phuong trinh vi phan co loi giai

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1

` TA
ˆ. P PHU.O.NG TR`INH VI PHAN
ˆ
BAI
1)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

2xy y” = y 2 − 1

2xpp = p2 − 1
√
dx
2pdp
✳
2
2
❱✓
♦ ✐ x(p − 1) = 0 t❛ ❝♦
C1 x + 1
=
⇔
p

−
1
=
C
⇔
p
=
±
✓✿
1
p2 − 1
x
dy √
2
3
p=
= C1 + 1 ⇒ y =
(C1 x + 1) 2 + 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
√
√
= 2 y + C1 ⇒
dp = √ ⇒ p = 2 y + C1 ⇔
y
dx

dy
dx = √

2 y + C1
✳
✖✓
♦ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t✿
❚✒
✉ ❞
✳ ♠ t❫
◆❣♦❛
✒✐ r❛

3)

y = c✿

x=

√

y−

C1
√
ln |2 y + C1 | + C2
2

✒ ♥❣ ❝✉
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
1
dy
ey + C1 − ey
=
dy
=
(y −
C1 = 0 t❛ ❝♦
✓✿
ey + C1
C1
ey + 1
C1
=0:

1
ln(ey + C1 )
C1


−e−y
dx
✳

♥❤✉ ✈❫
❛
=
1
✳ ②✿
ey + C1  (y − ln |ey + C1 |)
C1
✒ ♥❣ ❧❛
◆❣♦❛
✒✐ r❛ y = C : ❤✕
❛
✒ ♠❫
♦
❡
✳ t ♥❣❤✐❫
✳♠

5)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

xy = y(1 + ln y − ln x)

nˆe´u C1 = 0
nˆe´u C1 = 0.

✳

✈✓
♦✐

y(1) = e

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ey dy
y
)
=
−
ey + C1
C1


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2

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 = 0 ⇒
z ln z
x
x
x
y(1) = e → C = 1. ❱❫
❛
✳ ② y = xe
HD gia’i:

6)

✲ ✉✳❛ ♣❤✉✳♦✳♥❣ tr✏
✒
❉
✒♥❤ ✈❫
❡✿

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

HD gia’i:

✲ ❛
❉
✕
✳t

y =

y”(1 + y) = y 2 + y

y = z(y) ⇒ z = z

dz
dy

✳ ✳
t❤❛② ✈❛
✒♦ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿

⇒ z + 1 = C1 (y + 1) ⇒ z = C1 y + C1 − 1 ⇔
• C1 = 0 ⇒ (∗)

❝❤♦

• C1 = 0 ⇒ (∗)

❝❤♦

◆❣♦❛

✒✐ r❛

y=C

dy
= dx (∗)
C1 y + C1 − 1

y =C −x
1
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 −

1
ln |C1 y + C1 − 1| = x + C2
C1

2
x2

2
2
✓♥ ❞
✒
HD gia’i: ❇✐❫
❡
✖♦
❫✬✐ ✭✸✮ ✈❫
❡ ❞❛
✳ ♥❣✿ x y = (xy) − 2 (∗)

✲ ❛
❉
✕
✒♦ (∗) s✉② r❛✿
✳ t z = xy ⇒ z = y + xy t❤❛② ✈❛

xz = z 2 + z − 2 ⇔
❱❫
❛
✳ ② ❚P❚◗✿

8)

dx
dz
=
⇔
+z−2
x

3

z−1
= Cx
z+x

xy − 1
= Cx3 .
xy + 2

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

HD gia’i:

z2

✲ ❛
❉
✕
✳t

yy” + y 2 = 1

y = z(y) ⇒ y” = z.

dz
dy

z
C1
dy
⇔ z2 = 1 + 2
dz =
2
1−z
y
y

dy
C1
dy
⇒
=± 1+ 2 ⇔±
= dx ⇒ y 2 + C1 = (x + C2 )2
dx
y
C1
1+ 2
y
2
✬♥❣ q✉❛
◆❣❤✐❫
❡
♠
t❫
♦
✓
t✿
y
+
C
=
(x
+
C2 )2
1
✳
✳ ✳

