➔♥❤ tê♥❣ ❝→❝ ♣❤➙♥ t❤ù❝ ✤ì♥ ❣✐↔♥✳ ●✐↔✐✿ F (x) t❤➔♥❤✿ A B C F (x) = + + x + x + (x + 2)2 A(x + 2)2 + B(x + 1)(x + 2) + C(x + 1) = (x + 1)(x + 2) ❚❛ ❜✐➳♥ ✤ê✐ ❚❤❡♦ ❜✐➸✉ t❤ù❝ ✤➣ ❝❤♦ t❛ ❝â ✿ A(x + 2)2 + B(x + 1)(x + 2) + C(x + 1) = − (x + 2)(4x + 10) ❈❤♦ ❈❤♦ ❈❤♦ x = −2 t❛ ❝â C = −8 x = −1 t❛ ❝â A = x = t❛ ❝â B = −6 ❉♦ ✤â✿ F (x) = − − x + x + (x + 2)2 ✶✳✷ ❚➼❝❤ ♣❤➙♥ s✉② rë♥❣ ✈➔ sü ❤ë✐ tö ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ✭❚➼❝❤ ♣❤➙♥ s✉② rë♥❣✮ ●✐↔ sû tr➯♥ ❦❤♦↔♥❣ [a, +∞) ✈➔ ❦❤↔ t➼❝❤ tr➯♥ ♠å✐ ✤♦↕♥ f ❧➔ ♠ët ❤➔♠ sè ①→❝ [a, b] ✈ỵ✐ b > a ◆➳✉ ✤à♥❤ b lim f (x)dx = I, b→+∞ a tr♦♥❣ ✤â rë♥❣ ❝õ❛ I ∈ R✱ I = +∞ ❤♦➦❝ I = −∞ t❤➻ I ✤÷đ❝ f tr➯♥ ❦❤♦↔♥❣ [a, +∞) ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ +∞ f (x)dx a ✻ ❣å✐ ❧➔ t➼❝❤ ♣❤➙♥ s✉② +∞ f (x)dx ❑❤✐ ✤â t❛ õ r tỗ t I ỳ tự ❧➔ I ∈R t❤➻ a +∞ f (x)dx t❛ ♥â✐ r➡♥❣ ❧➔ ❤ë✐ tư✳ ❚➼❝❤ ♣❤➙♥ ❦❤ỉ♥❣ ❤ë✐ tư ❣å✐ ❧➔ ♣❤➙♥ ❦ý✳ a ❱➼ ❞ö ✶✳✷✳ +∞ e−x dx ❚➼♥❤ ❱ỵ✐ ♠å✐ sè t❤ü❝ ❜ ❃ ✵✱ t❛ ❝â✿ b e−x dx = (−e−x ) |b0 = − e−b b e−x dx = lim (1 − e−b ) = lim b→+∞ b→+∞ +∞ e−x dx ❉♦ ✤â ❤ë✐ tö ✈➔ +∞ e−x dx = ❱➼ ❞ö ✶✳✸✳ +∞ b dx = lim + x2 b→+∞ ❱➼ ❞ö ✶✳✹✳ dx π = lim arctan b = + x2 b→+∞ +∞ b dx = lim b = +∞ dx = lim b→+∞ b→+∞ +∞ dx õ t s rở tỗ t ữ ỵ + f (x)dx ❝→❝ t➼❝❤ ♣❤➙♥ s✉② rë♥❣ a ✼ +∞ g(x)dx ✈➔ a ❤ë✐ +∞ [f (x) + g(x)]dx tö t❤➻ t➼❝❤ ♣❤➙♥ s✉② rë♥❣ ❝ơ♥❣ ❤ë✐ tư ✈➔ a +∞ +∞ [f (x) + g(x)]dx = a +∞ f (x)dx + g(x)dx a a +∞ f (x)dx ❜✮ ◆➳✉ t➼❝❤ ♣❤➙♥ λ ❤ë✐ tö ✈➔ ❧➔ ♠ët ❤➡♥❣ sè t❤ü❝ a + + f (x)dx t t ỵ λf (x)dx = λ ❤ë✐ tö ✈➔ a +∞ a f (x)dx a f ❧➔ ♠ët ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥ [a, b) ✈ỵ✐ b > a✳ ◆➳✉ f (x) ≥ ✈ỵ✐ ●✐↔ sû ❦❤↔ t➼❝❤ tr➯♥ ♠å✐ ✤♦↕♥ [a, +∞)✱ x ∈ [a, +∞) ❦❤♦↔♥❣ ♠å✐ +∞ f (x)dx t t ổ ổ tỗ t ỳ ❤♦➦❝ ❜➡♥❣ +∞✮✳ a b ❈❤ù♥❣ ♠✐♥❤✿ ✣➦t f (x)dx✱ b ≥ a✳ F (b) = b ≥b ◆➳✉ t❤➻✿ a b b f (x)dx = F (b ) = a t tr b lim [a, +) tỗ t f (x)dx tự tỗ t a a + b f (x)dx = a sup F (b) = b∈[a,+∞) sup f (x)dx b∈[a,+∞) a +∞ f (x)dx ❍✐➸♥ ♥❤✐➯♥ b ❤ë✐ tö ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ b b→ sup b≥a a ❤➔♠ sè lim F (b) = b→+∞ +∞ f (x)dx b→+∞ b b b → F (b) f (x)dx ≥ F (b) f (x)dx = F (b) + f (x)dx + a ❍➔♠ sè b b f (x)dx ❜à ❝❤➦♥ tr➯♥ [a, +∞)✳ a ✽ f (x)dx < + a tự ỵ s s→♥❤✮ ●✐↔ sû f ✈➔ [a, +∞) ✈➔ ❦❤↔ t➼❝❤ tr➯♥ ♠å✐ ✤♦↕♥ ≤ f (x) ≤ g(x) ✈ỵ✐ ♠å✐ x ∈ [a, +∞) t❤➻ tr➯♥ ❦❤♦↔♥❣ ◆➳✉✿ +∞ g ❧➔ ❤❛✐ ❤➔♠ sè [a, b] ✈ỵ✐ b > a✳ ①→❝ ✤à♥❤ +∞ f (x)dx ≤ a g(x)dx a ❚ø ✤â s✉② r❛✿ +∞ +∞ g(x)dx ◆➳✉ a +∞ ❤ë✐ tö✳ a +∞ f (x)dx ◆➳✉ f (x)dx ❤ë✐ tö t❤➻ ❍➺ q✉↔ ✶✳✶✳ g(x)dx ♣❤➙♥ ❦➻ t❤➻ a ♣❤➙♥ ❦➻✳ a ●✐↔ sû f g ❧➔ ♥❤ú♥❣ ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥ ❦❤♦↔♥❣ [a, +∞) ✤♦↕♥ [a, b] ✈ỵ✐ ❜ ❃ ❛✳ ◆➳✉ f (x) ≥ 0✱ g(x) ≥ tr➯♥ ✈➔ ❦❤↔ t➼❝❤ tr➯♥ ♠å✐ ✈➔ +∞ [a, +∞) f ∼g ✈➔ ❦❤✐ x → +∞ +∞ f (x)dx t❤➻ ❝→❝ t➼❝❤ ♣❤➙♥ g(x)dx ✈➔ a a ❝ị♥❣ ❤ë✐ tư ❤♦➦❝ ❝ị♥❣ ♣❤➙♥ ❦➻✳ ❱➼ ❞ư ✶✳✺✳ +∞ e−x dx ❳➨t t➼♥❤ ❤ë✐ tö ❝õ❛ t➼❝❤ ♣❤➙♥ +∞ ❚❛ ❝â✿ < e−x ≤ e−x ✈ỵ✐ ♠å✐ x ≥ 1✳ e−x dx ❚❛ ❜✐➳t r➡♥❣ ❤ë✐ tö✳ +∞ e−x dx ❉♦ ✤â tử ú ỵ ỵ ❤➺ q✉↔ ✶✳✶ ❝❤➾ ✤÷đ❝ →♣ ❞ư♥❣ ❝❤♦ tr÷í♥❣ ❤đ♣ số ữợ t ổ ợ tr ợ ỵ ❈❛✉❝❤② ✈➲ sü ❤ë✐ tö ❝õ❛ t➼❝❤ ♣❤➙♥✮ ♠ët ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥ [a, +∞) ✈➔ ❦❤↔ t➼❝❤ tr➯♥ ♠å✐ ✤♦↕♥ f ❧➔ [a, b]✱ b > a✳ ●✐↔ sû +∞ f (x)dx ❑❤✐ ✤â t➼❝❤ ♣❤➙♥ ❤ë✐ tö ợ ởt số ữỡ a t tỗ t ởt số tỹ b0 a s❛♦ ❝❤♦ b2 (∀b1 , b2 ∈ R) b2 ≥ b1 ≥ b0 ⇒ | f (x)dx| < ε b1 ❈❤ù♥❣ ♠✐♥❤✿ F ●å✐ ❧➔ ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥ [a, +∞) ❜ð✐ b b → F (b) = f (x)dx a t sỹ tỗ t↕✐ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❤➔♠ sè✱ ❦❤✐ ❝❤➾ ❦❤✐ ∀ε > 0, ∃b0 ≥ a ∃ lim F (b) b→+∞ s❛♦ ❝❤♦✿ ∀b1 , b2 ∈ R : b2 ≥ b1 ≥ b0 ⇒ |F (b2 ) − F (b1 )| < ε; b2 F (b2 ) − F (b1 ) = ♠➔ f (x)dx ♥➯♥ tø ✤â s✉② r❛ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ✭❚➼❝❤ ♣❤➙♥ ❤ë✐ tö t✉②➺t ✤è✐✮ b1 +∞ ❚❛ ♥â✐ r➡♥❣ t➼❝❤ ♣❤➙♥ +∞ f (x)dx |f (x)|dx ❤ë✐ tö t✉②➺t ✤è✐ ♥➳✉ ỵ a tử a tö t✉②➺t ✤è✐ t❤➻ ❤ë✐ tö✳ +∞ ❈❤ù♥❣ ♠✐♥❤✿ f (x)dx ●✐↔ sû t➼❝❤ ♣❤➙♥ ❤ë✐ tö t✉②➺t ✤è✐✱ tù❝ ❧➔ t➼❝❤ a +∞ |f (x)|dx ♣❤➙♥ ❤ë✐ tö✳ ❈❤♦ >0 tũ ỵ t t sỹ a tử t tỗ t b0 a s❛♦ ❝❤♦✿ b2 b2 ≥ b1 ≥ b0 ⇒ | |f (x)|dx| < ε b1 b2 ❉♦ ✤â | b2 f (x)dx| ≤ | b1 |f (x)|dx| < ε ✈ỵ✐ b2 ≥ b1 ≥ b0 b1 +∞ f (x)dx ❱➟② t➼❝❤ ♣❤➙♥ ❤ë✐ tö ✭t❤❡♦ t✐➯✉ ❝❤✉➞♥ ❈❛✉❝❤②✮✳ a ỵ ỵ tr f (0+) = lim sF (s) s→∞ ❈❤ù♥❣ ♠✐♥❤✿ ❚ø ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝õ❛ ✤↕♦ ❤➔♠ ✭❚➼♥❤ ❝❤➜t ✺✮✿ ▲➜② ❣✐ỵ✐ ❤↕♥ ❦❤✐ s → +∞ lim s→+∞ ♠➔ lim s→+∞ ▲{f (t)} = ▲{f (t)} = ▲{f (t)} = sF (s)−f (0+) lim [sF (s) − f (0+)] s→+∞ +∞ f (t)e−st dt = lim s→+∞ ❱➟② ▼➔ lim [sF (s) − f (0+)] = s→+∞ f (0+) ❧➔ ❤➡♥❣ sè ♥➯♥ f (0+) = lim sF (s) ỵ ỵ tr ố s F (+∞) = lim(sF (s)) s→0 ❈❤ù♥❣ ♠✐♥❤✿ ❚ø ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝õ❛ ✤↕♦ ❤➔♠ ✭❚➼♥❤ ❝❤➜t ✺✮✿ ▲➜② ❣✐ỵ✐ ❤↕♥ ❦❤✐ s→0 ▲ +∞ f (t)e−st dt = lim[sF (s) − f (0+)] lim {f (t)} = lim s→0 ▲{f (t)} = sF (s)−f (0+) s→0 s→0 +∞ ♠➔ +∞ f (t)e−st dt = lim lim s→0 df (t) = f (+∞) − f (0+) s→0 0 f (+∞) − f (0+) = lim[sF (s) − f (0+)] s→0 ❍❛② F (+∞) = lim(sF (s)) ❱➟② s→0 ✷✳✷ P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳ ▲{f (t)} = F (s) ⇒ f (t) = ▲ P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ −1 ✷✶ {F (s)} ✭✷✳✷✮ ✷✳ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❧❛♣❧❛❝❡ ♥❣÷đ❝ ❚➼♥❤ ❝❤➜t ✷✳✶✵✳ ▲ {c F (s) + c F (s)} = c f (t) + c f (t) ❚➼♥❤ ❝❤➜t ✷✳✶✶✳ ▲ {F (s + a)} = e f (t) = e ▲ {F (s)} ❚➼♥❤ ❝❤➜t ✷✳✶✷✳ ▲ {F (s)} = e ▲ {F (s − a)} ❚➼♥❤ ❝❤➜t ✷✳✶✸✳ ▲ {F (s − a)} = e ▲ {F (s)} ❱➼ ❞ö ✷✳✽✳ −1 1 2 −1 1 −at −1 −1 −1 −at −1 −at 2 at −1 ❳→❝ ✤à♥❤ ♣❤➨♣ ❜✐➸♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝ ❝õ❛ ❤➔♠ F (s) = s2 − 2s + ●✐↔✐✿ 1 = , F (s + 1) = 2 − 2s + (s − 1) + 22 s + 22 1 ⇒ L−1 {F (s + 1)} = sin 2t, L−1 {F (s)} = et sin 2t 2 F (s) = s2 ❚➼♥❤ ❝❤➜t ✷✳✶✹✳ ▲ {e F (s)} = f (t − a)u(t − a) ❚➼♥❤ ❝❤➜t ✷✳✶✺✳ ▲ {F (as)} = a1 f ( a1 ), a >  ▲ {F (s)} = (−1) t ▲ {F (s)} ❚➼♥❤ ❝❤➜t ✷✳✶✻✳  ▲ {F (s)} = −1 ▲ {F (s)} t s+1 ❱➼ ❞ö ✷✳✾✳ F (s) = ln s−1 −1 −as −1 −1 (n) n n n −1 −1 −1 (n) n ❚➻♠ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝ ❝õ❛ ❤➔♠ ●✐↔✐✿ ❚❛ ❝â✿ 1 − ⇒ L−1 {F (s)} =e−t − et = −2sht s+1 s− n (−1) −1 −1 2sht −1 (n) −1 L−1 {F (s)} = L {F (s)} ⇒ L {F (s)} = L {F (s)} = tn t t F (s) = ❚➼♥❤ ❝❤➜t ✷✳✶✼✳ ▲ +∞ −1 { F (u)du} = t {F (s)} s ú ỵ r ❜✐➳♥ ✤ê✐ ♥❣÷đ❝✱ ❝→❝ ♠➝✉ sè ❝➛♥ ✤÷đ❝ ❝❤✉②➸♥ ✈➲ ❞↕♥❣ ♥❤÷ ❜↔♥❣ s❛✉✿ ✷✷ ❉↕♥❣ ❝õ❛ ♠➝✉ sè ❈❤✉②➸♥ ✈➲ ❞↕♥❣ ❝õ❛ ♣❤➙♥ t❤ù❝ A ax + b A1 A2 Ak + + + ax + b (ax + b) (ax + b)k A2 x + B2 Ak x + Bk A1 x + B1 + + + ax2 + bx + c (ax2 + bx + c)2 (ax2 + bx + c)k ax + b (ax + b)k (ax2 + bx + c)k ✷✳✸ ✣à♥❤ ỵ t P ♥❤➙♥ ❝❤➟♣ ✭t➼❝❤ ❝❤➟♣✮ ❝õ❛ ❤❛✐ ❤➔♠ ❤✐➺✉ ❣✐ú❛ ❝❤ó♥❣ ❜ð✐ ❞➜✉ ✯ ❧➔ ❤➔♠ f ∗g f (t) ✈➔ g(t) tũ ỵ ữủ ỵ s t [f g](t) = f (t − τ )g(τ )dτ ; ❤♦➦❝ ❜✐➸✉ ❞✐➵♥ q✉❛ ♠ët t➼❝❤ ♣❤➙♥ ✈æ ❤↕♥ ✤è✐ ✈ỵ✐ ❤❛✐ ❤➔♠ +∞ f ∗g = f✱ g ✈➔ h g tũ ỵ + f ( )g(t − τ )dτ = −∞ ◆➳✉ f g(τ )f (t )d tũ ỵ a ❧➔ ❤➡♥❣ sè✱ t➼❝❤ ❝❤➟♣ ❝õ❛ ❤❛✐ ❤➔♠ sè ❝â t➼♥❤ ❝❤➜t✿ f ∗ g = g ∗ f ; f ∗ (g ∗ h) = (f ∗ g) ∗ h; f ∗ (g + h) = (f ∗ g) + (f ∗ h); a(f ∗ g) = (af ) ∗ g = f ∗ (ag) ▲➜② ✤↕♦ ❤➔♠ t➼❝❤ ❝❤➟♣✱ t❛ ❝â✿ (f ∗ g) = f ∗ g = f ∗ g ✱ ❤♦➦❝ ❧➜② ♣❤➙♥ t❛ ❝â✿ +∞ +∞ +∞ (f ∗ g)dt = −∞ [ f (u)g(t − u)du]dt −∞ −∞ +∞ = +∞ f (u)[ −∞ +∞ =[ g(t − u)dt]du −∞ +∞ f (u)du][ −∞ −∞ ✷✸ g(t)dt], t➼❝❤ tr♦♥❣ ✤â t➼❝❤ ♣❤➙♥ ♥➔② ❦❤æ♥❣ t❤❛② ✤ê✐ ❣✐→ trà ❦❤✐ ✤ê✐ ❝❤é ❝→❝ ❤➔♠ ✈➔ g(t)✱ f (t) ❞♦ ✤â t➼❝❤ ❝❤➟♣ ❝õ❛ ❤❛✐ ❤➔♠ ✤è✐ ①ù♥❣ ♥❤❛✉ ✤è✐ ✈ỵ✐ ❝→❝ ❤➔♠ ♥❤➙♥ ❝❤➟♣✳ ❱➼ ❞ö ✷✳✶✵✳ ❚➼❝❤ ❝❤➟♣ ❝õ❛ cos t sin t ✈➔ ❧➔✿ t (cos t) ∗ (sin t) = cos τ sin(t − τ )dτ ⑩♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❧÷đ♥❣ ❣✐→❝ cos A sin B = [sin(A + B) − sin(A − B)] ❚❛ ❝â✿ t [sin t − sin(2τ − t)] (cos t) ∗ (sin t) = 1 = [τ sin t + cos(2τ − t)]tτ =0 , 2 ❞♦ ✤â✿ ỵ (cos t) (sin t) = t sin t f (t) ✈➔ g(t) ❧➔ ♥❤ú♥❣ ❤➔♠ ❧✐➯♥ ct tư❝ ✈ỵ✐ t ≥ ✈➔ |f (t)|✱ |g(t)| ✤➲✉ ❜à ❝❤➦♥ ❜ð✐ M e ✈ỵ✐ t → ∞✳ ❑❤✐ ✤â ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ t➼❝❤ f g tỗ t s > c ❤ì♥ ♥ú❛✿ ✭❳❡♠ t➔✐ ❧✐➺✉ ❬✼❪✮ ●✐↔ sû ▲{f (t) ∗ g(t)} = ▲{f (t)}▲{g(t)} = F (s).G(s) ▲ ✈➔ −1 {F (s).G(s)} = f (t) ∗ g(t) ◆❤÷ ✈➟② ❞ị♥❣ t➼❝❤ ❝❤➟♣ ❝❤ó♥❣ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝õ❛ F (s).G(s) ♥❤÷ s❛✉✿ ▲ t −1 {F (s).G(s)} = f (τ )g(t − τ )dτ ✷✹ ❱➼ ❞ö ✷✳✶✶✳ f (t) = sin 2t g(t) = e ▲ { (s − 1)(s2 + 4) } = sin(2t) ∗ e = e ❱ỵ✐ t ✈➔ ✱ t❛ ❝â✿ t −1 t t−τ sin 2τ dτ t e−τ sin 2τ dτ = et [ = et ❱➟②✿ ▲ e−τ (− sin 2τ − cos 2τ )]t0 −1 { t 2 } = e − sin 2t − cos 2t (s − 1)(s2 + 4) 5 ✷✺ ữỡ ệ PP P Pì ❚❘➐◆❍ ❱■ P❍❹◆ ✸✳✶ Ù♥❣ ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ✈ỵ✐ ❤➺ sè ❤➡♥❣ sè✿ ay (t) + by (t) + cy(t) = f (t), tr♦♥❣ ✤â a✱ b ✱ c ✭✸✳✶✮ ❧➔ ❝→❝ ❤➡♥❣ sè✱ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉✿ y(0) = y0 , y (0) = y0 , ð ✤➙② y0 ✈➔ y0 ❧➔ ❝→❝ ❤➡♥❣ sè ①→❝ ✤à♥❤ ❣✐→ trà t=0 trữợ ỹ ✤ê✐ ▲❛♣❧❛❝❡ ❤❛✐ ✈➳ ❝õ❛ ✭✸✳✶✮ t❛ ❝â✿ aL{y (t)} + bL{y (t)} + cL{y(t)} = L{f (t)} ⇒ a[s2 Y (s) − sy(0) − y (0)] + b[sY (s) − y(0)] + cY (s) = F (s) ⇒ (as2 + bs + c)Y (s) − (as + b)y0 − ay = F (s) F (s) + (as + b)y0 + ay ⇒ Y (s) = as2 + bs + c ◆❤÷ ✈➟②✱ ♥➳✉ ❜✐➳t f (t)✱ t❛ s✉② r❛ F (s)✳ ❉♦ ✤â ❤♦➔♥ t♦➔♥ ①→❝ ✤à♥❤ ữủ Y (s) r ữỡ tr ợ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ✭✸✳✷✮ ❧➔✿ ▲ {Y (s)} ◆❤÷ ữỡ tr t t ợ ❤➺ y(t) = −1 ✷✻ sè ❤➡♥❣ sè ❜➡♥❣ ♣❤➨♣ ữủ tỹ t ữợ s ữợ ũ t ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➸ ❜✐➳♥ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜✐➳♥ t t❤➔♥❤ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t s ữợ ữỡ tr số tr t s ữợ ũ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝ ✤➸ t➻♠ y(t) = −1{Y (s)} ▲ ❚r♦♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿ ●✐↔ sû f, f , , f (n−1) ❧➔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ ✈➔ ▲{f (t)} = s F (s) − s f (0) − − s ▲{f (t)} = s F (s) − sf (0) − sf (0); ▲{f (t)} = sF (s) − f (0) f (n) ❧➔ ❤➔♠ ❧✐➯♥ tư❝ tø♥❣ ❦❤ó❝✱ t❤➻✿ (n) n n−1 n−2 f (0) − f (n−1) (0); ✣✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ t↕✐ t = 0✱ y(0) ✈➔ y (0) ❝â t❤➸ t❤❛② ✈➔♦ tr♦♥❣ ❝→❝ ❜✐➸✉ t❤ù❝ tr➯♥✳ ❱➼ ❞ư ✸✳✶✳ ●✐↔✐✿ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ y − 10y + 9y = 5t, y(0) = −1, y (0) = ❚❤ü❝ ❤✐➺♥ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❤❛✐ ✈➳ t❛ ❝â✿ ▲{y − 10y + 9y} = ▲{5t} ⇔s2 Y (s) − sy(0) − y (0) − 10[sY (s) − y(0)] + 9Y (s) = s2 ❚❤❛② ✤✐➲✉ ❦✐➺♥ ✤➛✉ ②✭✵✮ ❂ ✲✶✱ ②✬✭✵✮ ❂ ✷ t❛ ✤÷đ❝✿ s2 Y (s) + s − − 10[sY (s) + 1] + 9Y (s) = s2 + 12s2 − s3 ⇒ Y (s) = s (s − 9)(s − 1) A B C D ⇒ Y (s) = + + + s s s−9 s−1 As(s − 9)(s − 1) + B(s − 9)(s − 1) + Cs2 (s − 1) + Ds2 (s − 9) ⇒ Y (s) = s2 (s − 9)(s − 1) ⇒ 5+12s2 −s3 = As(s−9)(s−1)+B(s−9)(s−1)+Cs2 (s−1)+Ds2 (s−9) 31 ❑❤✐ s = ⇒ B = ✱ ❦❤✐ s = ⇒ D = −2✱ ❦❤✐ s = ⇒ C = ✱ ❦❤✐ 81 ✷✼ 50 81 s=2⇒A= ⇒ Y (s) = 50/81 5/9 31/81 + + − s s s−9 s−1 ❚→❝ ✤ë♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝ ✈➔♦ ❝↔ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ▲ tr➯♥✿ ❱➼ ❞ư ✸✳✷✳ −1 ▲ 50/81 5/9 31/81 + + − } s s s−9 s−1 50 31 ⇒ y(t) = + t + e9t − 2et 81 81 {Y (s)} = −1 { ❉ị♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✿ 2y + 3y + y = 8e−2t ✈ỵ✐ y(0) = −4✱ y (0) = ●✐↔✐✿ ⑩♣ ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➔♦ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❛ ✤÷đ❝✿ 2s2 Y (s) − 2sy(0) − 2y (0) + 3sY (s) − 3y(0) + Y (s) = ❚❤❛② ❝→❝ ❣✐→ trà ✤➛✉ y(0) = −4✱ y (0) = s+2 ✈➔♦ t❛ ✤÷đ❝✿ − 8s + − 12 s+2 4[1 − (s + 2)(s + 1)] ⇒Y (s) = (s + 21 )(s + 1)(s + 2) (2s2 + 3s + 1)Y (s) = ⑩♣ ❞ö♥❣ ❝→❝ ♣❤➙♥ t➼❝❤ r✐➯♥❣ ♣❤➛♥ t❛ ❝â✿ Y (s) = ❉♦ ✤â t❛ ❝â✿ ❱➼ ❞ư ✸✳✸✳ ●✐↔✐✿ ✤÷đ❝✿ y= ●✐↔✐ 3s + − 8 + s + 3s + ▲ {Y (s)} = e−t/2 − 8e−t + e−2t 3 t ♣❤÷ì♥❣ tr➻♥❤✿ y + 4y = 2e ✱ ✈ỵ✐ y0 = 0, y0 = −1 ❚→❝ ✤ë♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➔♦ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ t➻♥❤ tr➯♥ t❛ +1 s−1 ⇒Y (s) = + (s − 1)(s2 + 4) s2 + (s2 + 4)Y (s) = ✷✽ ▲↕✐ t→❝ ✤ë♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝ t❛ ❝â✿ ▼➔ t❛ ❝â✿ ▲ ▲ ▲ −1 } + { } (s − 1)(s2 + 4) s2 + t 2 −1 } = e − sin 2t − cos 2t ✭❚❤❡♦ ✈➼ ❞ö ✷✳✽✱ { (s − 1)(s2 + 4) 5 y= −1 { ỵ t 1 } = sin 2t s2 + 2 t r❛✿ y = e + sin 2t − cos 2t 10 t ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✿ y − 3y + 3y − y = t e −1 ❉♦ ✤â s✉② { ❱➼ ❞ö ✸✳✹✳ ●✐↔✐✿ ❚→❝ ✤ë♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➔♦ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ t➻♥❤ tr➯♥ t❛ ✤÷đ❝✿ s3 F (s) − s2 f (0) − sf (0) − f (0) − 3[s2 F (s) − sf (0) − f (0)] 2! + 3[sF (0) − f (0)] − F (s) = (s − 1)3 s − 3s + 1 + f (0) + f (0) + f (0) ⇒ F (s) = (s − 1)6 (s − 1)3 (s − 1)2 (s − 1)3 C1 ⇒ f (t) = t5 et + t2 et + C2 tet + C3 et 60 ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✿ ty − ty + y = 2✱ y(0) = 2✱ y (0) = −4 ❱➼ ❞ö ✸✳✺✳ ●✐↔✐✿ ⑩♣ ❞ö♥❣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❤❛✐ ✈➳ t❛ ❝â✿ s ⇔ (s − s2 )Y (s) + (2 − 2s)Y (s) + = s 2(1 − s) ⇔ s(s − 1)Y (s) + 2(1 − s)Y (s) = s 2 ⇔ Y (s) + Y (s) = s s −s2 Y (s) − 2sY (s) + y(0) − [−sY (s) − Y (s)] + Y (s) = P❤÷ì♥❣ tr➻♥❤ tr➯♥ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✶ ♥➯♥ t❛ s✉② r❛ ♥❣❤✐➺♠✿ Y (s) = e− ⇒ s ds ( e s2 s ds ds + C) = y(t) = + Ct ⇒ y(t) = − 4t y (0) = −4 ✷✾ 1 C (2s + C) = + 2 s s s ❱➼ ❞ö ✸✳✻✳ ●✐↔✐✿ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✿ y + 3ty − 6y = 2, y(0) = 0, y (0) = ⑩♣ ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➔♦ ❝↔ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ t❛ ✤÷đ❝✿ s2 Y (s) − sy(0) − y (0) + 3[−sY (s) − Y (s)] − 6Y (s) = ⇔ − 3sY (s) + (s2 − 9)Y (s) = s ⇔Y (s) + ( − )Y (s) = s 3s s − ( − )ds s [ ⇒ Y (s) = e s s s −2 ( − )ds e s ds + C] 3s2 −s2 e = +c s s ❚❛ ❝â✿ lim s→+∞ s3 + c −s2 = ⇒ c = ✭❚❤❡♦ ❝æ♥❣ t❤ù❝ ✹✳✻✼ tr♦♥❣ t➔✐ ❧✐➺✉ e s3 ❬✶❪✮ ❉♦ ✤â✿ Y (s) = ⇒ y(t) = t2 s ✸✳✷ Ù♥❣ ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝❤♦ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❚r♦♥❣ ♠ö❝ ♥➔② t❛ s➩ →♣ ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✤➸ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ✤÷đ❝ ♥➯✉ ❧➯♥ ð ♠ư❝ tr➯♥ ❝❤õ ②➳✉ ❜➡♥❣ ợ ữỡ tr➻♥❤✿ x1 = 3x1 − 3x2 + x2 = −6x1 − t x1 (0) = 1, x2 (0) = −1 ●✐↔✐✿ t❛ ❝â✿ ⑩♣ ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝❤♦ ❤❛✐ ✈➳ ❝õ❛ ❝↔ ❤❛✐ ♣❤÷ì♥❣ tr➻♥❤    sX1 (s) − x1 (0) = 3X1 (s) − 3X2 (s) + s   sX2 (s) − x2 (0) = −6X1 (s) − s2 ✸✵ ❚❤❛② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ✈➔ rót  ❣å♥ t❛ ✤÷đ❝✿ 2+s   (s − 3)X1 (s) + 3X2 (s) = 3s2 + s ⇒ (s − 3s − 18)X (s) = + s + s2 +  s2  6X1 (s) + sX2 (s) = s2 133 28 18 s + 5s + = ( − + − ) ⇒ X1 (s) = s (s + 3)(s − 6) 108 s − s + s s ❚❤ü❝ ❤✐➺♥ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ♥❣÷đ❝✱ t❛ ❝â✿ x1 (t) = (133e6t −28e−3t +3−18t) 108 ✣➸ t➻♠ ♥❣❤✐➺♠ t❤ù ❤❛✐✱ t❛ ❝â✿ x( t) = −6x1 − t ⇒ x2 = (−6x1 − t)dt = − ⇒ x2 (t) = − ❚❤❛② ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ 18 (133e6t − 28e−3t + 3) (133e6t − 56e−3t + 18t) + C 108 x2 (0) = −1✱ t❛ ✤÷đ❝✿ C= ❱➟② ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ♥❣❤✐➺♠ ❧➔✿ ❱➼ ❞ư ✸✳✽✳ ✈ỵ✐    x1 (t) = (133e6t − 28e−3t + − 18 108   x2 (t) = − (133e6t − 56e−3t + 18t) + 108 ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✿ x = 2x − 3y y = y − 2x x(0) = 8, y(0) = ●✐↔✐✿ ❚❤ü❝ ❤✐➺♥ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❤❛✐ ✈➳ ❝õ❛ ❝↔ ❤❛✐ ♣❤÷ì♥❣ tr➻♥❤ t❛ ❝â✿ sX − = 2X − 3Y ⇒ sY − = Y − 2X (s − 2)X + 3Y = 2X + (s − 1)Y = ❉ò♥❣ q✉② t➢❝ ❈r❛♠❡r ❣✐↔✐ ❤➺ t❛ ✤÷đ❝✿ + s+1 s−4 ⇒  Y = − s+1 s−4   X= ữỡ tr ợ x = 5e−t + 3e4t y = 5e−t − 2e4t x + 4y + 3x = 15e−t , y − 4x + 3y = 15 sin 2t, x(0) = 35, x (0) = −48, y(0) = 27, y (0) = −55 ✸✶ ●✐↔✐✿ ❚❤ü❝ ❤✐➺♥ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❤❛✐ ✈➳ ❝õ❛ ❝↔ ❤❛✐ ♣❤÷ì♥❣ tr➻♥❤ t❛ ❝â✿ ❱✐➳t ❧↕✐  15   s2 X − 35s + 48 + sY − 27 + 3X = , s+1 30   s2 Y − 27s + 55 − 4(sX − 35) + 3Y = s2 + t❤➔♥❤ ❤➺ t✉②➳♥ t➼♥❤ t❤❡♦ X, Y ✿  15   (s2 + 3)X + 4sY = 35s − 21 + , s+1 30   −4sX + (s2 + 3)Y = 27s − 197 + s + ❉ị♥❣ q✉② t➢❝ ❈r❛♠❡r ❣✐↔✐ ❤➺ t❛ ✤÷đ❝✿  30s 45 2s  X= + + + s2 + s2 + s + s2 + 30 60 2s  Y = − − + s2 + s2 + s + s2 + ⇒ ❱➼ ❞ö ✸✳✶✵✳ x(t) = 30 cos t − 15 sin 3t + 3e−t + cos 2t, y = 30 cos 3t − 60 sin t − 3e−t + sin 2t ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✿    x −x+y−z =0 x+y −y+z =0   x+y+z −z =0 ❚❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➛✉✿ x(0) = 1, y(0) = z(0) = x (0) = y (0) = z (0) = ●✐↔✐✿ Ù♥❣ ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➔♦ ✷ ✈➳ ❝õ❛ ❝↔ ✸ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ t❤❛② ✤✐➲✉ ❦✐➺♥ ✤➛✉ ✈➔♦ t❛ ✤÷đ❝ ❤➺ s❛✉✿    (s − 1)X + Y + Z = s X + (s2 − 1)Y + Z =   X + Y + (s2 − 1)X = ●✐↔✐ ❤➺ ♥➔② t❛ ✤÷đ❝✿ s3 X= (s + 2)(s2 − 2) s Y =Z= (s + 1)(s2 − 2) ✸✷ ◆❤÷ ✈➟②✿ x(t) = √ cosh( 2t) + cos t, 3 √ 1 y(t) = z(t) = − cosh( 2t) + cos t 3 ✸✳✸ ◆❤ú♥❣ ÷✉ ✤✐➸♠ ✈➔ ♥❤÷đ❝ ✤✐➸♠ ❝õ❛ ✈✐➺❝ →♣ ❞ư♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ tr♦♥❣ ✈✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❉ị♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝â ♥❤ú♥❣ ÷✉ ✤✐➸♠ ♥❤÷ ✿ ✲ ❚❤❛② ❝→❝ ♣❤➨♣ t♦→♥ ✤↕♦ ❤➔♠ tr♦♥❣ ♠✐➲♥ ❝õ❛ sè ✭❝ë♥❣✱ trø✱ ♥❤➙♥✱ ❝❤✐❛✮ tr♦♥❣ ♠✐➲♥ ❝õ❛ s✳ t ❜➡♥❣ ❝→❝ ♣❤➨♣ t♦→♥ ✤↕✐ ❉♦ ✤â ✈✐➺❝ t➼♥❤ t♦→♥ trð ♥➯♥ ❞➵ ❞➔♥❣ ❤ì♥✳ ✲ ❚ü ✤ë♥❣ ✤÷❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ✈➔♦ q✉→ tr➻♥❤ ❣✐↔✐✱ ❦❤æ♥❣ ❝➛♥ ♣❤↔✐ t➻♠ ♥❣❤✐➺♠ tê♥❣ q✉→t✱ s õ ợ ữ ✤à♥❤ ❝→❝ ❤➺ sè✳ ✲ ◆❤ú♥❣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ổ ữợ tr t ổ t ❣✐↔✐ ✤÷đ❝ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♠❛ tr➟♥ ♥❤÷♥❣ ❝â t❤➸ ❣✐↔✐ ✤÷đ❝ ❜➡♥❣ ❝→❝❤ ❞ị♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✭♥❤÷ ✈➼ ❞ư ✸✳✺✮ ❇➯♥ ❝↕♥❤ ✤â ❞ị♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝ơ♥❣ ❝á♥ ❝â ♥❤ú♥❣ ♠➦❝ ❤↕♥ ❝❤➳ ♥❤÷✿ ✲ ❈❤➾ →♣ ❞ư♥❣ ✤÷đ❝ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ❤➔♠ ∞ e−st f (t) f (t) ❝â t➼❝❤ ♣❤➙♥ s✉② rë♥❣ ❤ë✐ tö✳ ✲ ❑❤✐ ❜✐➳♥ ✤ê✐ t❤➔♥❤ ❤➔♠ ↔♥❤ ✤➸ ❧➜② ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➲ ❤➔♠ ❣è❝ ❝á♥ ❤↕♥ ❝❤➳ ✈➻ ❝â ♥❤ú♥❣ ❤➔♠ ❦❤ỉ♥❣ t❤➸ →♣ ❞ư♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ♥❣÷đ❝ ✤÷đ❝✳ ✸✸ ❑➌❚ ▲❯❾◆ ✣➲ t➔✐ ✏P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➔ ỳ ự tr ỵ tt ữỡ tr ♣❤➙♥✑ ✤➣ t❤ü❝ ❤✐➺♥ ✤÷đ❝ ♠ư❝ ✤➼❝❤ ✤➣ ✤➲ r❛✳ ❈ö t❤➸ ❧➔✿ ✶✴ ❈õ♥❣ ❝è ✈➔ ❤➺ t❤è♥❣ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ t➼❝❤ ♣❤➙♥ s✉② rë♥❣ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ✷✴ ❚➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝ị♥❣ ❝→❝ ✈➼ ❞ư ♠✐♥❤ ❤å❛✳ ✸✴ ❚➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝ ❝ị♥❣ ❝→❝ ✈➼ ❞ư ♠✐♥❤ ❤å❛✳ ✹✴ ⑩♣ ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ✺✴ ❚ø ✤â rót r❛ ✤÷đ❝ ♥❤ú♥❣ ÷✉ ✤✐➸♠ ✈➔ ♥❤÷đ❝ ✤✐➸♠ ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ tr♦♥❣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ❳✐♥ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝ỉ ❣✐→♦ t❤✉ë❝ ❦❤♦❛ ❚♦→♥ ❚r÷í♥❣ ✣↕✐ ❍å❝ ❙÷ P❤↕♠ ✕ ✣↕✐ ữợ t ✤✐➲✉ ❦✐➺♥ ✤➸ ❡♠ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ✤➲ t➔✐ ♥➔②✳ ❱ỵ✐ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ♥❤➜t✱ ❡♠ ①✐♥ ❣û✐ ✤➳♥ t❤➛② ❣✐→♦✱ ❚❙ ▲➯ ❍↔✐ ❚r✉♥❣ ✤➣ t➟♥ t➙♠ ữợ q tứ tr ợ ụ ♥❤÷ ♥❤ú♥❣ ❜✉ê✐ ♥â✐ ❝❤✉②➺♥✱ t❤↔♦ ❧✉➟♥ ✈➲ ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐✳ ▼➦❝ ❞ò ✤➣ ❝è ❣➢♥❣ ❤➳t sù❝✱ s♦♥❣ ❞♦ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤â tr→♥❤ ❦❤ä✐ t❤✐➳✉ sât✳ ❚→❝ ❣✐↔ r➜t ♠♦♥❣ ✤÷đ❝ sü ✤â♥❣ ❣â♣ ❝õ❛ t❤➛② ❝ỉ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✸✹ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❚✐➳♥❣ ❱✐➺t ❬✶❪ P❤❛♥ ❍✉② ❚❤✐➺♥❀ P❤÷ì♥❣ ❚r➻♥❤ ❱✐ P❤➙♥ ❀ ◆❤➔ ①✉➜t ❜↔♥ ❣✐→♦ ❞ö❝ ❱✐➺t ◆❛♠✱ ✷✵✶✵✳ ●✐↔✐ ❚➼❝❤ t➟♣ ✷ ❀ ◆❤➔ ①✉➜t ❜↔♥ ❣✐→♦ ❞ö❝✱ ✷✵✵✾✳ ❬✸❪ ▲➯ ❍↔✐ ❚r✉♥❣✱ ●✐→♦ tr➻♥❤ P❤÷ì♥❣ ❚r➻♥❤ ❱✐ P❤➙♥ ❀ ✣➔ ◆➤♥❣✱ ✷✵✶✶✳ ❬✹❪ ❍♦➔♥❣ ❚ö②✱ ●✐↔✐ ❚➼❝❤ ❍✐➺♥ ✣↕✐✱ r tr ỵ t ữợ ❋✳ ❚r❡♥❝❤✱ ❊❧❡♠❡♥t❛r② ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ❲✐t❤ ❇♦✉♥❞✲ ❛r② ❱❛❧✉❡ Pr♦❜❧❡♠s✱ ❆♥❞r❡✇ ●✳ ❈♦✇❧❡s ❉✐st✐♥❣✉✐s❤❡❞ Pr♦❢❡ss♦r ❊♠❡r✐t✉s ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❚r✐♥✐t② ❯♥✐✈❡rs✐t② ❙❛♥ ❆♥t♦♥✐♦✱ ❚❡①❛s✱ ❯❙❆✱ ✷✵✶✶✳ ❬✼❪ ❈✳ ❍❡♥r② ❊❞✇❛r❞s ✫ ❉❛✈✐❞ ❊✳ P❡♥♥❡②✱ ❊❧❡♠❡♥t❛r② ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛✲ t✐♦♥s✱ ❚❤❡ ❯♥✐✈❡rs✐t② ♦❢ ●❡♦r❣✐❛✳ ❬✽❪ ❏♦❡❧ ▲✳ ❙❝❤✐❢❢ ✱ ❚❤❡ ▲❛♣❧❛❝❡ ❚r❛♥s❢♦r♠✿ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✶✾✾✾✳ ✸✺