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MATH-EDUCARE www.VNMATH.com ` TA ˆ P PHU.O.NG TR`INH VI PHAN ˆ BAI 1) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ●✐❛ 2xy y” = y − 2xpp = p2 − √ dx 2pdp ✳ 2 ❱✓ ♦ ✐ x(p − 1) = t❛ ❝♦ C1 x + = ⇔ p − = C ⇔ p = ± ✓✿ p2 − x dy √ p= = C1 + ⇒ y = (C1 x + 1) + C2 dx 3C1 ✲ ❛ ❉ ✕ ✳t HD gia’i: 2) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ ✲ ❛ ❉ ✕ ✳t HD gia’i: ✳ ❱✓ ♦✐ y =p: p=0 √ y.y” = y y = p ⇒ y” = p dp dy ✳ ✳ ✬✳ t❤❛ ✒♥❤ tr♦ ✭❤❛ ✒♠ t❤❡♦ ②✮✳ P❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ✳ ✳ ✳ ✳ t❛ ❞ ✖✉ ♦ ✒♥❤✿ ✳ ❝ ♣❤✉ ♦ ♥❣ tr✏ √ yp dp =p dy dy dy √ √ = y + C1 ⇒ dp = √ ⇒ p = y + C1 ⇔ y dx dy dx = √ y + C1 ✳ ✖✓ ♦ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ❚✒ ✉ ❞ ✳ ♠ t❫ ◆❣♦❛ ✒✐ r❛ 3) y = c✿ x= √ y− C1 √ ln |2 y + C1 | + C2 ✒ ♥❣ ❝✉ ⑦ ♥❣ ❧❛ ❤✕ ❛ ✒ ♥❣❤✐❫ ❡ ✳ ♠✳ ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ●✐❛ a(xy + 2y) = xyy HD gia’i: a(xy + 2y) = xyy ⇒ x(a − y)y = −2ay ✓✉ ◆❫ ❡ y = 0✱ ◆❣♦❛ ✒✐ r❛ 4) y=0 ✳ ❱✓ ♦ ✐p ✲ ❛ ❉ ✕ ✳t 2a a−y dy = − dx ⇔ x2a y a e−y = C y x ⑦ ♥❣ ❧❛ ❝✉ ✒ ♥❣❤✐❫ ❡ ✳ ♠✳ ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ●✐❛ HD gia’i: ✳ ❱✓ ♦✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t❛ ❝♦ ✓ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉ ♦ ♥❣ ❞ ✖✉ ♦ ♥❣ ✈✓ ♦✐ y” = y ey y = p ⇒ y” = p dp dy ✳ ✳ t❤❛② ✈❛ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ p dp = pey dy dy dy dp = ey ⇔ p = ey + C1 ⇒ = ey + C1 ⇔ y = dx dy dx e + C1 1 dy ey + C1 − ey = dy = (y − C1 = t❛ ❝♦ ✓✿ ey + C1 C1 ey + C1 =0: ln(ey + C1 ) C1 −e−y dx ✳ ♥❤✉ ✈❫ ❛ = ✳ ②✿ ey + C1 (y − ln |ey + C1 |) C1 ✒ ♥❣ ❧❛ ◆❣♦❛ ✒✐ r❛ y = C : ❤✕ ❛ ✒ ♠❫ ♦ ❡ ✳ t ♥❣❤✐❫ ✳♠ 5) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ xy = y(1 + ln y − ln x) nˆe´u C1 = nˆe´u C1 = ✳ ✈✓ ♦✐ y(1) = e www.matheducare.com ey dy y ) = − ey + C1 C1 MATH-EDUCARE www.VNMATH.com y y ✳ ✳ (1 + ln )✱ ❞✖❛ ✕ ✖✉ ♦ ✳ t y = zx ❞ ✳ ❝✿ xz = z ln z x x dx y dz = ⇒ ln z = Cx ❤❛② ln = Cx ⇔ y = xeCx • z ln z = ⇒ z ln z x x x y(1) = e → C = ❱❫ ❛ ✳ ② y = xe HD gia’i: 6) ✲ ✉✳❛ ♣❤✉✳♦✳♥❣ tr✏ ✒ ❉ ✒♥❤ ✈❫ ❡✿ ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ HD gia’i: ✲ ❛ ❉ ✕ ✳t y = y”(1 + y) = y + y y = z(y) ⇒ z = z dz dy ✳ ✳ t❤❛② ✈❛ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ⇒ z + = C1 (y + 1) ⇒ z = C1 y + C1 − ⇔ • C1 = ⇒ (∗) ❝❤♦ • C1 = ⇒ (∗) ❝❤♦ ◆❣♦❛ ✒✐ r❛ y=C dy = dx (∗) C1 y + C1 − y =C −x ln |C1 y + C1 − 1| = x + C2 C1 ❧❛ ✒ ♥❣❤✐❫ ❡ ✳ ♠✳ ❚♦ ✓ ♠ ❧❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ✐ ♥❣❤✐❫ ✳ ♠ t❫ 7) dy dz = z+1 y+1 ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ●✐❛ y = C, y = C − x; y = y2 − ln |C1 y + C1 − 1| = x + C2 C1 x2 2 ✓♥ ❞ ✒ HD gia’i: ❇✐❫ ❡ ✖♦ ❫✬✐ ✭✸✮ ✈❫ ❡ ❞❛ ✳ ♥❣✿ x y = (xy) − (∗) ✲ ❛ ❉ ✕ ✒♦ (∗) s✉② r❛✿ ✳ t z = xy ⇒ z = y + xy t❤❛② ✈❛ xz = z + z − ⇔ ❱❫ ❛ ✳ ② ❚P❚◗✿ 8) dx dz = ⇔ +z−2 x z−1 = Cx z+x xy − = Cx3 xy + ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ●✐❛ HD gia’i: z2 ✲ ❛ ❉ ✕ ✳t yy” + y = y = z(y) ⇒ y” = z dz dy z C1 dy ⇔ z2 = + dz = 1−z y y dy C1 dy ⇒ =± 1+ ⇔± = dx ⇒ y + C1 = (x + C2 )2 dx y C1 1+ y ✬♥❣ q✉❛ ◆❣❤✐❫ ❡ ♠ t❫ ♦ ✓ t✿ y + C = (x + C2 )2 ✳ ✳ ✳ ✓♥ ❞ ✒ ❇✐❫ ❡ ✖♦ ❫✬✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ✈❫ ❡✿ 9) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ HD gia’i: y − √ 2x(1 + x)y − (3x + 4)y + 2x + x = 3x + y = − √ ; x = 0, x = −1 2x(x + 1) x+1 ✒ ✓t✿ ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✳ ♠ t❫ dy = y 3x + Cx2 dx = ( − )dx ⇔ y = √ 2x(x + 1) x 2(x + 1) x+1 www.matheducare.com MATH-EDUCARE www.VNMATH.com ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓✿ ❇✐❫ ❡ ❡♥ ❤✕ ❛ ♦ 1 ⇒ C = − + ε x x x2 y=√ ( + ε) x+1 x C =− ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ 10) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ HD gia’i: ✲ ❛ ❉ ✕ ✳t y” = e2y z = y → y” = z dz dy ✬ t❤♦❛ y(0) = y (0) = ✳ ✳ ✬✳ t❤❛ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ✒♥❤ tr♦ z z2 e2y dz = e2y ⇔ = +ε dy 2 ✳ 2y ❛ − ❚✒ ✉ ❞ ✖✓ ♦✿ y (0) = y(0) = ⇒ ε = − ❱❫ ✳② z = e √ dy √ 2y dy √ = e −1⇒ z= = x + ε d¯ˆo’i biˆe´n t = e2y − dx e2y − √ arctg e2y − = x + ε ✒ ✬ ❞ ln(tg x + 1) y(0) = ⇒ ε = ❱❫ ❛ ❡ ❡♥❣ t❤♦❛ ✖✐❫ ❡✉ ❦✐❫ ❡ ✖✒ ❫ ❡ ❜❛ ✒✐✿ y = ✳ ② ♥❣❤✐❫ ✳ ♠ r✐❫ ✳♥ ❞ 11) ✳ ✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ ❝✉ ✳ ♠ r✐❫ ⑦♥ ❞ ✒ ✒ ✬ ♠❛ t❤♦❛ ✖✐❫ ❡✉ ❦✐❫ ❡ ✖❫ ❛✉ ✳♥ ❞ HD gia’i: ✓t ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ ❧❛ ❱✐❫ ❡ ✳ ✐✿ x(1 − y)y = −2y ❀ 1−y dx dy = −2 y x ✳ ✳ ✓♥✿ ✒♥❤ t❛ ✓ ❝❤ ❜✐❫ ❡ ♣❤✉ ♦ ♥❣ tr✏ t✏ ✓❝❤ ♣❤❫ ❛♥ t❫ ♦✬♥❣ q✉❛ ✓ t✿ ✒ r✐❫ ❡♥❣ ❝❫ ❛♥ t✏ ✒♠ ❧❛ ✒✿ 12) xy + 2y = xyy y(−1) = 1✳ x2 ye−y = C ✳ ❞♦ y(−1) = ✳ ✳ ✒ ❚❤❛② ❞ ✖✐❫ ❡✉ ❦✐❫ ❡ ✒♦ t❛ ❞ ✖✉ ♦ ✳ ♥ ✈❛ ✳❝ C= x2 ye1−y = 1✳ ✒ ❇✕ ❛ ♥❣ ❝❛ ✓ ❝❤ ❞ ✖✕ ❛ ✳t y = ux✱ ✳ ✳ ⑦ ② ❣✐❛ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ❤❛ xdy − ydx − ✳ ✳ ✲ ✕t y = ux; du = udx + xdu t❤❛② ✈❛ ✒♥❤ ✈❛ ✒ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ ✳ √ HD gia’i: ❉❛ ✳ ✳ ✳ ⑦ r❛ ❛ ♣❤✉ ♦ ♥❣ − u2 dx = 0✳ ❘♦ ✒♥❣ u − ±1 ❧❛ ✒ ♥❣❤✐❫ ❡ ♠✳ ❦❤✐ u ≡ ±1 ❞ ✖ ✉ ✳ du dx ✳ ❚P❚◗✿ arcsin u − ln x = C ✭❞♦ x > 0✮✳ = − u2 x y ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ❱❫ ❛ = ln x + C ✳ ✒♥❤✿ y = ±x; arcsin ✳ ② ◆❚◗ ❝✉ x 13) ✳ ✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ ❝✉ ✳ ♠ r✐❫ ⑦♥ ❞ ✒ ✒ ✬ ♠❛ t❤♦❛ ✖✐❫ ❡✉ ❦✐❫ ❡ ✖❫ ❛✉ ✳♥ ❞ xy = y(1) = 0✳ x2 − y + y HD gia’i: xy = ❞ ✖❛ ✕ ✳t u= y x ❤❛② y = ux ✳ ✳ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t❤❛ ✒♥❤✿ x2 − y + y ⇐⇒ y = 1− y2 y + x2 x y = xu + u √ du dx xu = − u2 ⇐⇒ √ = x − u2 s✉② r❛ www.matheducare.com ♥❫ ❡♥ ✳ e y ≡ 0✳ ✲ ✉✳❛ ✈❫ ✒ ❉ ❡ ❱❫ ❛ ✓❝❤ ♣❤❫ ❛♥ ✳ ② t✏ x2 − y dx = (x > 0) ✳ ✬ ♥ ✉✳♦ ❣✐❛ ✓❝ x✿ xdu − ✒ ✓♥✿ tr✏ ✒♥❤ ✈❫ ❡ t❛ ✓ ❝❤ ❜✐❫ ❡ MATH-EDUCARE www.VNMATH.com ⇐⇒ arcsin u = ln Cx ⑦♥ ❞ ✒ ✬ ♠❛ t❤♦❛ ✖✐❫ ❡✉ ❦✐❫ ❡ ✖✒ ❛ ❫✉ ✳♥ ❞ 14) y(1) = ❦❤✐ C = 1✳ ❱❫ ❛ ❡ ✳ ② ♥❣❤✐❫ ✳♠ y = ±x✳ ✳ ✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ ❝✉ ✳ ♠ r✐❫ ⑦♥ ❞ ✒ ✒ ✬ ♠❛ t❤♦❛ ✖✐❫ ❡✉ ❦✐❫ ❡ ✖❫ ❛✉ ✳♥ ❞ y sin x = y ln y π y( ) = e✳ HD gia’i: y sin x = y ln y ⇐⇒ ⑦♥ ❞ ✒ ✬ ♠❛ t❤♦❛ ✖✐❫ ❡✉ ❦✐❫ ❡ ✳♥ 15) dx dy = y ln y sin x x C tan x ⇐⇒ ln y = C tan ⇐⇒ y = e x tan π 2✳ ❛ ❞ ✖✒ ❛ ❫✉ y( ) = e ❦❤✐ C = 1✳ ❱❫ ✳② y = e ✳ ✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ ❝✉ ✒♥❤✿ ✳ ♠ r✐❫ (x + y + 1)dx + (2x + 2y − 1)dy = y(0) = 1✳ ⑦♥ ❞ ✒ ✒ ✬ ♠❛ t❤♦❛ ✖✐❫ ❡✉ ❦✐❫ ❡ ✖❫ ❛✉ ✳♥ ❞ ✲ ❛ ❉ ✕ ✳t x + y = ✳ ✳ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t❤❛ ✒♥❤✿ HD gia’i: z =⇒ dy = dz − dx (2 − z)dx + (2z − 1)dz = 0❀ x + 2y + ln |x + y − 2| = C ⑦♥ ❞ ✒ ✬ ♠❛ t❤♦❛ ✖✐❫ ❡✉ ❦✐❫ ❡ ✖✒ ❛ ❫✉ y(0) = ❦❤✐ C = 2✳ ✳♥ ❞ 16) ✲ ❛ ❉ ✕ ✳t y = (z − x2 )dz + 2zxdx = 0❀ ⇐⇒ ⇐⇒ ln |x| + ln t❤❛② u= 17) xy ❱❫ ❛ ✳② y= ✳ ✳ ❞ ✖✉ ♦ ✳ ❝✿ z (u2 − 1)(udx + xdu) + 2udx = HD gia’i: x − 2z − ln |z − 2| = C ✳ ⑦ ✒ ✬ r❫ ♦✐ ❞ ✖✕ ❛ ✳ t z = ux✱❤❛ ② ❣✐❛✐ z (x2 y − 1)dy + 2xy dx = ✒ ❇✕ ❛ ♥❣ ❝❛ ✓ ❝❤ ❞ ✖✕ ❛ ✳t ✳ ✳ ✒♥❤✿ ♣❤✉ ♦ ♥❣ tr✏ ✬ ✐ r❛ ❣✐❛ ✳ ✳ ❞ ✖✉ ♦ ❡ ✳ ❝ ♥❣❤✐❫ ✳♠ ✒ r❫ ♦✐ ❞ ✖❛ ✕ ✳t z = ux✱ dx u2 − + du = x u +u u2 + x(u2 + 1) = ln C ⇐⇒ =C |u| u + x2 y = Cy ✳ ✳ ✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ s❛✉✿ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✳ ♠ t❫ y − xy = x + x3 HD gia’i: ✳ ✳ ✲ ❛ ✓♥ t✏ ✓♣ ✶ ✈❛ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤ ❝❫ ❛ ✒ ❝♦ ✓ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❧❛ ✒ ✳ ♠ t❫ x2 y = Ce x2 +1 ✳ www.matheducare.com ✳ ✳ ❞ ✖✉ ♦ ✳❝ MATH-EDUCARE www.VNMATH.com 18) ✳ ✳ ✬ ❛ ❝❛ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✓ ❝ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ s❛✉✿ ✳ ♠ t❫ HD gia’i: y − y = y2 ✳ ✳ ✲ ❛ ✓♥ ✈❛ ✒♥❤ t❛ ✓ ❝❤ ❜✐❫ ❡ ✒ ❝♦ ✓ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❧❛ ✒ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✳ ♠ t❫ ln | 19) y | = x + C y+1 ✳ ✳ ✬ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ✓ ❝ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ s❛✉✿ ✳ ♠ ❝✉❛ ❝❛ y + y = ex x HD gia’i: ✳ ✳ ✲ ❛ ✓♥ t✏ ✓♣ ✶ ✈❛ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤ ❝❫ ❛ ✒ ❝♦ ✓ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❧❛ ✒ ✳ ♠ t❫ 20) ✳ ✳ ✬ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ✓ ❝ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ s❛✉✿ ✳ ♠ ❝✉❛ ❝❛ HD gia’i: ex C x y = +e − ✳ x x y − y = y3 ✳ ✳ ✲ ❛ ✓♥ ✈❛ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t❛ ✓ ❝❤ ❜✐❫ ❡ ✒ ❝♦ ✓ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❧❛ ✒ ✳ ♠ t❫ C + x = ln |y| − arctgy 21) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ y = y y + sin ✱ x x HD gia’i: y = zx ⇒ y = z x + z ✱ z x = sin x ⇔ ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ ❱❫ ❛ ✳ ②✿ 22) tg y = x✳ 2x ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ HD gia’i: ✲ ❛ ❉ ✕ ✳t 23) y y (x − y cos )dx + x cos dy = x x ✳ ✳ ✳ ✳ ✳ ✒ ✖✉ ❛ ✈❫ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖✉ ♦ ❡ ❞❛ ✳❝ ❞ ✳ ♥❣✿ cos zdz = − dx + C ⇔ sin z = − ln |x| + C x y = − ln |x| + C x ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ HD gia’i: π ✳ ✳ ✬✳ t❤❛ ✒♥❤✿ ✒♥❤ tr♦ ♣❤✉ ♦ ♥❣ tr✏ y =z ⇒y =zx+z x sin y(1) = dz dx z z = ⇔ ln |tg | = ln |x| + ln C ⇔ tg = Cx sin z x 2 y π tg = Cx; y(1) = ⇒ C = 2x x cos z.z + = ⇔ ❱❫ ❛ ✳ ② ❚P❚◗✿ ✳ ✈✓ ♦✐ (y − 1)x2 y + y (x4 − y ) = ✳ ✳ ✳ ✬ ♥❣ ❝❫ ✓♣ ♥❤✉✳♥❣ ❣✐❛ ✬ ✐ ❦❤❛ ▲❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖✕ ❛ ❛ ✓ ♣❤✓ ✉ ❝ t❛ ✳ ♣✳ www.matheducare.com MATH-EDUCARE www.VNMATH.com ✳ ✳ ✳ ✓✐ ✈✓ ✒♥❤ ❜❫ ❛ ✖♦ ❫ ♦✐ ❳❡♠ ♣❤✉ ♦ ♥❣ tr✏ ✳ ❝ ❤❛✐ ❞ ✳ ✖✓ ♦ ❝♦ ✓ ❤❛✐ ❤♦ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ❚✒ ✉ ❞ ✳ ♥❣❤✐❫ ✳ ♠ t❫ 24) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ●✐❛ HD gia’i: 25) y + x2 y = xyy ✓t ♣❤✉✳♦✳♥❣ tr✏ ❱✐❫ ❡ ✒♥❤ ❧❛ ✳✐ ✳ ✳ ❡ ♦✬♥❣ q✉❛ ✓ t✿ r❛ ❞ ✖✉ ♦ ✳ ❝ ♥❣❤✐❫ ✳ ♠ t❫ y2 x2 = (x4 + y )2 ⇒ y1 = ; y2 = − x y x 3 ; x + y = C2 y= C1 x + y✿ y = y y2 x2 y x −1 ✳ ✳ ✒ ✓t✱ ❣✐❛ ✬✐ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ❞ ✖❛ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ y = Cxe x ✳ ✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ ❝✉ ✒♥❤✿ ✳ ♠ r✐❫ (x + y − 2)dx + (x − y + 4)dy = y(1) = 0✳ ⑦♥ ❞ ✒ ✒ ✬ ♠❛ t❤♦❛ ✖✐❫ ❡✉ ❦✐❫ ❡ ✖❫ ❛✉ ✳♥ ❞ HD gia’i: ✲ ❛ ❉ ✕ ✳t x y =u−1 = v + (u + v)du + (u − v)dv = 0✱ u + 2uv − v = C ✳ ✳ ✳ ✳ ✳ t❤❛② ✈❛ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖✉ ♦ ✳ ❝✿ ✳ ✳ ✒ ✓t ❝♦ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✓ t✏ ✓❝❤ ♣❤❫ ❛♥ t❫ ♦✬♥❣ q✉❛ ✓ t ❧❛ ✒✿ ❞ ✖❛ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ❱❫ ❛ ✓❝❤ ♣❤❫ ❛♥ t❫ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✒♥❤ ❜❛♥ ❞ ✖✒ ❛ ❫✉ ❧❛ ✒✿ ✳ ② t✏ 26) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤ HD gia’i: ✲ ❛ ❉ ✕ ✳t x y x2 + 2xy − y − 4x + 8y = C (x + y − 2)dx + (x − y + 4)dy = 0✳ =X −1 ✱ =Y +3 ✳ ✳ ✒♥❤ t❤❛ ✒♥❤✿ ♣❤✉ ♦ ♥❣ tr✏ (X + Y )dX + (X − Y )dY = ❞ ✖❛ ✕ ✳t Y = uX ✬ ✐ r❛ ●✐❛ 27) 1−u dX + du = 0✳ X + 2u − u2 x2 + 2xy − y − 4x + 8y = C ✳ ✳ ✳ ✳ ✒ ❞ ✖✉ ❛ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ✈❫ ❡ X (1 + 2u − u2 ) = C ❤❛② ✳ ✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ s❛✉✿ ❚✏ ✒♠ t✏ ✓ ❝❤ ♣❤❫ ❛♥ t❫ ♦✬♥❣ q✉❛ ✓ t ❝✉ HD gia’i: ✳ ✳ ✲ ❛ ✬ ♥❣ ❝❫ ✓♣✱ t❛ ❞ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖✕ ❛ ❛ ✖❛ ✕ ✳t b) y = z= y ✳ z 2xy − y2 x2 ✳ ✳ ❑❤✐ ❞ ✖✓ ♦ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ tr❫ ❡♥ z(1 + z ) 2z dx xz = ✳ ❙✉② r❛ ♥❣❤✐❫ ❡ ✳ ❍❛② ( − )dz = ✳♠ 2 1−z z 1+z x z ♥❛ ✒② ❧❛ ✒ = Cx, C = + z2 2 ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ ❞ ✖⑦ ❛ ❝❤♦ ❧❛ ✒ x + y = C1 y, C1 = ❱❫ ❛ ❡ ✳ ② ♥❣❤✐❫ ✳ ♠ ❝✉ ✬✳ t❤❛ ✒♥❤ tr♦ 28) ✳ ✳ ✬ ❛ ❝❛ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✓ ❝ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ s❛✉✿ ✳ ♠ t❫ HD gia’i: ✲ ❛ ❉ ✕ ✳t u = 2x + y ✳ ✳ ✳ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖✉ ❛ ✈❫ ❡ ❞❛ ✳ ♥❣ 5u + du = dx 2u + www.matheducare.com y = ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ❝✉ ✒♥❤ 2x + y − 4x + 2y + MATH-EDUCARE www.VNMATH.com ✳ ✳ ✬ ✐ ♣❤✉✳♦✳♥❣ tr✏ ●✐❛ ✒♥❤ ♥❛ ✒② t❛ ❞ ✖✉ ♦ ❡ ✳ ❝ ♥❣❤✐❫ ✳ ♠ 10u + ln |5u + 9| = ✳ ✳ ⑦ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖❛ ❝❤♦ ❧❛ ✒ 10y + ln |10x + 5y ❱❫ ❛ ❡ ✳ ② ♥❣❤✐❫ ✳ ♠ ❝✉ 29) 25x + C = 9| − 5x = C ✳ ✳ ✬ ❛ ❝❛ ❚✏ ✒♠ t✏ ✓ ❝❤ ♣❤❫ ❛♥ t❫ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✓ ❝ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ s❛✉✿ (x − y + 4)dy + (y + x − 2)dx = HD gia’i: ✳ ✳ ✳ ✳ ✳ ✲ ❛ ✬ ♥❣ ❝❫ ✒ ✒ ✓♣ ❞ ✒♥❤ ❞ ✖✉ ❛ ✈❫ ❡ ❞❛ ✖✕ ❛ ❛ ✖✉ ♦ ❛ ♥❣ ❝❛ ✓ ❝❤ ❞ ✖✕ ❛ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✳ ♥❣ ❞ ✳ ❝ ❜✕ ✳t u + 1, y = v − 3, tr✏ ✒♥❤ ❧❛ ✒ ✳ ✳ t❛ ❞ ✖✉ ♦ ✳❝ v − 2uv − v = C u+v dv = ✳ du −u + v ✬ ✐ ♣❤✉✳♦✳♥❣ tr✏ ✬ ❛ ♣❤✉✳♦✳♥❣ ●✐❛ ✒♥❤ t❛ ❝♦ ✓ ♥❣❤✐❫ ❡ ✳ ♠ ❝✉ ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ ❞ ✖⑦ ❛ ❝❤♦ ❧❛ ✒ ❱❫ ❛ ❡ ✳ ② ♥❣❤✐❫ ✳ ♠ ❝✉ 30) x = y − x2 − 2xy − 8y + 4x = C1 ✳ ✳ ✒ ✬ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ❛✮ ❚✏ ✒♠ ♠✐❫ ❡♥ ♠❛ ✒ tr♦♥❣ ❞ ✖♦ ✓ ♥❣❤✐❫ ❡ ✒✐ t♦❛ ✓ ♥ ❈❛✉❝❤② ❝✉ ✒♥❤ ✳ ♠ ❝✉❛ ❜❛ y = √ ✒ ✓t s❛✉ ❞ ✖❫ ❛② t❫ ♦♥ t❛ ✒ ❞✉② ♥❤❫ ❛ ✳ ✐ ✈❛ ✳ ✳ ✬ ❛ ❝❛ ✒♥❤ s❛✉✿ ❜✮ ❚✏ ✒♠ t✏ ✓ ❝❤ ♣❤❫ ❛♥ t❫ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✓ ❝ ♣❤✉ ♦ ♥❣ tr✏ x − y (x2 − y )dy − 2xydx = HD gia’i: ✓t ♥❣❤✐❫ ✒ ❛✮ ❇❛ ✒✐ t♦❛ ✓ ♥ ❈❛✉❝❤② ❝♦ ✓ ❞✉② ♥❤❫ ❛ ❡ ❡♥ ✳ ♠ tr♦♥❣ ♠✐❫ ✳ ✒② ✓ ②✳ D = {(x, y) ∈ R |x − y ≥ δ} ✈✓ ♦ ✐ δ > t✉ ✲ ✉✳❛ ♣❤✉✳♦✳♥❣ tr✏ ✒ ❜✮ ❉ ✒♥❤ ✈❫ ❡ ❞❛ ✳ ♥❣ z= y ✳ x dy xy ✳ = dx x − y2 ✳ ✳ ✲ ❛ ✬ ♥❣ ❝❫ ✓♣✱ t❛ ❞ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖❛ ✕ ❛ ✖❛ ✕ ✳t ✳ ✳ ✬✳ t❤❛ ❑❤✐ ❞ ✖✓ ♦ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ✒♥❤ tr❫ ❡♥ tr♦ z(1 + z ) xz = − z2 ❍❛② dx 2z )dz = ( − ✳ z 1+z x z = Cx, C = + z2 2 ❧❛ ✒ x + y = C1 y, C1 = ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ ♥❛ ✒② ❧❛ ✒ ❙✉② r❛ ♥❣❤✐❫ ❡ ✳ ♠ ❝✉ ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ❱❫ ❛ ❡ ✒♥❤ ❞ ✖⑦ ❛ ❝❤♦ ✳ ② ♥❣❤✐❫ ✳ ♠ ❝✉ ✳ ✳ 2x 2x ✒ ❛ ♥❣ ❤❫ ❡ ✓ ❝ ✈❡❝t♦ ❛✮ ❈❤✓ ✉ ♥❣ ♠✐♥❤ r✕ ✳ ❝❛ ✳ ✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ s❛✉✿ ❜✮ ❚✏ ✒♠ t✏ ✓ ❝❤ ♣❤❫ ❛♥ t❫ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✓♥ t✏ {e , xe , x } ❧❛ ✒ ❤❫ ❡ ✖❫ ♦ ❛ ❡ ✓ ♥❤✳ ✳ ❞ ✳ ❝ ❧❫ ✳ ♣ t✉②❫ (x − y)dy − (x + y)dx = 0; 31) HD gia’i: ⑦❛ ❦✐❫ ✓♥ t✏ ❛✮ ❉✉ ✒♥❣ ❞ ✖✳✐♥❤ ♥❣❤✏ ❡✬♠ tr❛ ❤❫ ❡ ✖♦ ❫ ❛ ❡ ✓♥❤ ✳ ✳ ❞ ✳ ❝ ❧❫ ✳ ♣ t✉②❫ ✲ ✉✳❛ ♣❤✉✳♦✳♥❣ tr✏ ✒ ❜✮ ❉ ✒♥❤ ✈❫ ❡ ❞❛ ✳ ♥❣ z= y ✳ x y = x+y ✳ x−y ✳ ✳ ✲ ❛ ✬ ♥❣ ❝❫ ✓♣✱ t❛ ❞ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖✕ ❛ ❛ ✖✕ ❛ ✳t ✳ ✳ ✬✳ t❤❛ ✒♥❤ ❑❤✐ ❞ ✖✓ ♦ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ tr❫ ❡♥ tr♦ xz = + z2 1−z ✳ ✳ ✬ ✐ ♣❤✉✳♦✳♥❣ tr✏ ●✐❛ ✒♥❤ ♥❛ ✒② t❛ ❞ ✖✉ ♦ ✳❝ y x2 + y = Cearctg x ✳ ✳ ✒ ✓♥ t✏ ❛✮ ❈❤✓ ✉ ♥❣ ♠✐♥❤ r✕ ❛ ♥❣ ❤❫ ❡ ✓ ❝ ✈❡❝t♦ ❧❛ ✒ ❤❫ ❡ ♦ ❡ ✓ ♥❤✳ ✳ ❝❛ ✳ ♣❤✉ ✳ t❤✉❫ ✳ ❝ t✉②❫ ✳ ✬ ❛ ❝❤✉ ❚✏ ✓ ♥❤ ❞ ✖✐ ♥❤ t❤✓ ✉ ❝ ❲r♦♥s❦✐ ❝✉ ✓ ♥❣✳ ✳ ✳ ✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ❜✮ ❚✏ ✒♠ t✏ ✓ ❝❤ ♣❤❫ ❛♥ t❫ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✒♥❤ s❛✉✿ 32) {cos 2x, sin 2x, 2} (x − 2y + 1)dy − (x + y)dx = www.matheducare.com MATH-EDUCARE www.VNMATH.com HD gia’i: 2 ✓♥ t✏ ❛✮ ❍❫ ❡ ✒② ♣❤✉ ♦ ❡ ✓♥❤ ✈✏ ✒ cos 2x + sin 2x − = 0✳ ✳ ♥❛ ✳ t❤✉❫ ✳ ❝ t✉②❫ ✳ ✳ ✳ ✳ ✳ ✬ ♥❣ ❝❫ ✒ ✓♣✱ t❛ ❞ ❜✮ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ ♥❛ ✒② ❝♦ ✓ t❤❫ ❡✬ ❞ ✖✉ ❛ ✈❫ ❡ ❞❛ ✖❛ ✕ ❛ ✖✉ ♦ ✳ ♥❣ ❞ ✳❝ y = ✲ ❛ ❉ ✕ ✳t 1 u=x− , v =y+ ✱ 3 x+y x − 2y + ✳ ✳ ✬✳ t❤❛ ❦❤✐ ❞ ✖✓ ♦ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ✒♥❤ tr❫ ❡♥ tr♦ v = ✳ ✳ ✬ ✐ ♣❤✉✳♦✳♥❣ tr✏ ●✐❛ ✒♥❤ ♥❛ ✒② t❛ ❞ ✖✉ ♦ ✳❝ ❍❛② 33) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ●✐❛ HD gia’i: y = C : y + x2 y = xyy ✳ y = y y2 x2 y x −1 ✳ ✳ ✒ ✓t✱ ❣✐❛ ✬✐ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ❞ ✖❛ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ y = Cxe x y” cos y + (y )2 sin y = y y = p ⇒ y” = p dp cos y + p sin y = 1✿ dy dp dy ✭❤❛ ✒♠ t❤❡♦ t✏ ✓❝❤ ♣❤❫ ❛♥ 36) p = C cos y dy dy = sin y + C1 cos y ⇔ = dx dx sin y + C1 cos y y 1 tg + + − C1 C1 ✓♥✿ ❞ ✖✐ ❞ ✖❫ ❡ ln = x + C2 y 1 C1 + −tg + + + C1 C1 ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ HD gia’i: y✮ ✳ ✳ ✓♥ t✏ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤✳ ✳ ✳ ✒ ✓t ❝♦ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✓ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ♠ t❫ ✳ ✳ ✒ ✓ ✓ ❜✐❫ ❡ ♥ t❤✐❫ ❡♥ ❤✕ ❛ ♥❣ s❫ ♦ ❞ ✖✉ ♦ ✳ ❝ C = t❣y ✰ C1 ✳ p= ✒ ♥❣ ❧❛ ❤✕ ❛ ✒ ♠❫ ♦ ❡ ✳ t ♥❣❤✐❫ ✳ ♠✳ ✲ ❛ ✒ ♥❣✮✳ ❉ ✭❤✕ ❛ ✕ ✳t t❤❛② ✈❛ ✒♦ ✭✷✮✿ ✳ t✒ ✉ ❞ ✖✓ ♦ 2u ) v y = zx → y = z x + z dx z−1 dz = → z − ln |z| = ln |x| + C z x y y − ln | | = ln |x| + C x x ✓t ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ ❧❛ ❱✐❫ ❡ ✳✐ ✳ ✳ r❛ ❞ ✖✉ ♦ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ❝ ♥❣❤✐❫ ✳ ♠ t❫ y=C √ ✳ ✳ ✒ ✓t✿ ❞ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✖❛ ✕ ✳t ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤ HD gia’i: √1 arctg( y + x2 y = xyy ✳ ✳ ✬✳ t❤❛ ✒♥❤ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ tr♦ 35) √ = Ce u2 + 2v √ 3x−1 √ arctg( ) 3y+1 (3x − 1)2 + 2(3y + 1)2 = C1 e HD gia’i: 34) u+v u − 2v ❈♦✐ x = x(y) y + =0 2x − y ✬❛ ❧❛ ✒ ❤❛ ✒♠ ❝✉ y t❛ ❝♦ ✓✿ y = x ✳ ✳ t❤❛② ✈❛ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ www.matheducare.com MATH-EDUCARE www.VNMATH.com 1 + = ⇔ x + 2x = y : x 2x − y ✳ ✳ ✓♥ t✏ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤✳ ✒ ✓t✿ ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✳ ♠ t❫ x = Ce−2y 1 ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓✿ C (y) = y e2y ⇒ C(y) = y e2y − ye2y + e2y + C ❇✐❫ ❡ ❡♥ ❤✕ ❛ ♦ 2 1 ✳ ✳ −2y ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ x = Ce + y2 − y + ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ 2 37) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ HD gia’i: ✲ ❛ ❉ ✕ ✳t y = p✱ xy” = y + x2 ✬✳ t❤❛ ✭✶✮ tr♦ ✒♥❤✿ xp − p = x2 ✓♥ t✏ t✉②❫ ❡ ✓♥❤ ✒ ✓t✿ ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✳ ♠ t❫ ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓ → ❇✐❫ ❡ ❡♥ ❤✕ ❛ ♦ C(x) = x + C1 ❙✉② r❛✿ 38) dy = x(x + C1 ) dx ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ●✐❛ HD gia’i: ⇔p+y ✲ ❛ ❉ ✕ ✳t →y= y=0 ①❡ ✓t x3 x2 + C1 + C2 y + yy” = yy p = y (p = 0)✱ dp = y✱ dy p = Cx ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✒♥❤ t✉ ♦ ♥❣ ❞ ✖✉ ♦ ♥❣ ✈✓ ♦ ✐✿ ♣❤✉ ♦ ♥❣ tr✏ ✳ ✳ ✳ ✒ ✒♥❤ ✈❫ ❡✿ ❞ ✖✉ ❛ ♣❤✉ ♦ ♥❣ tr✏ ✒ ✓t✿ ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ◆❚◗ ❝✉ p= ⇒ C(y) = C ✱ y dp p + =1 dy y p2 + yp dp = yp dy ✓♥ t✏ ✭t✉②❫ ❡ ✓♥❤✮ ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓ ❜✐❫ ❡ ❡♥ ❤✕ ❛ ♦ y2 + C1 dy y + 2C1 2ydy y + 2C1 ⇒ = ⇒ = dx 2y dx 2y y + 2C1 ⇒ y = A1 ex + A2 x x x ✓ tr❛ ❈❤✉ ✓ ✓ ② ✿ ❱❫ ❡ ✓ ✐ (yy ) = yy ⇔ yy = C1 e ⇔ ydy = C1 e dx ⇔ y = 2C1 e + C2 ✳ ❛ ◆❤✉ ✈❫ ✳ ②✿ 39) p= ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ HD gia’i: yx = xy yey = y (y + 2xey ) ✳ ✈✓ ♦✐ ✳ ✳ ✓♥ ❞ ✒ ❜✐❫ ❡ ✖♦ ❫✬✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ✈❫ ❡✿ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ♠ t❫ y(0) = −1 x − x = y e−y y x = y (C − e−y ) y(0) = −1 ⇒ C = e −y ❱❫ ❛ ✳ ② x = y (e − e ) 40) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ HD gia’i: ✲ ❛ ❉ ✕ ✳t y = p; ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ♠ t❫ xy” = y + x p − p=1 x ✓ s❫ ♦ ✿ C = ln |x| + C1 ✳ ✳ ✬✳ t❤❛ ✒♥❤✿ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ tr♦ p = Cx ✒ ♥❣ ✓♥ t❤✐❫ ❜✐❫ ❡ ❡♥ ❤✕ ❛ www.matheducare.com MATH-EDUCARE www.VNMATH.com 10 ⇒p= dy = (ln |x| + C1 )x ⇒ y = dx (ln |x| + C1 )xdx + C2 = C1 x2 + 41) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ●✐❛ x2 x2 ln |x| − + C2 y + xy = x3 ✒ ✓t ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✳ ♥ t❫ x2 ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓✿ C(x) = (x2 − 2)e− + ε ❜✐❫ ❡ ❡♥ ❤✕ ❛ ♦ HD gia’i: ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ 42) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ x2 y = Ce− x2 y = εe− + x2 − (x2 − y)dx + xdy = ✳ ✳ ✳ ✳ ✓t ❧❛ ✒ ✓t✿ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ ✈✐❫ ❡ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✳ ✐✿ xy − y = −x ✱ ♣❤✉ ♦ ♥❣ tr✏ ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓ s✉② r❛ C = −x + ε ❡♥ ❤✕ ❛ ✓ t✿ y = Cx ❜✐❫ ❡ ♦ ❝♦ ✓ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✳ ♠ t❫ ✬ ❱❫ ❛ ❡ ♦ ♥❣ q✉❛ ✓ t ✿ y = −x + εx ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ HD gia’i: 43) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ●✐❛ HD gia’i: ✳ ✳ ✓♥ t✏ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤✿ y = εx2 − ; x ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ●✐❛ HD gia’i: ✲ ❛ ❉ ✕ ✳t ❳❡ ✓t ✳ ✈✓ ♦✐ y(1) = y = Cx2 ; C = ⇒C =− +ε x x y(1) = ⇒ ε = ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ 44) y − y= x x xy − y = y = 0, y = 2x2 − x (x + 1)(y + y ) = −y y = −y x+1 ✒ tr✏ ✒♥❤ ✈❫ ❡ z − z = x+1 ✓t✿ z = C1 (x + 1) ❜✐❫ ✓♥ t❤✐❫ ♥❤❫ ❛ ❡ ❡♥ ✳ ✳ ✒ ✓♥ ❞ ✒♥❤ ✈❫ ❡ ❞❛ ❜✐❫ ❡ ✖♦ ❫✬✐ ♣❤✉ ♦ ♥❣ tr✏ ✳ ♥❣ z = z ⇒ y = − = −y z y z ✳ ✳ ✳ ❞ ✖✉ ❛ ♣❤✉ ♦ ♥❣ ✒ ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ t❤✉❫ ❛♥ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✳ ♠ t❫ y + ✒ ♥❣ s❫ ✓ ❤✕ ❛ ♦ C1 = ln |x + 1| + ε ❱❫ ❛ ❡ ✳ ② ♥❣❤✐❫ ✳ ♠✿ z = (x + 1)(ln |x + 1| + ε) ⑦ ♥❣ ❧❛ ♥❣♦❛ ✒✐ r❛ y = ❝✉ ✒ ♥❣❤✐❫ ❡ ✳ ♠✳ ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ 45) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ HD gia’i: y= (x + 1)(ln |x + 1| + ε) 2xy + y = ✈❛ ✒ y=0 ♥❣❤✐❫ ❡ ✒ ❞✐ ✳ ♠ ❦✏ ✳✳ 1−x ✲ ✉✳❛ ♣❤✉✳♦✳♥❣ tr✏ ✒ ❉ ✒♥❤ ✈❫ ❡ ❞❛ ✳ ♥❣ y + 1 y = 2x 2x(1 − x) ✓♣ ✶ t✏ ✓♥❤ ❝❫ ❛ www.matheducare.com ✳ ✳ ✓♥ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ MATH-EDUCARE www.VNMATH.com 11 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖⑦ ❛ ❝❤♦ t✉ ♦ ♥❣ ❞ ✖✉ ♦ ♥❣ ✈✓ ♦ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ x y” + 3xy + y = ✳ ✳ ✳ ✳ ✳ ✳ ✲ ❛ ✒ ✓♥ t✏ ✓ ✒♥❤ ❊✉❧❡r ♥❫ ❡♥ t❛ ❝♦ ✓ t❤❫ ❡✬ ❞ ✖✉ ❛ ✈❫ ❡ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤ ✈✓ ♦ ✐ ❤❫ ❡ ♦ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✳ s❫ ✳ ✳ ✳ t ⑦ ✒ ✒ ✬ ✒♥❤ yt ” + 2yt + y = ✒♥❤ ❞ ✖❛ ❝❤♦ tr♦ t❤❛ ❤✕ ❛ ♥❣ ❜✕ ❛ ♥❣ ❝❛ ✓ ❝❤ ❞ ✖❛ ✕ ✖✓ ♦ ♣❤✉ ♦ ♥❣ tr✏ ✳ t x = e ❑❤✐ ❞ ✳ ✳ −t −t ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ ♥❛ ✒② ❝♦ ✓ ♥❣❤✐❫ ❡ ♠ ❧❛ ✒ y = C e + C te ❱❫ ❛ ② ♥❣❤✐❫ ❡ ✒♥❤ P❤✉ ♦ ♥❣ tr✏ ✳ ✳ ✳ ♠ ❝✉ HD gia’i: ❞ ✖⑦ ❛ ❝❤♦ ❧❛ ✒ 145) y= ln |x| C1 + C2 ✳ x x ✳ ✳ ✳ ✒ ✓♥ t✏ ✓♣ ✷ ✈✓ ✓ ❤✕ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤ t✉②❫ ❡ ✓ ♥❤ ❝❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ♥❣ ✳ s❫ a) y” − 3y + 2y = 2e2x ✳ ✳ ✳ ✲ ❛ ✒ ♥❣✳ ✓♥ t✏ ✓♣ ✷ ❦❤❫ ✒ ✓t ✈✓ ✓ ❤✕ ✒♥❤ t✉②❫ ❡ ✓♥❤ ❝❫ ❛ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✳ s❫ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❧❛ ✒✿ ✳ ♠ t❫ HD gia’i: y = C1 ex + C2 e2x + 2e2x 146) ✳ ✳ ✳ ✒ ✓♥ t✏ ✓♣ ✷ ✈✓ ✓ ❤✕ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓ ♥❤ ❝❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ♥❣ ●✐❛ ✳ s❫ a) y” + y = HD gia’i: cos2 x ✒ ✓t ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ◆❣❤✐❫ ❡ ✳ ♠ ❝✉ ✳ ✳ ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓ t❛ ❞ ♣❤❛ ✓ ♣ ❜✐❫ ❡ ❡♥ ❤✕ ❛ ♦ ✖✉ ♦ ✳❝ y = C1 cos x + C2 sin x sin x ✒ C2 (x) = C1 (x) = − ✈❛ cos x cos x ✳ ✳ ❉✉ ✒♥❣ ♣❤✉ ♦ ♥❣ ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ❱❫ ❛ ❡ ✒♥❤ ❧❛ ✒ ✳ ② ♥❣❤✐❫ ✳ ♠ ❝✉ y = C1 cos x + C2 sin x − + 147) sin x + sin x ln | | − sin x ✳ ✳ ✳ ✒ ✓♥ t✏ ✓♣ ✷ ✈✓ ✓ ❤✕ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓ ♥❤ ❝❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ♥❣ ●✐❛ ✳ s❫ y” − 2y + 2y = x + ex ✳ ✳ ✳ ✲ ❛ ✒ ♥❣✳ ✓♥ t✏ ✓♣ ✷ ❦❤❫ ✒ ✓t ✈✓ ✓ ❤✕ ✒♥❤ t✉②❫ ❡ ✓♥❤ ❝❫ ❛ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✳ s❫ ✬ ◆❣❤✐❫ ❡ ♦ ♥❣ q✉❛ ✓ t ❧❛ ✒✿ ✳ ♠ t❫ HD gia’i: y = ex (C1 cos x + C2 sin x) + (x + 1) + ex 148) ✳ ✳ ✳ ✒ ✓♥ t✏ ✓♣ ✷ ✈✓ ✓ ❤✕ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤ t✉②❫ ❡ ✓ ♥❤ ❝❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ♥❣ ✳ s❫ y” + y = cos2 x HD gia’i: ✳ ✳ ✳ ✲ ❛ ✒ ♥❣✳ ✓♥ t✏ ✓♣ ✷ ❦❤❫ ✒ ✓t ✈✓ ✓ ❤✕ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤ ❝❫ ❛ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ✳ s❫ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❧❛ ✒✿ ✳ ♠ t❫ 149) y = C1 cos x + C2 sin x + 1 − cos 2x xy” + y − y = 0, x a y1 = ✳ x ✳ ✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✒♥❤ ✳ ♠ t❫ ✓t ♠❫ ❦❤✐ ❜✐❫ ❡ ♦ ❡ ❡♥❣ ❝♦ ✓ ❞❛ ✳ t ♥❣❤✐❫ ✳ ♠ r✐❫ ✳ ♥❣ www.matheducare.com MATH-EDUCARE www.VNMATH.com 12 HD gia’i: y1 = x ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ❧❛ ✒ ♠❫ ♦ ❡ ✒♥❤✳ ❚❛ t✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ ✳ t ♥❣❤✐❫ ✳ ♠ ❝✉ ✳ ♠ r✐❫ ✳ ✳ ✳ ✳ ❚❤❛② ✈❛ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t❛ t✏ ✒♠ ❞ ✖✉ ♦ ✳❝ y2 = x ❧❛ ✒ y= 150) y2 = u(x) x ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ C1 + C2 x x ✳ ✳ ✳ ✒ ✓♥ t✏ ✓♣ ✷ ✈✓ ✓ ❤✕ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓ ♥❤ ❝❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ♥❣ ●✐❛ ✳ s❫ a) y” − 3y + 2y = 2ex HD gia’i: ✳ ✳ ✳ ✲ ❛ ✒ ♥❣✳ ◆❣❤✐❫ ✓♥ t✏ ✓♣ ✷ ❦❤❫ ✒ ✓t ✈✓ ✓ ❤✕ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤ ❝❫ ❛ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ❡ ✳ s❫ ✳♠ ✬ t❫ ♦ ♥❣ q✉❛ ✓ t ❧❛ ✒✿ y = C1 ex + C2 e2x − 2xex 151) ✳ ✳ ✳ ✒ ✓♥ t✏ ✓♣ ✷ ✈✓ ✓ ❤✕ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤ t✉②❫ ❡ ✓ ♥❤ ❝❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ♥❣ ✳ s❫ y” − y = sin x HD gia’i: ✳ ✳ ✳ ✲ ❛ ✒ ♥❣✳ ◆❣❤✐❫ ✓♥ t✏ ✓♣ ✷ ❦❤❫ ✒ ✓t ✈✓ ✓ ❤✕ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤ ❝❫ ❛ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ❡ ✳ s❫ ✳♠ ✬ t❫ ♦ ♥❣ q✉❛ ✓ t ❧❛ ✒✿ y = C1 + C2 ex + 152) 1 cos x − sin x 2 ✳ ✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✒♥❤ ✳ ♠ t❫ ✓t ♠❫ ❦❤✐ ❜✐❫ ❡ ♦ ❡ ❡♥❣ ❝♦ ✓ ❞❛ ✳ t ♥❣❤✐❫ ✳ ♠ r✐❫ ✳ ♥❣ HD gia’i: y1 = x ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤✳ ❚❛ t✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ ❧❛ ✒ ♠❫ ♦ ❡ ✳ ♠ r✐❫ ✳ t ♥❣❤✐❫ ✳ ♠ ❝✉ ✳ ✳ ✳ ✳ ✒♥❤ t❛ t✏ ✒♠ ❞ ✖✉ ♦ ❚❤❛② ✈❛ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ ✳❝ y2 = x4 ❧❛ ✒ y= 153) x2 y” − 2xy − 4y = 0, y1 = ✳ x y2 = u(x) x ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✒♥❤ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ C1 + C2 x4 x ✳ ✳ ✳ ✒ ✓♥ t✏ ✓♣ ✷ ✈✓ ✓ ❤✕ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤ t✉②❫ ❡ ✓ ♥❤ ❝❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ♥❣ ✳ s❫ y” + y = x + 2ex HD gia’i: ✳ ✳ ✳ ✲ ❛ ✒ ♥❣✳ ◆❣❤✐❫ ✓♥ t✏ ✓♣ ✷ ❦❤❫ ✒ ✓t ✈✓ ✓ ❤✕ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤ ❝❫ ❛ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ❡ ✳ s❫ ✳♠ ✬ t❫ ♦ ♥❣ q✉❛ ✓ t ❧❛ ✒✿ y = C1 cos x + C2 sin x + x + ex 154) ✳ ✳ ✳ ✒ ✓♥ t✏ ✓♣ ✷ ✈✓ ✓ ❤✕ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤ t✉②❫ ❡ ✓ ♥❤ ❝❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ♥❣ ✳ s❫ www.matheducare.com y” − y + y = x MATH-EDUCARE www.VNMATH.com 13 HD gia’i: ✳ ✳ ✳ ✲ ❛ ✒ ♥❣✳ ◆❣❤✐❫ ✓♥ t✏ ✓♣ ✷ ❦❤❫ ✒ ✓t ✈✓ ✓ ❤✕ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤ ❝❫ ❛ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ❡ ✳ s❫ ✳♠ t❫ ♦✬♥❣ q✉❛ ✓ t ❧❛ ✒✿ √ √ 3 y = e (C1 cos x + C2 sin x) + + x 2 x 155) ✳ ✳ ✳ ✒ ✓♥ t✏ ✓♣ ✷ ✈✓ ✓ ❤✕ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤ t✉②❫ ❡ ✓ ♥❤ ❝❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ♥❣ ✳ s❫ y” − 2y + y = x + ex ✳ ✳ ✳ ✲ ❛ ✒ ♥❣✳ ✓♥ t✏ ✓♣ ✷ ❦❤❫ ✒ ✓t ✈✓ ✓ ❤✕ ✒♥❤ t✉②❫ ❡ ✓♥❤ ❝❫ ❛ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✳ s❫ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❧❛ ✒✿ ✳ ♠ t❫ HD gia’i: y = C1 ex + C2 xex + + x2 ex 156) ✳ ✳ ✳ ✒ ✓♥ t✏ ✓♣ ✷ ✈✓ ✓ ❤✕ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓ ♥❤ ❝❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ♥❣ ●✐❛ ✳ s❫ y” + y = sin2 x ✳ ✳ ✳ ✲ ❛ ✒ ♥❣✳ ✓♥ t✏ ✓♣ ✷ ❦❤❫ ✒ ✓t ✈✓ ✓ ❤✕ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤ ❝❫ ❛ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ✳ s❫ ✬ ◆❣❤✐❫ ❡ ♦ ♥❣ q✉❛ ✓ t ❧❛ ✒✿ ✳ ♠ t❫ HD gia’i: y = C1 cos x + C2 sin x + 157) 1 + cos 2x ✳ ✳ ✓♥ t✏ ✓♣ ✷ s❛✉✿ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓ ♥❤ ❝❫ ❛ ●✐❛ xy” − y − y = x ✳ ✳ ✳ ✳ ✳ ✲ ❛ ✒ ✓♥ ✒♥❤ ❊✉❧❡r ♥❫ ❡♥ t❛ ❝♦ ✓ t❤❫ ❡✬ ❞ ✖✉ ❛ ✈❫ ❡ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✳ ✳ ✳ ✳ t ✒ ♥❣ ❜✕ ✒ ♥❣ ❝❛ ✓ ❤✕ ✬ t❤❛ ✒♥❤ t✏ ✓♥❤ ✈✓ ♦ ✐ ❤❫ ❡ ♦ ❛ ❛ ✓ ❝❤ ❞ ✖❛ ✕ ✖✓ ♦ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖⑦ ❛ ❝❤♦ tr♦ ✳ s❫ ✳ t x = e ❑❤✐ ❞ HD gia’i: yt ” − 2yt − y = ✳ ✳ ✒♥❤ ♥❛ ✒② ❝♦ ✓ ♥❣❤✐❫ ❡ ✒ P❤✉ ♦ ♥❣ tr✏ ✳ ♠ ❧❛ y = C1 e(1+ √ 2)t + C2 e(1− √ 2)t ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ❱❫ ❛ ❡ ✒♥❤ ❞ ✖⑦ ❛ ❝❤♦ ❧❛ ✒ ✳ ② ♥❣❤✐❫ ✳ ♠ ❝✉ y = C1 x1+ 158) √ √ + C2 x1− ✳ ✳ ✳ ✒ ✓♥ t✏ ✓♣ ✷ ✈✓ ✓ ❤✕ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤ t✉②❫ ❡ ✓ ♥❤ ❝❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ♥❣ s❛✉✿ ✳ s❫ y” − 3y + 2y = cos x ✳ ✳ ✳ ✲ ❛ ✒ ♥❣✳ ✓♥ t✏ ✓♣ ✷ ❦❤❫ ✒ ✓t ✈✓ ✓ ❤✕ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤ ❝❫ ❛ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ✳ s❫ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❧❛ ✒✿ ✳ ♠ t❫ HD gia’i: y = C1 ex + C2 e2x + 159) cos x − sin x 5 ✳ ✳ ✳ ✒ ✓♥ t✏ ✓♣ ✷ ✈✓ ✓ ❤✕ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤ t✉②❫ ❡ ✓ ♥❤ ❝❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ♥❣ s❛✉✿ ✳ s❫ www.matheducare.com y” − y = sin x + ex MATH-EDUCARE www.VNMATH.com 14 ✳ ✳ ✳ ✲ ❛ ✒ ♥❣✳ ✓♥ t✏ ✓♣ ✷ ❦❤❫ ✒ ✓t ✈✓ ✓ ❤✕ ❉ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤ ❝❫ ❛ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ♦ ✐ ❤❫ ❡ ♦ ❛ ✳ s❫ ✬ ◆❣❤✐❫ ❡ ♦ ♥❣ q✉❛ ✓ t ❧❛ ✒✿ ✳ ♠ t❫ HD gia’i: y = C1 + C2 ex + xex + 160) 1 cos x − sin x 2 z ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ ❞ ✖❫ ❡✬ ❣✐❛ x x2 y” + 4xy + (x2 + 2)y = ex ❉✉ ✒♥❣ ♣❤❡ ✓♣ ❞ ✖❫ ♦✬✐ ❤❛ ✒♠ y= tr✏ ✒♥❤ ✈✐ ♣❤❫ ❛♥✿ z z x − 2z z”x2 − 4z x + 6z ⇒ y = ; y” = x2 x3 x4 ex ✳ ✳ x ✬✳ t❤❛ ✒♥❤ tr♦ y = P❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ z” + z = e ❝♦ ✓ ♠❫ ♦ t ♥❣❤✐❫ ❡ ♠ r✐❫ ❡ ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✒ ✓t ❝♦ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✓ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖✕ ❛ ❝ tr✉ ♥❣ λ + = ⇔ λ = ±i ✳ ex ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ z = C1 cos x + C2 sin x + ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ ex sin x cos x + C2 + ❱❫ ❛ ✳ ② y = C1 x2 x 2x HD gia’i: y = 161) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ●✐❛ y” cos x + y sin x − y cos3 x = ✒ ✓♥ ❞ ❜✕ ❛ ♥❣ ♣❤❡ ✓ ♣ ❜✐❫ ❡ ✖❫ ♦✬✐ t = sin x HD gia’i: t = sin x : yx = yt tx = yt cos x y”xx = y”tt cos2 x − yt sin x ✳ ✳ t −t ✒♥❤✿ y”tt − y = → y = C1 e + C2 e = C1 esin x + C2 e− sin x ❚❤❛② ✈❛ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ 162) ✳ ✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ ❝✉ ✳ ♠ r✐❫ x x x (x + e y )dx + e y (1 − ) = y ✒ ✬ ❞ t❤♦❛ ✖✐❫ ❡✉ ❦✐❫ ❡ ✳ ♥ ②✭✵✮ ❂ ✷ ∂P ∂Q x x = = − ey , y = ∂y ∂x y y(0) = ⇒ C = HD gia’i: 163) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ✈✐ ♣❤❫ ❛♥ ●✐❛ HD gia’i: ❚P❚◗✿ x x2 + ye y = C y” + y tgx − y cos2 x = ✒ ✓♥ ❞ ❜✕ ❛ ♥❣ ♣❤❡ ✓ ♣ ❜✐❫ ❡ ✖❫ ♦✬✐ ✳ ✳ ✳ ❚✉ ♦ ♥❣ t✉ ✒✐ ✷ ✳ ❜❛ 164) ✳ ❈❤♦ ❜✐❫ ❡✬✉ t❤✓ ✉ ❝✿ ⑦ ② t✏ ✓ ❍❛ ✒♠ ❤❛ ✒♠ s❫ ♦ h(x) ✲ ❛ ❉ ✕ ✳t Q = h(x) x+y 1 − ln(x + y))dx + dy x+y x+y ✳ ✳ ✒ ✬ t❤❛ ✬❛ ✒♥❤ ✈✐ ♣❤❫ ❛♥ t♦❛ ✒♥ ♣❤❫ ❛♥ ❝✉ s❛♦ ❝❤♦ ❜✐❫ ❡✬✉ t❤✓ ✉ ❝ tr❫ ❡♥ tr♦ ♠❫ ♦ ✒♠ ✳ t ❤❛ HD gia’i: h(x) ( P = h(x) ✲ ✐❫ ✒ ✭❉ ❡✉ ❦✐❫ ❡ ✖❫ ❡✬ ✳ ♥ ①✰② ❃ ✵✮ ❞ F (x, y) ✓ ❞ ✈❛ ✒ t✏ ✒♠ ❤❛ ✒♠ s❫ ♦ ✖♦ ✓✳ ln (x + y) x+y P dx + Qdy ✒ ❧❛ ✒ ✈✐ ♣❤❫ ❛♥ t♦❛ ✒♥ ♣❤❫ ❛♥✿ ∂P ∂Q −h(x)(x + y + 1) h (x)(x + y) − h(x) = ⇔ = ∂y ∂x (x + y) (x + y)2 www.matheducare.com t = sin x MATH-EDUCARE www.VNMATH.com 15 ⇔ h (x + y) + h(x + y) = ⇔ h + h = ⇔ h(x) = e−x −x ❱❛ ✒ F (x, y) = e ln(x + y) 165) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤ ✈✐ ♣❤❫ ❛♥ ✿ ✒ ❜✕ ❛ ♥❣ ♣❤❡ ✓♣ ❞ ✖❫ ♦✬✐ HD gia’i: z = yx ⇔ y = 166) z zx−z ; y = = ; y” = x x2 ✳ ✳ ✳ t✉ ♦ ♥❣ t✉ ✒✐ ✶ ✳ ❜❛ P (x, y) = ex sin y + 2m2 x cos y; Q(x, y) = ex cos y + mx2 sin y ✳ ✒ ✓ ✬ ❛ ❤❛ ❞ ✖❫ ❡✬ P (x, y)dx + Q(x, y)dy ❧❛ ✒ ✈✐ ♣❤❫ ❛♥ t♦❛ ✒♥ ♣❤❫ ❛♥ ❝✉ ✒♠ s❫ ♦ ✓②✳ F (x, y) ♥❛ ✒♦ ❞ ✖♦ ✓ ✈❛ ✒ t✏ ✒♠ ❤❛ ✒♠ ❫ ❛ ❈❤♦ ❚✏ ✒♠ ♠ HD gia’i: 167) xy” + 2(1 − x)y + (x − 2)y = e−x ❫ ❛✬♥ ❤❛ ✒♠ z = yx ∂P ∂Q = ⇔ 2x sin y(m2 + m) = ∂y ∂x ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ●✐❛ x2 y” + 2xy + ❈❤♦ ✳♥ y =0 x2 m = 0V m = −1 ✒ ✓♥ ❞ ❜✕ ❛ ♥❣ ♣❤❡ ✓ ♣ ❜✐❫ ❡ ✖❫ ♦✬✐ x= t HD gia’i: 168) ❚✏ ✒♠ ❤❛ ✒♠ ❧❛ ✒ ✈✐ ♣❤❫ ❛♥ t♦❛ ✒♥ µ(x2 + y ) µ(x2 + y ) (x − y)dx + (x + y)dy ✒ ✬ ❛ ♠❫ ♣❤❫ ❛♥ ❝✉ ♦ ✒♠ F (x, y) ♥❛ ✓ ✳ ❚✏ ✒♠ ❤❛ ✒♠ F (x, y) ✳ t ❤❛ √ √✒♦ ❞✖♦ ✓✉ ❜✐❫ ✓t µ(1, 1) = 0; µ( ♥❫ ❡ ❡ 2, 2) = ln s❛♦ ❝❤♦ HD gia’i: P (x, y) = h(x2 + y )(x − y); Q(x, y) = h(x2 + y )(x + y) ∂P ∂Q ✲ ❫ ✒ ✬ ✐ ❝♦ ❉ ❡✬ h(x − y)dx + h(x + y)dy ❧❛ ✒ ✈✐ ♣❤❫ ❛♥ t♦❛ ✒♥ ♣❤❫ ❛♥ t❛ ♣❤❛ ✓✿ = ∂y ∂x 2 ✲ ❛ ❉ ✕ t t = x + y ⇒ h 2y(x − y) − h = h 2y(x + y) + h ✳ t t C1 C1 2 ⇒h= ⇔ −ht (x + y ) = h ⇔ ht t = h ⇒ h = t x + y2 y x x−0 y C1 x+y ⇒ F (x, y) = C1 ln(x2 + y ) + C2 dx + C dy = C1 arctg + 2 2 x + x + y 2 √ √ π F (1, 1) = 0; F ( 2, 2) = ln ❈❤♦✿ C1 = 2; C2 = −( + ln 2) 169) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤ x2 y” + xy + y = x ✒ ✓♥ ❜✕ ❛ ♥❣ ♣❤❡ ✓♣ ❞ ✖❫ ♦✬✐ ❜✐❫ ❡ x = et 1 yx = yt ; y”xx = (y”tt − yt ) x x ✳ ✳ t ❚❤❛② ✈❛ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ y”tt + y = e ✒ ✓t✿ y = C1 cos t + C2 sin t ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✳ ♠ t❫ t ❚✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ ❞❛ ✳ ♠ r✐❫ ✳ ♥❣✿ y + Ae ; A = x ✬ ❱❫ ❛ ❡ ♦ ♥❣ q✉❛ ✓ t✿ y = C1 cos (ln x) + C2 sin (ln x) + ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ HD gia’i: x = et t❛ ❝♦ ✓✿ www.matheducare.com MATH-EDUCARE www.VNMATH.com 16 170) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤ ✈✐ ♣❤❫ ❛♥✿ xy” − (x + 1)y − 2(x − 1)y + x2 = y1 = eαx ✳ ✳ ✳ ✳ ✳ ✒ ✒ ✓t t✉ ♦ ♥❣ ✉ ✓t r✕ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✓ ♥❣ ❝♦ ✓ ♠❫ ♦ ❡ ❡♥❣ ❜✐❫ ❡ ❛ ♥❣ ♣❤✉ ♦ ♥❣ tr✏ ✳ t ♥❣❤✐❫ ✳ ♠ r✐❫ ✳ ✒ ✓ ❝❫ ✒ ✈✓ ♦✐ ❧❛ ✒ ❤✕ ❛ ♥❣ s❫ ♦ ❛♥ ①❛ ✓❝ ❞ ✖✐ ♥❤✳ ✳ α HD gia’i: ❚❤❛② ♥❣❤✐❫ ❡ ✳♠ ✳ ✳ ✳ ✳ ✒ ✓t ❞ ✒♥❤ r❫ ♦✐ ❞ ✖✒ ♦ ❫♥❣ ♥❤❫ ❛ ✖✉ ♦ ✈❛ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ ✳❝ y1 = eαx ✲ ✉✳❛ ♣❤✉✳♦✳♥❣ tr✏ ✒ ❉ ✒♥❤ ✈❫ ❡ ❞❛ ✳ ♥❣✿ α=2 2(x − 1) x+1 y − y = −x; x = x x y” − 2(x − 1) x+1 ; q(x) = − ; f (x) = −x x x x+1 dx 2x x e−4x e ♥❣❤✐❫ ❡ ❡♥❣✿ y2 = e dx = − (3x + 1)e−x ✳ ♠ r✐❫ p(x) = − ❚✏ ✒♠ ✒ ✓t✿ ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ❙✉② r❛ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✳ ♠ t❫ y = C1 e2x + C2 (3x + 1)e−x ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓✿ ❇✐❫ ❡ ❡♥ ❤✕ ❛ ♦ C1 = (6x + 5)e−2x 36 → C2 = ex C1 = − (3x + 1)e−2x C2 = ex ⇒ ◆❚◗✳ 171) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ✈✐ ♣❤❫ ❛♥ ●✐❛ HD gia’i: x = et ✱ ✒ ✓♥ ❜✕ ❛ ♥❣ ♣❤❡ ✓♣ ❞ ✖❫ ♦✬✐ ❜✐❫ ❡ x = et ✳ 1 yx = yt , y”xx = (y”tt − yt ) x x t❤❛ ✒♥❤✿ y”tt − 5yt + 6y = ⇒ ◆❚◗✿ y = C1 x + C2 x t❛ ❝♦ ✓✿ ✳ ✳ ✬✳ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ tr♦ 172) x2 y − 4xy + 6y = ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ✈✐ ♣❤❫ ❛♥✿ ●✐❛ y” − (2ex + 1)y + e2x y = e3x x ✓♥ t = e ✳ ❜✐❫ ❡ ✒ ❜✕ ❛ ♥❣ ♣❤❡ ✓♣ ❞ ✖❫ ♦✬✐ ✲ ♦ ✓♥ t = ex ⇒ y = y ex , y”xx = y”tt e2x + y ex ❉ ❫✬✐ ❜✐❫ ❡ x t t ✳ ✳ ✒♥❤✿ y”tt − 2yt + y = t ❚❤❛② ✈❛ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ ✒ ✓t✿ y = et (C1 t + C2 ) ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✳ ♠ t❫ 3 ❚✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ ❞❛ ✳ ♠ r✐❫ ✳ ♥❣ y = At + Bt + Ct + D → y = t + 6t + x = ee (C1 ex + C2 ) + e3x + 6e2x + 18ex + 24 HD gia’i: y 173) α ✭ ✳ ❱✓ ♦✐ ✲ ✉✳❛ ♣❤✉✳♦✳♥❣ tr✏ ✒ ❉ ✒♥❤ ✈❫ ❡✿ p(x) = ✓t q✉❛ ✬ ❑❫ ❡ ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ✈✐ ♣❤❫ ❛♥✿ ●✐❛ ✳ ✳ ✓t ♠❫ ✬ ❛ ♣❤✉ ♦ ♥❣ ❜✐❫ ❡ ♦ ❡ ❡♥❣ ❝✉ ✳ t ♥❣❤✐❫ ✳ ♠ r✐❫ HD gia’i: 18t + 24 (x − 1)y” − xy + y = (x − 1)2 e2x ✳ ✳ ✳ αx ✒ ✓t t✉ ♦ ♥❣ ✉ tr✏ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✓ ♥❣ ❝♦ ✓ ❞❛ ✳ ♥❣ y = e ✒ ❝❫ ❛♥ ①❛ ✓❝ ❞ ✖✐ ✳♥❤✮✳ y” − x y + y = (x − 1)e2x x−1 x−1 x ; q(x) = ; f (x) = (x − 1)e2x x−1 x−1 ✳ ✳ ✳ ✒ ✓t t✉✳♦✳♥❣ ✉ ✒ ✓t s✉② r❛ ✈❛ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✓ ♥❣ r❫ ♦✐ ❞ ✖✒ ♦ ❫♥❣ ♥❤❫ ❛ x dx x −2x x−1 ❚✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ y2 = e e e dx = −x ✳ ♠ r✐❫ ❚❤❛② y1 = eαx www.matheducare.com α=1 MATH-EDUCARE www.VNMATH.com ⇒ ◆❚◗✿ 17 y = C1 ex + C2 (−x) ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓✿ ❇✐❫ ❡ ❡♥ ❤✕ ❛ ♦ x C1 = xe C2 = e2x ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ 174) x y = ( − 1)e2x + K1 ex − K2 x ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤ ✈✐ ♣❤❫ ❛♥✿ HD gia’i: C1 = xex − ex + K1 → C2 = e2x + K2 ✲ ✉✳❛ ♣❤✉✳♦✳♥❣ tr✏ ✒ ✒♥❤ ✈❫ ❡✿ ❉ x2 (x + 1)y” = 2y y − ✓t ♠❫ ❜✐❫ ❡ ♦ ❡ ✳ t ♥❣❤✐❫ ✳♠ y1 = + ✳ x y = 0; p(x) = 0; f (x) = + 1) x2 (x ❚✏ ✒♠ ◆❘ ❞❛ ✳ ♥❣ x2 1 e− 0dx dx = (1 + )(x − ln |x + 1| − ) (x + 1) x 1+x x+1 ln(x + 1)2 − =x+1− x x y2 = (1 + ) x ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ② ❂ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ 175) 1 x+1 C1 (1 + ) + C2 (x − − + ln(x + 1)2 + 1) x x x ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ✈✐ ♣❤❫ ❛♥ ●✐❛ (x2 + 1)y” − 2y = ✳ ✓✉ ❜✐❫ ✓t ♠❫ ✬ ♥❫ ❡ ❡ ♦ ❡ ✓ ❝♦ ✓ ❞❛ ✖❛ t❤✓ ✉ ❝✳ ✳ t ♥❣❤✐❫ ✳ ♠ ❝✉❛ ♥♦ ✳ ♥❣ ❞ HD gia’i: ⑦ ✓② ❉❫ ❡ t❤❫ ❛ y1 = x2 + ✬ ❛ ✭✶✮✳ ❧❛ ✒ ♠❫ ♦ ❡ ❡♥❣ ❝✉ ✳ t ♥❣❤✐❫ ✳ ♠ r✐❫ Nghiˆe.m th´ u hai: y2 = y1 ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ 176) ✲ ❛ ❉ ✕ ✳t p(x)dx dx = (x2 + 1) xy” + 2y − xy = ex z = xy ⇒ z = y + xy ; z = 2y + xy y = Axex → A = z 1 x x y = = (C1 + C2 e + xe ) x x xn+1 n=0 n! xf (x) − (x + 1)f (x) = 0✳ ✳ ✒ ✬ r✕ ❈❤✓ ✉ ♥❣ t♦ ❛ ♥❣ ❤❛ ✒♠✿ ✳ ✳ ✬ ❧❛ ✒ ♥❣❤✐❫ ❡ ✒♥❤ ✳ ♠ ❝✉❛ ♣❤✉ ♦ ♥❣ tr✏ z = xy ✳ ✳ ✳ ✳ ❚❤❛② ✈❛ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ◆❣❤✐❫ ❡ ❡♥❣ ❞❛ ✳ ♠ r✐❫ ✳ ♥❣✿ 177) dx + 1)2 ✒ ❜✕ ❛ ♥❣ ♣❤❡ ✓♣ ❞ ✖❫ ♦✬✐ ❤❛ ✒♠ z − z = ex → NTQ z = C1 + C2 ex ❱❫ ❛ ✳ ②✿ (x2 x = (x2 + 1)( + arctgx) x +1 x y = C1 (x2 + 1) + C2 (x2 + 1)( + arctgx) x +1 ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ✈✐ ♣❤❫ ❛♥ ●✐❛ HD gia’i: − e y12 ∞ f (x) = www.matheducare.com MATH-EDUCARE www.VNMATH.com 18 HD gia’i: ✳ ⑦✐ ✓t ❉✬❆❧❡♠❜❡rt ❞ ✬ ❝❤✉❫ ♦ ❉✉ ✒♥❣ t✏ ✓♥❤ ❝❤❫ ❛ ✖❫ ❡✬ ❝❤✓ ✉ ♥❣ t♦ xn+1 ✳ ①❛ ✓❝ ❞ ✖✳✐♥❤ ✈✓ ♦ ✐ ♠♦ ✳ ✐ x✳ n=0 n! ∞ xn ✳ ✳ = xex ✉ ❛✿ f (x) = x ❍♦ ♥ ♥⑦ n=0 n! ⇒ xf (x) − (x + 1)f (x) = x(x + 1)ex − (x + 1)xex = 0, ∀x ✳ ◆❤✉ ✈❫ ❛ ✒♠ ✳ ② ❤❛ 178) xn+1 n=0 n! ∞ ✳ ❤❫ ♦ ♦ ✐ ♠♦ ✳✐ ✳ ✐ t✉ ✳ ✈✓ x ∞ f (x) = ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ●✐❛ ✳ ✒ ✬ ✐ ❝❤✓ ❞ ✖✐❫ ❡✉ ♣❤❛ ✉ ♥❣ ♠✐♥❤✳ x(x2 + 6)y” − 4(x2 + 3)y + 6xy = ✳ ✒ ✓t r✕ ❜✐❫ ❡ ❛ ♥❣ ♥♦ ✓ ❝♦ ✓ ♥❣❤✐❫ ❡ ✖❛ t❤✓ ✉ ❝✳ ✳ ♠ ❞❛ ✳ ♥❣ ❞ ✳ ✳ ❚❛ t✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ ❞✉ ♦ ✓ ✐ ❞❛ ✳ ♠ r✐❫ ✳ ♥❣ y1 = Ax + Bx + C +3) − − 4(x dx ✳ x(x+6) dx ♥❣❤✐❫ ❡ ❡♥❣ t❤✓ ✉ ❤❛✐✿ y2 = y1 e ✳ ♠ r✐❫ y1 √ x2 (x2 + 6) x 2x √ = (x2 + 2) 2arctg dx = (x + 2)(x + + (x2 + 2)2 (x2 + 2) HD gia’i: ❱❫ ❛ ✳ ② ◆❚◗✿ 179) √ x y = C1 (x + 2) + C2 [x + 4x + 2(x + 2)arctg √ ] ⇒ y1 = x2 + ) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ ●✐❛ ♥♦ ✓ ❝♦ ✓ ❤❛✐ (2x + 1)y” + (2x − 1)y − 2y = x2 + x ❜✐❫❡✓t x2 + 4x − x2 + ♥❣❤✐❫ ❡ ❡♥❣ y1 = ; y2 = ✳ ✳ ♠ r✐❫ 2 ✒ r✕ ❛ ♥❣ ✳ ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✬❛ ❚✒ ✉ ❤❛✐ ♥❣❤✐❫ ❡ ❡♥❣ y1 , y2 ❝✉ ✒♥❤ t❛ s✉② r❛ ♥❣❤✐❫ ❡ ❡♥❣ ❝✉ ✳ ♠ r✐❫ ✳ ♠ r✐❫ ✳ ✳ ✒ ✓t ❧❛ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✒ y1 = y1 − y2 = 2x − ✳ ❤❛✐✿ ❙✉② r❛ ♥❣❤✐❫ ❡ ♠ t❤✓ ✉ ✳ HD gia’i: − e y1 y2 = y1 = 2(x − 1) 2x−1 e− 2x+1 dx dx (2x − 1) −x (2x + 1)e−x (2x + 1)e dx = + (2x − 1)[− (2x − 1)2 (2x − 1)2 p(x)dx dx = (2x − 1) e−x (1 − 2x) dx] 2x − = −e−x ❙✉② r❛ ◆❚◗✿ y = C1 (2x − 1) + C2 e−x ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ ❜❛♥ ❞ ✖✒ ❛ ❫✉✿ ❱❛ ✒ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✳ ♠ t❫ 180) x2 + 2 ✳ ✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ α s❛♦ ❝❤♦ y = eαx ❧❛ ✒ ♠❫ ♦ ❡ ❡♥❣ ❝✉ ✳ t ♥❣❤✐❫ ✳ ♠ r✐❫ ✳ ✳ ✬ ✓②✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ y” + 4xy + (4x + 2)y = 0✳ ❚✏✒♠ ♥❣❤✐❫❡✳♠ t❫ ♦ ♥❣ q✉❛ ✓ t ❝✉ ✒♥❤ ❫ ❛ ✒ ✓ ❳❛ ✓❝ ❞ ✖✐ ❛ ♥❣ s❫ ♦ ✳♥❤ ❤✕ HD gia’i: ❚❛ −x2 r✐❫ ❡♥❣ y1 = e ✳ ✳ t✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ ❞✉ ♦ ✓ ✐ ❞❛ ✳ ♠ r✐❫ ✳ ♥❣ ✳ ◆❣❤✐❫ ❡ ✉ ❤❛✐✿ ✳ ♠ t❤✓ ❱❫ ❛ ✳ ② ◆❚◗✿ 181) y = C1 (2x − 1) + C2 e−x + y = eαx − P (x)dx e dx = e−x y12 + C2 xe−x ✳ y2 = y1 y = C1 e−x ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✒♥❤✿ ✳ ♣❤✉ ♦ ♥❣ tr✏ ✳ ✳ t❤❛② ✈❛ ✒♦ ❞ ✖✉ ♦ ✳❝ e2x e− dx = 3x − y dt dy = 4x − y dt www.matheducare.com 4xdx α = −1 dx = xe−x ✳ ✈✐ ♣❤❫ ❛♥✿ ✈❛ ✒ ♥❣❤✐❫ ❡ ✳♠ MATH-EDUCARE www.VNMATH.com 19 3−λ −1 = (λ − 1)2 = ⇔ λ = ✭❜❫ ♦ ✳ ✐ ✷✮ −1 − λ a = 3a − c a + b = 3b − d t (at + b)e x ✳ ✳ ✒ ✓t ❞ t❤❛② ✈❛ ✒♦ ❤❫ ❡ ♦✐ ❞ ✖✒ ♦ ❫♥❣ ♥❤❫ ❛ ✖✉ ♦ = ❚✏ ✒♠ ♥❣❤✐❫ ❡ ✳ r❫ ✳ ♠ ❞❛ ✳ ♥❣ ✳ ❝✿ (ct + d)et y c = 4a − c c + d = 4b − d ❈❤♦ a = C1 , b = C2 ⇒ c = 2C1 , d = 2C2 − C1 x = (C1 t + C2 )et ✬ ❱❫ ❛ ❡ ♦ ♥❣ q✉❛ ✓ t✿ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ y = (2C1 t + 2C2 − C1 )et HD gia’i: 182) ✳ ✳ ✳ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖✕ ❛ ✳ ❝ tr✉ ♥❣ ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✒♥❤✿ ✳ ♣❤✉ ♦ ♥❣ tr✏ dx = 2x + y dt dy = 4y − x dt ✳ ✳ ✳ ✳ ✳ ✳ ❚✉ ♦ ♥❣ t✉ ✒♥❤ ❞ ✖✕ ❛ ✓ ♥❣❤✐❫ ❡ ♦ ✒✐ ✶✮✱ ♣❤✉ ♦ ♥❣ tr✏ ✳ ❝ tr✉ ♥❣ ❝♦ ✳ ♠ λ = ✭❜❫ ✳ ✐ ✷✮ ✳ ❜❛ (at + b)e3t ❚✏ ✒♠ ♥❣❤✐❫ ❡ ⇒ a = C1 , c = C1 , b = C2 , d = C1 + C2 3t ✳ ♠ ❞❛ ✳ ♥❣ HD gia’i: (ct + d)e ❱❫ ❛ ✳ ② ◆❚◗✿ 183) x = (C1 t + C2 )e3t y = (2C1 t + C1 + C2 )e3t ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✒♥❤✿ ✳ ♣❤✉ ♦ ♥❣ tr✏ HD gia’i: − λ −2 −1 −1 − λ = ⇔ λ(λ2 − λ − 2) = −1 − λ − λi −2 −1 P1i −1 − λi P2i = −1 − λi P3i ✳ ✳ ✳ ✒♥❤ ❞ ✖✕ ❛ P❤✉ ♦ ♥❣ tr✏ ✳ ❝ tr✉ ♥❣ ⇔ λ1 = 0, λ2 = −1, λ3 = ✳ ✓❝ ❱✓ ♦ ✐ ❝❛ dx = x − 2y − z dt dy =y−x+z dt dz = x − z dt λi ; i = 1, 2, ✬ ✐ ❤❫ ❣✐❛ ❡ ✳✿ ✳ ✳ ✳ ✳ ✳ ✬ ✲ ❫ ✓ ♥❣✳ ❚✒ ✉ ❞ ✖✓ ♦ s✉② r❛ ❤❫ ❡ ❡ ♥✿ ❉ ❡✬ t✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ t✉ ♦ ♥❣ ✉ ✳ ♠ r✐❫ ✳ ♥❣❤✐❫ ✳ ♠ ❝♦ ❜❛ −t −t 2t x1 = 1, y1 = 0, z1 = 1; x2 = 0, y2 = e , z2 = −2e ; x3 = 3e , y3 = −2e−2t , z3 = e2t ✳ 2t x = C1 + 3C3 e ❱❫ ❛ ❡ ❡ ♦✬♥❣ q✉❛ ✓ t✿ y = C2 e−t − 2C3 e2t ✳ ② ❤❫ ✳ ♥❣❤✐❫ ✳ ♠ t❫ z = C − 2C e−t + C e2t 184) ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✒♥❤✿ ✳ ♣❤✉ ♦ ♥❣ tr✏ HD gia’i: dx − 5x − 3x = dt dy + 3x + y = dt ✳ ✳ ✳ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖✕ ❛ ✳ ❝ tr✉ ♥❣ 5−λ =0⇔λ=2 −3 −λ − www.matheducare.com ✭❜❫ ♦ ✳ ✐ ✷✮ MATH-EDUCARE www.VNMATH.com 