✓♥ ❞
✒
❇✐❫
❡
✖♦
❫✬✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈❫
❡✿

9)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

HD gia’i: y −

√
2x(1 + x)y − (3x + 4)y + 2x 1 + x = 0

3x + 4
1
.y = − √
; x = 0, x = −1
2x(x + 1)
x+1

✒
✓t✿

✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
✳ ♠ t❫

dy
=
y

3x + 4
2
1
Cx2
dx = ( −
)dx ⇔ y = √
2x(x + 1)
x 2(x + 1)
x+1

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✒ ♥❣ s❫

✓♥ t❤✐❫
✓✿
❇✐❫
❡
❡♥ ❤✕
❛
♦

1
1
⇒ C = − + ε.
2
x
x
x2
1
y=√
( + ε)
x+1 x

C =−

❱❫
❛
❡
♦✬♥❣ q✉❛
✓ t✿
✳ ② ♥❣❤✐❫
✳ ♠ t❫

10)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

HD gia’i:

✲ ❛
❉
✕
✳t

3

y” = e2y

z = y → y” = z.

dz
dy

✬
t❤♦❛

y(0) = 0
y (0) = 0

✳ ✳

✬✳ t❤❛
♣❤✉ ♦ ♥❣ tr✏
✒♥❤
✒♥❤ tr♦

z.

z2
e2y
dz
= e2y ⇔
=
+ε
dy
2
2

1
✳
2
2y
❛
− 1. ❚✒
✉ ❞
✖✓
♦✿
y (0) = y(0) = 0 ⇒ ε = − . ❱❫
✳② z = e
2
√

dy √ 2y
dy
√
= e −1⇒
z=
= x + ε. d¯ˆo’i biˆe´n t = e2y − 1
dx
e2y − 1
√
arctg e2y − 1 = x + ε
1
✒
✬ ❞
ln(tg 2 x + 1).
y(0) = 0 ⇒ ε = 0. ❱❫
❛
❡
❡♥❣ t❤♦❛
✖✐❫
❡✉ ❦✐❫
❡
✖✒
❫
❡ ❜❛
✒✐✿ y =
✳ ② ♥❣❤✐❫
✳ ♠ r✐❫
✳♥ ❞
2

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) = 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❛
❛
♣❤✉
♦
♥❣
1 − u2 dx = 0✳ ❘♦
✒♥❣ u − ±1 ❧❛
✒ ♥❣❤✐❫
❡
♠✳
❦❤✐
u

≡
±1
❞
✖
✉
✳
du
dx
✳ ❚P❚◗✿ arcsin u − ln x = C ✭❞♦ x > 0✮✳
=
1 − u2
x
y
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
❱❫
❛
= ln x + C ✳
✒♥❤✿ y = ±x; arcsin
✳ ② ◆❚◗ ❝✉
x

13)

✳ ✳
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
❚✏
✒♠ ♥❣❤✐❫
❡
❡♥❣ ❝✉

✳ ♠ r✐❫
⑦♥ ❞
✒
✒
✬ ♠❛
t❤♦❛
✖✐❫
❡✉ ❦✐❫
❡
✖❫
❛✉
✳♥ ❞

xy =
y(1) = 0✳

x2 − y 2 + y

HD gia’i:
xy =
❞
✖❛
✕
✳t

u=

y
x

❤❛②

y = ux

✳ ✳
♣❤✉ ♦ ♥❣ tr✏
✒♥❤ t❤❛
✒♥❤✿

x2 − y 2 + y ⇐⇒ y =

1−

y2 y
+
x2 x

y = xu + u
√
du
dx
xu = 1 − u2 ⇐⇒ √
=
x
1 − u2
s✉② r❛

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♥❫

❡♥

1
✳
e

y ≡ 0✳

✲ ✉✳❛ ✈❫
✒
❉
❡

❱❫
❛
✓❝❤ ♣❤❫
❛♥
✳ ② t✏

x2 − y 2 dx = 0. (x > 0)
✳
✬ ♥ ✉✳♦
❣✐❛
✓❝

x✿ xdu −

✒
✓♥✿
tr✏

✒♥❤ ✈❫
❡ t❛
✓ ❝❤ ❜✐❫
❡


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4

⇐⇒ arcsin u = ln Cx
⑦♥ ❞
✒
✬ ♠❛
t❤♦❛
✖✐❫
❡✉ ❦✐❫
❡
✖✒
❛
❫✉
✳♥ ❞

14)

y(1) = 0

❦❤✐

C = 1✳

❱❫
❛
❡
✳ ② ♥❣❤✐❫
✳♠

y = ±x✳

✳ ✳
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
❚✏
✒♠ ♥❣❤✐❫
❡
❡♥❣ ❝✉
✳ ♠ r✐❫
⑦♥ ❞
✒
✒
✬ ♠❛
t❤♦❛
✖✐❫
❡✉ ❦✐❫
❡
✖❫
❛✉
✳♥ ❞

y sin x = y ln y
π
y( ) = e✳
2

HD gia’i:
y sin x = y ln y ⇐⇒

⑦♥ ❞
✒
✬ ♠❛
t❤♦❛
✖✐❫
❡✉ ❦✐❫
❡
✳♥

15)

dx
dy
=
y ln y
sin x

x
C tan
x
2
⇐⇒ ln y = C tan

⇐⇒ y = e
2
x
tan
π
2✳
❛
❞
✖✒
❛
❫✉ y( ) = e ❦❤✐ C = 1✳ ❱❫
✳② y = e
2

✳ ✳
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
❚✏
✒♠ ♥❣❤✐❫
❡
❡♥❣ ❝✉
✒♥❤✿
✳ ♠ r✐❫

(x + y + 1)dx + (2x + 2y − 1)dy = 0
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 + 3 ln |x + y − 2| = C
⑦♥ ❞
✒
✬ ♠❛
t❤♦❛
✖✐❫
❡✉ ❦✐❫
❡
✖✒

❛
❫✉ y(0) = 1 ❦❤✐ C = 2✳
✳♥ ❞

16)

✲ ❛
❉
✕
✳t

y =

(z 2 − x2 )dz + 2zxdx = 0❀

⇐⇒
⇐⇒ ln |x| + ln
t❤❛②

u=

17)

1
xy

❱❫
❛
✳②

y=

1
✳ ✳
❞
✖✉ ♦
✳ ❝✿
z
(u2 − 1)(udx + xdu) + 2udx = 0
HD gia’i:

x − 2z − 3 ln |z − 2| = C ✳

1
⑦
✒
✬
r❫
♦✐ ❞
✖✕
❛
✳ t z = ux✱❤❛ ② ❣✐❛✐
z
(x2 y 2 − 1)dy + 2xy 3 dx = 0

✒
❇✕
❛ ♥❣ ❝❛
✓ ❝❤ ❞
✖✕

❛
✳t

✳ ✳
✒♥❤✿
♣❤✉ ♦ ♥❣ tr✏

✬ ✐ r❛
❣✐❛

✳ ✳
❞
✖✉ ♦
❡
✳ ❝ ♥❣❤✐❫
✳♠

✒
r❫
♦✐ ❞
✖❛
✕
✳t

z = ux✱

dx u2 − 1
+ 3
du = 0
x

u +u

u2 + 1
x(u2 + 1)
= ln C ⇐⇒
=C
|u|
u

1 + x2 y 2 = Cy ✳

✳ ✳
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ s❛✉✿
❚✏
✒♠ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✳ ♠ t❫

y − xy = x + x3

HD gia’i:
✳ ✳
✲ ❛
✓♥ t✏
✓♣ ✶ ✈❛
❉
❫② ❧❛

✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ t✉②❫
❡
✓♥❤ ❝❫
❛
✒ ❝♦
✓ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❧❛
✒
✳ ♠ t❫
x2

y = Ce 2 .

x2
+1
2

✳

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✳ ✳
❞
✖✉ ♦
✳❝

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18)

✳ ✳
✬ ❛ ❝❛
❚✏
✒♠ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ s❛✉✿
✳ ♠ t❫

HD gia’i:

y − y = y2.

✳ ✳
✲ ❛
✓♥ ✈❛
✒♥❤ t❛
✓ ❝❤ ❜✐❫
❡
✒ ❝♦
✓ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❧❛

✒
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✳ ♠ t❫

ln |

19)

5

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 = 0
x
x
✳ ✳
✳ ✳
✳

✒
✖✉ ❛ ✈❫
♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ❞
✖✉ ♦
❡ ❞❛
✳❝ ❞
✳ ♥❣✿

cos zdz = −

dx
+ C ⇔ sin z = − ln |x| + C
x

y
= − ln |x| + C
x

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

HD gia’i:

π
2

✳ ✳

✬✳ 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
2
y
π
tg
= Cx; y(1) = ⇒ C = 1.
2x
2

x cos z.z + 1 = 0 ⇔
❱❫
❛
✳ ② ❚P❚◗✿

✳
✈✓
♦✐

(y 2 − 1)x2 y 2 + y (x4 − y 4 ) = 0

✳ ✳
✳
✬ ♥❣ ❝❫
✓♣ ♥❤✉✳♥❣ ❣✐❛
✬ ✐ ❦❤❛
▲❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ❞
✖✕
❛
❛
✓ ♣❤✓
✉ ❝ t❛
✳ ♣✳

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6

✳ ✳
✳
✓✐ ✈✓
✒♥❤ ❜❫
❛
✖♦
❫
♦✐
❳❡♠ ♣❤✉ ♦ ♥❣ tr✏
✳ ❝ ❤❛✐ ❞
✳
✖✓
♦ ❝♦
✓ ❤❛✐ ❤♦
❡
♦✬♥❣ q✉❛
✓ t✿
❚✒
✉ ❞
✳ ♥❣❤✐❫
✳ ♠ t❫

24)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏

✒♥❤✿
●✐❛

HD gia’i:

25)

y 2 + x2 y = xyy

✓t ♣❤✉✳♦✳♥❣ tr✏
❱✐❫
❡
✒♥❤ ❧❛
✳✐

✳ ✳
❡
♦✬♥❣ q✉❛
✓ t✿
r❛ ❞
✖✉ ♦
✳ ❝ ♥❣❤✐❫
✳ ♠ t❫

y2
x2
= (x4 + y 4 )2 ⇒ y1 = 2 ; y2 = − 2 .
x
y
x

3
3
; x + y = C2
y=
C1 x + 1

y✿

y =
y

y2
x2
y
x

−1

✳ ✳
✒
✓t✱ ❣✐❛
✬✐
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
❞
✖❛
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏

y 2 = Cxe x

✳ ✳
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
❚✏
✒♠ ♥❣❤✐❫
❡
❡♥❣ ❝✉
✒♥❤✿
✳ ♠ r✐❫

(x + y − 2)dx + (x − y + 4)dy = 0
y(1) = 0✳

⑦♥ ❞
✒
✒
✬ ♠❛
t❤♦❛
✖✐❫
❡✉ ❦✐❫
❡
✖❫
❛✉
✳♥ ❞

HD gia’i:

✲ ❛
❉

✕
✳t

x
y

=u−1
= v + 3.