20 ⇒ at + b 2t e ct + d ♥❣❤✐❫ ❡ ✓ ❞❛ ✳ ♠ ❝♦ ✳ ♥❣ ❈❤♦ ❱❫ ❛ ✳② t❤❛② ✈❛ ✒♦ ❤❫ ❡ ✳ a − 3b = 3d ⇒ a+c=0 c + 3b = −3d C1 a = C1 , b = C2 ⇒ c = −C1 , d = − C2 x = (C1 t + C2 )e2t ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t✿ C ✳ ♠ t❫ y = (−C1 t + − C2 )e2t 185) dx = 2x − 3y dt dy = x − 2y + sin t dt ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✒♥❤✿ ✳ ♣❤✉ ♦ ♥❣ tr✏ HD gia’i: ✳ ✳ ✳ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖✕ ❛ ✓ ❤❛✐ ♥❣❤✐❫ ❡ ✳ ❝ tr✉ ♥❣ ❝♦ ✳♠ ✰ λ1 = −1 ✰ λ2 = 0 ✬ ✐ ❤❫ ❣✐❛ ❡ ✳✿ ✬ ✐ ❤❫ ❣✐❛ ❡ ✳✿ = −3 −3 −3 −1 γ21 = γ22 λ1,2 = ±1 γ11 ⇒ γ11 = γ12 = γ12 ⇒ γ21 = 3; γ22 = ✳ ✬ ✳ ✒ ✓t t✉✳♦✳♥❣ ✉ ✬ ❛ ❤❫ ♥ ❝✉ ❡ ❛♥ ♥❤❫ ❛ ✓ ♥❣ ❧❛ ✒✿ ❍❫ ❡ ❡ ✳ t❤✉❫ ✳ ♥❣❤✐❫ ✳ ♠ ❝♦ ❜❛ x1 = e−t y1 = e−t ✒ ✓t✿ ✬ ❛ ❤❫ ❱❫ ❛ ❡ ❛♥ ♥❤❫ ❛ ✳ ② ◆❚◗ ❝✉ ✳ t❤✉❫ ; x2 = 3et y2 = et x(t) = C1 e−t + 3C2 et y(t) = C1 e−t + C2 et ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓✿ ❇✐❫ ❡ ❡♥ ❤✕ ❛ ♦ C1 e−t + 3C2 et = C1 e−t + C2 et = sin t ❱❫ ❛ ✳ ② ◆❚◗✿ 186) ⇒ C1 = 3et sin t C2 = e−t sin t C1 (t) = et (sin t − cos t) ⇒ C2 (t) = − e−t (sin t + cos t) x(t) = C1 e−t + 3C2 et − cos t y(t) = C1 e−t + C2 et + sin t − cos t ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✒♥❤✿ ✳ ♣❤✉ ♦ ♥❣ tr✏ dx = 2x − y + z dt dy = x + 2y − z dt dz = x − y + 2z dt ✳ HD gia’i: P❤✉✳♦✳♥❣ tr✏✒♥❤ ❞✖✕ ❛ ✓ ✸ ♥❣❤✐❫ ❡ ✳ ❝ tr✉ ♥❣✿ (λ − 1)(λ − 2)(λ − 3) = ❝♦ ✳ ♠ λ1 = 1; λ2 = 2; λ3 = 3✳ − λi −1 P1i ✳ ✳ ✬ ✐ ❤❫ − λ −1 P 0 = ✉ ✓ ♥❣ ✈✓ ♦ ✐ λi ❣✐❛ ❡ ✿ i 2i ✳ −1 − λi P3i 2t 3t 1 e e ✳ ✬ t 2t ✲ ✉✳♦✳❝ 1 ; 1 ; 0✳ ❙✉② r❛ ❤❫ e e 0 ❉ ❡ ♥❣❤✐❫ ❡ ♠ ❝♦ ❜❛ ♥ ; ; ✳ ✳ ✳ t 2t 1 e e e3t www.matheducare.com MATH-EDUCARE www.VNMATH.com ❱❫ ❛ ✳ ② ◆❚◗✿ 187) 2t 3t x = C2 e + C3 e y = C1 et + C2 e2t z = C et + C e2t + C e3t ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✒♥❤✿ ✳ ♣❤✉ ♦ ♥❣ tr✏ dx = y − cos t dt dy = 2x + y dt ✳ ✳ ✬✳✿ ❉✉ ✒♥❣ ♣❤✉ ♦ ♥❣ ♣❤❛ ✓ ♣ ❦❤✉ HD gia’i: y” = 2x + y 21 ✳ ✳ ✳ ✓② ❞ ▲❫ ❛ ✖❛ ✒♠ t❤❡♦ t ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t❤✓ ✉ ❤❛✐✿ ✳ ♦ ❤❛ ✳ ✳ ✳ ✲ ❫ ✒ ✒♥❤ ❞ ✖✒ ❫ ❛✉✱ ❞ ✖✉ ❛ ✈❫ ❡✿ y” = 2(y − cos t) + y ⇔ y” − y − 2y = −10 cos t ❉ ❡✬ ✓ ② ♣❤✉ ♦ ♥❣ tr✏ ✳ ✳ ✳ ✳ ✲ ✓♥ t✏ ✓♣ ❤❛✐✱ ❣✐❛ ✬ ✐ r❛ ❞ ❉❛ ❫② ❧❛ ✒ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t✉②❫ ❡ ✓♥❤ ❝❫ ❛ ✖✉ ♦ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ❝ ♥❣❤✐❫ ✳ ♠ t❫ y = C1 e2t + C2 e−t + cos t + sin t x = C1 e2t − C2 e−t − cos t − sin t 2t −t x = A1 e + A2 e − cos t − sin t y = 2A1 e2t − A2 e−t + cos t + sin t ✳ ✳ ✒♥❤ ❞ ✖✒ ❫ ❛✉✿ ❚❤❛② ✈❛ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ ❱❫ ❛ ✳ ② ◆❚◗✿ 188) ✳ ✳ ✬ ✐ ❤❫ ✒♥❤✿ ●✐❛ ❡ ✳ ♣❤✉ ♦ ♥❣ tr✏ HD gia’i: ✳ ✳ ✳ ◆❣❤✐❫ ❡ ✒♥❤ ❞ ✖❛ ✕ ✳ ♠ ♣❤✉ ♦ ♥❣ tr✏ ✳ ❝ tr✉ ♥❣ ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓✿ ❇✐❫ ❡ ❡♥ ❤✕ ❛ ♦ 189) C1 = e4x C2 = ex ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✒♥❤✿ ✳ ♣❤✉ ♦ ♥❣ tr✏ HD gia’i: y = 3y + 2z + 4e5x z = y + 2z → ◆❚◗ y = C1 ex + C2 e−x z = C1 ex + 3C2 e−x y ∗ = xex z ∗ = (x + 1)ex y = C1 ex + C2 e−x + xex z = C1 ex + 3C2 e−x + (x + 1)ex ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✒♥❤✿ ✳ ♣❤✉ ♦ ♥❣ tr✏ y = C1 ex + 2C2 e4x z = −C1 ex + C2 e4x y = C1 ex + 2C2 e4x + 3e5x z = −C1 ex + C2 e4x + e5x ✒ ✓t✿ ✬ ❛ ❤❫ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ❡ ❛♥ ♥❤❫ ❛ ✳ ♠ t❫ ✳ t❤✉❫ ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ ◆❚◗✿ y = 2y − z + 2ex z = 3y − 2z + 4ex ✒ ✓t✿ ✬ ❛ ❤❫ ◆❣❤✐❫ ❡ ❡♥❣ ❝✉ ❡ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ✳ ♠ r✐❫ ✳ ❦❤❫ 190) λ1 = 1; λ2 = 4❀ y = 2y − 4z + 4e−2x z = 2y − 2z www.matheducare.com MATH-EDUCARE www.VNMATH.com 22 HD gia’i: 191) ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ♠ t❫ ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✳ ♣❤✉ ♦ ♥❣ HD gia’i: y = C1 (cos 2x − sin 2x) + C2 (cos 2x + sin 2x) z = C1 cos 2x + C2 sin 2x + e−2x dy dx tr✏ ✒♥❤✿ dz dx ✳ ✳ ✳ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖❛ ✕ ✳ ❝ tr✉ ♥❣ =y+z = z − 4y λ2 − 2λ + = ❑❤✐ ❞ ✖✓ ♦ λ1 = + 2i, λ2 = − 2i ✒ ✓t ❝♦ ❍❫ ❡ ❛♥ ♥❤❫ ❛ ✓ ♥❣❤✐❫ ❡ ✳ t❤✉❫ ✳♠ y = ex (C1 cos 2x + C2 sin 2x), z = 2ex (C2 cos 2x − C1 sin 2x) 192) ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✳ ♣❤✉ ♦ ♥❣ HD gia’i: dy dx tr✏ ✒♥❤✿ dz dx ✳ ✳ ✳ ✒♥❤ ❞ ✖❛ ✕ P❤✉ ♦ ♥❣ tr✏ ✳ ❝ tr✉ ♥❣ = y + z + ex = z − 4y λ2 − 2λ + = ❑❤✐ ❞ ✖✓ ♦ λ1 = + 2i, λ2 = − 2i ✒ ✓t ❝♦ ❍❫ ❡ ❛♥ ♥❤❫ ❛ ✓ ♥❣❤✐❫ ❡ ✳ t❤✉❫ ✳♠ y = ex (C1 cos 2x + C2 sin 2x), z = 2ex (C2 cos 2x − C1 sin 2x) ✒ ✓t ✬ ❛ ❤❫ ❱❛ ✒ ♥❣❤✐❫ ❡ ❡ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ✳ ♠ ❝✉ ✳ ❦❤❫ y = ex (C1 cos 2x + C2 sin 2x), z = 2ex (C2 cos 2x − C1 sin 2x) − ex 193) ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✳ ♣❤✉ ♦ ♥❣ dy dx tr✏ ✒♥❤✿ dz dx ✳ ✳ ✳ ✒♥❤ ❞ ✖❛ ✕ P❤✉ ♦ ♥❣ tr✏ ✳ ❝ tr✉ ♥❣ ✒ ✓t ❝♦ ❍❫ ❡ ❛♥ ♥❤❫ ❛ ✓ ♥❣❤✐❫ ❡ ✳ t❤✉❫ ✳♠ HD gia’i: = 2y − z = 2z + 4y + e2x λ2 − 4λ + = ❑❤✐ ❞ ✖✓ ♦ λ1 = + 2i, λ2 = − 2i y = e2x (C1 cos 2x + C2 sin 2x), z = −2e2x (C2 cos 2x − C1 sin 2x) ✒ ✓t ✬ ❛ ❤❫ ❱❛ ✒ ♥❣❤✐❫ ❡ ❡ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ✳ ♠ ❝✉ ✳ ❦❤❫ y = e2x (C1 cos 2x + C2 sin 2x) − e2x , z = −2e2x (C2 cos 2x − C1 sin 2x) 194) ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✳ ♣❤✉ ♦ ♥❣ dy dx tr✏ ✒♥❤✿ dz dx = 2y + z + ex = z − 4y HD gia’i: ✳ ✳ ✳ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖✕ ❛ ✳ ❝ tr✉ ♥❣ λ2 − 2λ + = ❑❤✐ ❞ ✖✓ ♦ λ1 = + 2i, www.matheducare.com MATH-EDUCARE www.VNMATH.com λ2 = − 2i 23 ✒ ✓t ❝♦ ❍❫ ❡ ❛♥ ♥❤❫ ❛ ✓ ♥❣❤✐❫ ❡ ✳ t❤✉❫ ✳♠ y = ex (C1 cos 2x + C2 sin 2x), z = 2ex (C2 cos 2x − C1 sin 2x) ✒ ✓t ✬ ❛ ❤❫ ❱❛ ✒ ♥❣❤✐❫ ❡ ❡ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ✳ ♠ ❝✉ ✳ ❦❤❫ y = ex (C1 cos 2x + C2 sin 2x), z = 2ex (C2 cos 2x − C1 sin 2x) − ex 195) ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✒♥❤✿ ✳ ♣❤✉ ♦ ♥❣ tr✏ HD gia’i: dx dt dy dt = x + 2y = x − sin t ✳ ✳ ✒ ✓t✿ ✬ ❛ ❤❫ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ❡ ✳ ♠ t❫ ✳ ♣❤✉ ♦ ♥❣ tr✏ x y = C1 e−t + 2C2 e2t = −C1 et + C2 e2t ✳ ✳ ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓ ❞ ❇✐❫ ❡ ❡♥ ❤✕ ❛ ♦ ✖❫ ❡✬ ❞ ✖✉ ♦ ❡ ✳ ❝ ♥❣❤✐❫ ✳ ♠✿ x y 196) = C1 e−t + 2C2 e2t + 38 sin t + 43 cos t = −C1 et + C2 e2t + cos t − sin t ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✒♥❤✿ ✳ ♣❤✉ ♦ ♥❣ tr✏ dx dt dy dt = x − 2y + et = x + 4y + e2t ✳ ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ ❞ ✖❛ ✕ ◆❣❤✐❫ ❡ ✳ ❝ tr✉ ♥❣✿ r1 = 2; r2 ✳ ♠ ❝✉ x = 2C1 e2t + C2 e3t ✳ ✳ ✒ ✓t ❧❛ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✒✿ ❤❫ ❡ ✳ ♣❤✉ ♦ ♥❣ tr✏ y = −C1 e2t − C2 e3t HD gia’i: = 3❀ ✳ ✳ ✳ ✬❛ t✒ ✉ ❞ ✖✓ ♦ ❞ ✖✉ ♦ ✳ ❝ ◆❚◗ ❝✉ ✳ ✳ ✒ ♥❣ s❫ ✒ ✓t✿ ✓♥ t❤✐❫ ✓ ❞ ✬ ❛ ❤❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ❡ ♦♥❣ t❤✉❫ ❛♥ ♥❤❫ ❛ ❇✐❫ ❡ ❡♥ ❤✕ ❛ ♦ ✖❫ ❡✬ ❞ ✖✉ ♦ ✳ ❝ ♥❣❤✐❫ ✳ ♠ t❫ ✳ ❦❤❫ x y 197) ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✒♥❤✿ ✳ ♣❤✉ ♦ ♥❣ tr✏ HD gia’i: ◆❣❤✐❫ ❡ ✳♠ 3t x = (λ1 + µ1 t)e y = (λ2 + µ2 t)e3t ✳ ❚✓ ✉ ❝ ❧❛ ✒✿ = 2C1 e2t + C2 e3t − et + 2te2t 2t 3t = −C1 e − C2 e + et − (t + 1)e2t x y dx dt dy dt = 2x + y = 4y − z ✳ ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ❝✉ ✒♥❤ ❞ ✖❛ ✕ ✳ ❝ tr✉ ♥❣✿ ✳ ✈✓ ♦✐ r1 = r2 = 3✳ λ2 = λ + µ1 ; µ2 = µ1 = (C1 + C2 t)e3t = (C1 + C2 + C2 t)e3t www.matheducare.com ❱❫ ❛ ✓ ❞❛ ✳ ② ◆❚◗ ❝♦ ✳ ♥❣✿ MATH-EDUCARE www.VNMATH.com 24 dx = 3x + 8y ✳ ✳ dt 198) ❚✏✒♠ ♥❣❤✐❫❡✳♠ ❝✉✬❛ ❤❫❡✳ ♣❤✉ ♦ ♥❣ tr✏✒♥❤✿ dy = −x − 3y dt ⑦ ♥ ❝❛ ✒ ✬ ❛ ♠❛ t❤♦ ✓❝ ❞ ✖✐❫ ❡✉ ❦✐❫ ❡ x(0) = 6; y(0) = −2 ✳ ♥✿ HD gia’i: dy ✓② ❞ ✓✱ r❫ ✒ − 3y ✱ ❧❫ ❛ ✖❛ ✒♠ t❤❡♦ t ❤❛✐ ✈❫ ❡ ♦✐ ✳ ♦ ❤❛ dt d2 y ✳ ✳ − y = 0✱ ❣✐❛✬ ✐ r❛✿ y = C1 et − C2 e−t ✱ ❞ ✖✉ ♦ ✳ ❝✿ dt ✳ ✳ ✳ ✳ ❚✒ ✉ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t❤✓ ✉ ❤❛✐✿ x=− ✳ ✳ ✳ ✓t ❝✉ ✬ ❛ ❤❫ ✒♥❤ t❤✓ ✉ ♥❤❫ ❛ ❡ t❤❛② ✈❛ ✒♦ ♣❤✉ ♦ ♥❣ tr✏ ✳ t −t s✉② r❛ x = −4C1 e − 2C2 e ⑦ ♥ ❝❛ ✒ ✬ ❛ ♠❛ t❤♦ ✓❝ ❞ ✖✐❫ ❡✉ ❦✐❫ ❡ ✳ ♥ x(0) = 6; y(0) = −2✱ C1 = C2 = −1✳ s✉② r❛ ✬❛ ❱❫ ❛ ❡ ✳ ② ♥❣❤✐❫ ✳ ♠ ❝✉ ❤❫ ❡ ✳✿ x y 199) ✳ ✳ ✬ ✐ ❤❫ ✒♥❤✿ ●✐❛ ❡ ✳ ♣❤✉ ♦ ♥❣ tr✏ dx dt dy dt dz dt = 4et + 2e−t = −et − e−t = 3x − y + z = −x + 5y − z = x − y + 3z ✳ HD gia’i: P❤✉✳♦✳♥❣ tr✏✒♥❤ ❞✖❛ ✕ ✳ ❝ tr✉ ♥❣✿ λ − 11λ + 36λ − 36 = 0✱ ✳ ✳ ✳ ✳ ✬ ✖✓ ♦ ❞ ✖✉ ♦ 3; λ3 = 6✳ ❚✒ ✉ ❞ ❡ ❡ ✳ ❝ ❜❛ ❤❫ ✳ ♥❣❤✐❫ ✳ ♠ ❝♦ ❜❛ ♥✿ 2t −e2t e e3t ; e3t ; e3t e6t −2e6t e6t ✬ ✐ r❛ ❣✐❛ λ1 = 2; λ2 = ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ x y z 200) = C1 e2t + C2 e3t + C3 e6t = C2 e3t − 2C3 e6t = −C1 e2t + C2 e2t + C3 e6t dy dx dz dx ✳ ✳ ✬ ❛ ❤❫ ✒♥❤✿ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ❡ ✳ ♠ t❫ ✳ ♣❤✉ ♦ ♥❣ tr✏ ✳ HD gia’i: P❤✉✳♦✳♥❣ tr✏✒♥❤ ❞✖❛ ✕ ✳ ❝ tr✉ ♥❣✿ ✳ ✳ ✳ ✬ ❞ ✖✉ ♦ ❡ ❡ ✳ ❝ ❜❛ ❤❫ ✳ ♥❣❤✐❫ ✳ ♠ ❝♦ ❜❛ ♥✿ ex −ex (λ − 1)(λ − 2) = 0✱ ; 2e2x −3e2x ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ y z = C1 ex + 2C2 e3x = −C1 ex − 3C2 e2x www.matheducare.com =y+z = z − 4y ✬ ✐ r❛ ❣✐❛ λ1 = 1; λ2 = 2✳ ✳ ❚✒ ✉ ❞ ✖✓ ♦ MATH-EDUCARE www.VNMATH.com 201) ✳ ✳ ✬ ✐ ❤❫ ●✐❛ ❡ ✒♥❤✿ ✳ ♣❤✉ ♦ ♥❣ tr✏ dx dt dy dt = 2x − 3y = x − 2y + sin t ✳ ✳ ✳ ✒♥❤ ❞ ✖✕ ❛ ✓ ❝❛ ✓ ❝ ♥❣❤✐❫ ❡ P❤✉ ♦ ♥❣ tr✏ ✳ ❝ tr✉ ♥❣ ❝♦ ✳♠ −t t e 3e ✳ ✬ ♥✿ ; ♥❣❤✐❫ ❡ −t t ✳ ♠ ❝♦ ❜❛ HD gia’i: e λ1 = −1; λ2 = 1✳ ✳ ✳ ✳ ❚✒ ✉ ❞ ✖✓ ♦ ❞ ✖✉ ♦ ❡ ✳ ❝ ❤❫ ✳ e ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓✿ ❇✐❫ ❡ ❡♥ ❤✕ ❛ ♦ 25 x y = C1 e−t + 3C2 et = C1 e−t + C2 et C e−t + 3C et C e−t + C et =0 = sin t ⇐⇒ C C = 3et sin t = e−t sin t C1 (t) = et (sin t − cos t) ✬ ✐ r❛✿ ●✐❛ C2 (t) = − e−t (sin t + cos t) x(t) = C1 e−t + 3C2 et − cos t ✬♥❣ q✉❛ ✬ ❱❫ ❛ ② ♥❣❤✐❫ ❡ ♠ t❫ ♦ ✓ t ❝✉ ❛ ❤❫ ❡ ✿ ✳ ✳ ✳ y(t) = C1 e−t + C2 et + sin t − cos t www.matheducare.com [...]... ( ) = x cos x 3x2 sin x cos x C1 = ⇒ C1 = + K1 ✬ ●✐❛ ✐ r❛✿ 3 3 C = x3 cos x ⇒ C = x3 sin x + 3x2 cos x − 6x sin x + 6 cos x + K 2 2 2 2 1 K2 x sin x − (x3 sin x + 3x2 cos x − 6x sin x + 6 cos x) + K1 x2 − ❱❫ ❛ ✳ ② ◆❚◗✿ y = 3 3x 3x 106) 2 cotgx y” + y + y = x x sin x ✒ ✓t ❧❛ tr✏ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ✒ y1 = x ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ●✐❛ ✳ ✳ ✓t ♠❫ ✬ ❜✐❫ ❡ ♦ ❡ ✳ t ♥❣❤✐❫ ✳ ♠ ❝✉❛ ♣❤✉ ♦ ♥❣ x cotgx ✳... sin x + cos 2x www.matheducare.com MATH-EDUCARE www.VNMATH.com 7 ✳ ✳ ✳ 2 P❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖✕ ❛ ✳ ❝ tr✉ ♥❣ λ + 1 = 0 ⇔ λ = ±i ✒ ✓t✿ y = C1 ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✳ ♠ t❫ HD gia’i: cos x + C2 sin x ❚✏ ✒♠ ♥❣❤✐❫ ❡ ❡♥❣ ❞❛ ✳ ♠ r✐❫ ✳ ♥❣✿ y = x(A cos x + B sin x) + C cos 2x + D sin 2x 1 1 A = − ; B = 0; C = − ; D = 0 2 3 1 1 y = C1 cos x + C2 sin x − x cos x − cos 2x 2... α1 = (sin x − cos x) + ln |x| 2 x ⇒ x 2 α2 = −[ xe (sin x − cos x) + e cos x] − x 2 2 4 2 −x cos x x e y = e−x (C1 x + C2 ) + xe−x ln |x| − − ✳ 2 4 www.matheducare.com =x MATH-EDUCARE www.VNMATH.com 6 119) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ y” + y = 1 sin x ✳ ✳ ✳ 2 P❤✉ ♦ ♥❣ tr✏ ✒♥❤ ❞ ✖✕ ❛ ✳ ❝ tr✉ ♥❣ λ + 1 ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t✿ y = A1 cos x + A2 sin x ✳ ♠ t❫ HD gia’i: A1 = −1 A2 = cotgx ✒ ♥❣ s❫ ✓♥... ✉ ❤❛✐✿ ✳ ♠ r✐❫ 2 x sin x cos x sin x 1 − p(x)dx x2 − 2 dx dx x dx = y2 = y1 e = − e dx = 2 2 y12 x x x sin x sin x sin x cos x ✳ ✳ ✒ ✓t✿ y = C1 ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤ t❤✉❫ ❛♥ ♥❤❫ ❛ ◆❚◗ ❝✉ − C2 x x cos x sin x C1 + C2 ( )=0 x x ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓✿ ❇✐❫ ❡ ❡♥ ❤✕ ❛ ♦ x cos x − sin x cotgx x sin x + cos x C1 + C2 = 2 2 x x x HD gia’i: p(x) = ⇒ C1 = cos2 x ⇒ C1 (x) = sin x = cos2 x 1 − sin2 x dx + K1... 2xdy = cos2 y HD gia’i: ✲ ✉✳❛ ♣❤✉✳♦✳♥❣ tr✏ ✒ ✒♥❤ ✈❫ ❡ ❞❛ ❉ ✳ ♥❣ ✲ ❛ ❉ ✕ ✳t 1 z = x2 t❛ ❝♦ ✓ 1 1 z = x + x− 2 x 2 ✒ ✬ ❞ t❤♦❛ ✖✐❫ ❡✉ ❦✐❫ ❡ ✳♥ 2 2 1 x + x= x 2 2 y cos y t❤❛② ✈❛ ✒♦ r❛✿ y(0) = π ✭❇❡r♥♦✉❧❧✐✮ (∗) (∗) 1 1 z + z= y cos2 y ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ♠ t❫ z= c y C = ❱❫ ❛ ✳② Z = tgy + ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓✿ ❜✐❫ ❡ ❡♥ ❤✕ ❛ ♦ y ⇒ C(y) = ytgy + ln | cos y| + ε cos2 y 1 ε ln | cos y| + y y ε √ 1 ln | cos y|... (y cos2 x − sin x)dy = y cos x(y sin x + 1)dx ∂P ∂Q = = y sin 2x + cos x ∂y ∂x www.matheducare.com (∗) MATH-EDUCARE www.VNMATH.com ◆❚◗✿ x 19 y y2 Q(x, y)dy = C ⇔ y sin x − cos2 x = C P (x, y0 )dx + 2 y0 =0 x0 =0 83) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ ✳ ✳ ✒ P❤✉ ♦ ♥❣ tr✏ ✒♥❤ ✈✐ ♣❤❫ ❛♥ t♦❛ ✒♥ ♣❤❫ ❛♥✿ HD gia’i: 84) (2x + 3x2 y)dx = (3y 2 − x3 )dy ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿ ( x2 + x3 y − y 3 = C (x2 + 1) cos... cos y + sin y)dy = 0 x2 + 2(x sin y − cos y) = C www.matheducare.com MATH-EDUCARE www.VNMATH.com 20 90) ✳ ✳ ⑦ ② t✏ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ❍❛ ✒♠ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✒♥❤✿ ✳ ♠ t❫ 1 y2 − x (x − y)2 HD gia’i: 91) 1 x2 − 2 (x − y) y dx + dy = 0 P❚❱P❚P ❝♦ ✓ t✏ ✓❝❤ ♣❤❫ ❛♥ t❫ ♦✬♥❣ q✉❛ ✓ t✿ ln xy x + =C y x−y ✳ ✳ ✬ ❛ ♣❤✉ ♦ ♥❣ tr✏ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✒♥❤ ✈✐ ♣❤❫ ❛♥✿ ✳ ♠ t❫ (sin xy + xy cos xy)dx + x2 cos... ✒♥❤✿ ●✐❛ y” − 4y + 5y = e2x + cos x ✳ ✳ ✳ 2 ✒♥❤ ❞ ✖✕ ❛ P❤✉ ♦ ♥❣ tr✏ ✳ ❝ tr✉ ♥❣✿ λ − 4λ + 5 2x ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t✿ y = e (C1 cos x + C2 sin x) ✳ ♠ t❫ ✳ ❚✏ ✒♠ ♥❣❤✐❫ ❡ ♠ r✐❫ ❡ ♥❣ ❞❛ ♦ ✐ y1 = ✳ ✳ ♥❣✿ y = y1 + y2 ✈✓ HD gia’i: = 0 ⇔ λ1 = 2 − i; λ2 = 2 + i Ae2x ; y2 = A cos x + B sin y ⇒ y1 = 1 1 cos x − sin x 8 8 1 2x 2x ◆❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t✿ y = e (C1 cos x + C2 sin x) + e + (cos x − sin x) ✳ ♠ t❫ 8 e2x... α2 (x) sin x ✒ ♥❣ ❝❛ ✒ ♥❣ s❫ ✓♥ t❤✐❫ ✓ ❇✕ ❛ ✓ ❝❤ ❜✐❫ ❡ ❡♥ ❤✕ ❛ ♦ α1 cos x + α2 sin x = 0 1 α1 (− sin x) + α2 cos x = sin x ❱❫ ❛ ❡ ♦✬♥❣ q✉❛ ✓ t✿ ✳ ② ♥❣❤✐❫ ✳ ♠ t❫ 114) ⇒ α1 = −1 cos x α2 = sin x α1 = −x α2 = ln sin x ⇒ y = C1 cos x + C2 sin x − x cos x + sin x ln sin x ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ✒♥❤✿ ●✐❛ y” − 3y + 2y = 2x2 − 5 + 2ex cos x 2 HD gia’i: λ2 − 3λ + 2 = 0 ⇔ λ1 = 1; λ2 = 2 x 2x ◆❚◗✿ y = C1 e +... x α1 ex + α2 (2e2x ) = 2x2 − 5 + 2ex cos 2 α1 = −e−x (2x2 − 5) − 2 cos x 2 α2 = e−2x (2x2 − 5) + 2e−x cos x 2 x α1 = e−x (2x2 − 4x − 1) − 4 sin 2 ⇒ α2 = − 1 [e−2x (2x2 − 5) + 2(xe−2x + 1 e−2x )] + 8 (−e−2x cos x + 1 e−x sin x ) 2 2 3 2 2 2 ✳ ✬ ❛ ♣❤✉✳♦✳♥❣ tr✏ ❚✒ ✉ ❞ ✖✓ ♦ ❝♦ ✓ ♥❣❤✐❫ ❡ ♦✬♥❣ q✉❛ ✓ t ❝✉ ✒♥❤✳ ✳ ♠ t❫ www.matheducare.com MATH-EDUCARE www.VNMATH.com 115) ✳ ✳ ✬ ✐ ♣❤✉ ♦ ♥❣ tr✏ ●✐❛ ✒♥❤✿