(u + v)du + (u − v)dv = 0✱
2
u + 2uv − v 2 = 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 2 − 4x + 8y = C

(x + y − 2)dx + (x − y + 4)dy = 0✳

=X −1
✱
=Y +3

✳ ✳
✒♥❤ t❤❛
✒♥❤✿
♣❤✉ ♦ ♥❣ tr✏

(X + Y )dX + (X − Y )dY = 0
❞
✖❛
✕
✳t

Y = uX

✬ ✐ r❛
●✐❛

27)

1−u
dX
+
du = 0✳
X
1 + 2u − u2
x2 + 2xy − y 2 − 4x + 8y = C ✳

✳
✳ ✳
✒
❞
✖✉ ❛ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈❫
❡

X 2 (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 2 )
2z
1
dx
xz =
✳ ❙✉② r❛ ♥❣❤✐❫
❡
✳ ❍❛② ( −
)dz =
✳♠
2
2
1−z
z 1+z
x
z
♥❛

✒② ❧❛
✒
= Cx, C = 0.
1 + z2
2
2
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤ ❞
✖⑦
❛ ❝❤♦ ❧❛
✒ x + y = C1 y, C1 = 0.
❱❫
❛
❡
✳ ② ♥❣❤✐❫
✳ ♠ ❝✉
✬✳ t❤❛
✒♥❤
tr♦

28)

✳ ✳
✬ ❛ ❝❛
❚✏
✒♠ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏

✒♥❤ s❛✉✿
✳ ♠ t❫

HD gia’i:

✲ ❛
❉
✕
✳t

u = 2x + y

✳ ✳
✳
✒
♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ❞
✖✉ ❛ ✈❫
❡ ❞❛
✳ ♥❣

5u + 9
du
=
.
dx
2u + 5

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y =

✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
❝✉
✒♥❤

2x + y − 1
.
4x + 2y + 5


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7

✳ ✳
✬ ✐ ♣❤✉✳♦✳♥❣ tr✏
●✐❛
✒♥❤ ♥❛
✒② t❛ ❞
✖✉ ♦
❡
✳ ❝ ♥❣❤✐❫
✳ ♠ 10u + 7 ln |5u + 9| =
✳
✳
⑦
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ❞

✖❛ ❝❤♦ ❧❛
✒ 10y + 7 ln |10x + 5y
❱❫
❛
❡
✳ ② ♥❣❤✐❫
✳ ♠ ❝✉

29)

25x + C.
= 9| − 5x = C.

✳ ✳
✬ ❛ ❝❛
❚✏
✒♠ t✏
✓ ❝❤ ♣❤❫
❛♥ t❫
♦✬♥❣ q✉❛
✓ t ❝✉
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ s❛✉✿

(x − y + 4)dy + (y + x − 2)dx = 0

HD gia’i:

✳ ✳
✳

✳ ✳
✲ ❛
✬ ♥❣ ❝❫
✒
✒
✓♣ ❞
✒♥❤ ❞
✖✉ ❛ ✈❫
❡ ❞❛
✖✕
❛
❛
✖✉ ♦
❛ ♥❣ ❝❛
✓ ❝❤ ❞
✖✕
❛
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✳ ♥❣ ❞
✳ ❝ ❜✕
✳t

u + 1, y = v − 3,
tr✏
✒♥❤ ❧❛
✒

✳ ✳

t❛ ❞
✖✉ ♦
✳❝

v 2 − 2uv − v 2 = C.

u+v
dv
=
✳
du
−u + v

✬ ✐ ♣❤✉✳♦✳♥❣ tr✏
✬ ❛ ♣❤✉✳♦✳♥❣
●✐❛
✒♥❤ t❛ ❝♦
✓ ♥❣❤✐❫
❡
✳ ♠ ❝✉

✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤ ❞
✖⑦
❛ ❝❤♦ ❧❛
✒
❱❫
❛
❡
✳ ② ♥❣❤✐❫

✳ ♠ ❝✉

30)

x =

y 2 − 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 2 )dy − 2xydx = 0.

HD gia’i:
✓t ♥❣❤✐❫
✒
❛✮ ❇❛
✒✐ t♦❛
✓ ♥ ❈❛✉❝❤② ❝♦
✓ ❞✉② ♥❤❫
❛

❡
❡♥
✳ ♠ tr♦♥❣ ♠✐❫
✳
2
✒② ✓
②✳
D = {(x, y) ∈ R |x − y ≥ δ} ✈✓
♦ ✐ δ > 0 t✉
✲ ✉✳❛ ♣❤✉✳♦✳♥❣ tr✏
✒
❜✮ ❉
✒♥❤ ✈❫
❡ ❞❛
✳ ♥❣

z=

y
✳
x

dy
xy
✳
= 2
dx
x − y2

✳ ✳

✲ ❛
✬ ♥❣ ❝❫
✓♣✱ t❛ ❞
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ❞
✖❛
✕
❛
✖❛
✕
✳t

✳ ✳
✬✳ t❤❛
❑❤✐ ❞
✖✓
♦ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤
✒♥❤ tr❫
❡♥ tr♦

z(1 + z 2 )
.
xz =
1 − z2
❍❛②

dx

1
2z
)dz =
( −
✳
2
z 1+z
x

z
= Cx, C = 0.
1 + z2
2
2
❧❛
✒ x + y = C1 y, C1 = 0.

✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤ ♥❛
✒② ❧❛
✒
❙✉② r❛ ♥❣❤✐❫
❡
✳ ♠ ❝✉
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
❱❫
❛
❡
✒♥❤ ❞
✖⑦

❛ ❝❤♦
✳ ② ♥❣❤✐❫
✳ ♠ ❝✉

✳
✳
2x
2x
2
✒
❛ ♥❣ ❤❫
❡
✓ ❝ ✈❡❝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 =

1 + z2
.
1−z

✳ ✳
✬ ✐ ♣❤✉✳♦✳♥❣ tr✏
●✐❛
✒♥❤ ♥❛
✒② t❛ ❞
✖✉ ♦
✳❝
y

x2 + y 2 = Cearctg x .
2
✳

✳
2
✒
✓♥ 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 = 0.

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8

HD gia’i:
2
2

✓♥ t✏
❛✮ ❍❫
❡
✒② ♣❤✉
♦
❡
✓♥❤ ✈✏
✒ 2 cos 2x + 2 sin 2x − 2 = 0✳
✳ ♥❛
✳ t❤✉❫
✳ ❝ t✉②❫
✳ ✳
✳
✳ ✳
✬ ♥❣ ❝❫
✒
✓♣✱ t❛ ❞
❜✮ P❤✉ ♦ ♥❣ tr✏
✒♥❤ ♥❛
✒② ❝♦
✓ t❤❫
❡✬ ❞
✖✉ ❛ ✈❫
❡ ❞❛
✖❛
✕
❛
✖✉ ♦
✳ ♥❣ ❞
✳❝

y =
✲ ❛
❉
✕
✳t

1
1
u=x− , v =y+ ✱
3
3

x+y
.
x − 2y + 1

✳ ✳
✬✳ t❤❛
❦❤✐ ❞
✖✓
♦ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤
✒♥❤ tr❫
❡♥ tr♦

v =
✳ ✳
✬ ✐ ♣❤✉✳♦✳♥❣ tr✏
●✐❛

✒♥❤ ♥❛
✒② t❛ ❞
✖✉ ♦
✳❝
❍❛②

33)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

HD gia’i: y = C :

y 2 + x2 y = xyy ✳

y =
y

y2
x2
y
x

−1

✳ ✳
✒
✓t✱ ❣✐❛
✬✐
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
❞
✖❛
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏

y 2 = 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
1
tg + 1 + 2 −
1
2
C1
C1
✓♥✿
❞
✖✐ ❞
✖❫
❡
ln
= x + C2
2
y

1
1
C1 + 1
−tg + 1 + 2 +
2
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(
2

y 2 + x2 y = xyy

✳ ✳
✬✳ t❤❛
✒♥❤
P❤✉ ♦ ♥❣ tr✏
✒♥❤ tr♦

35)

√

2 = Ce
u2 + 2v
√ 3x−1
1

√ arctg( 2
)
3y+1
(3x − 1)2 + 2(3y + 1)2 = C1 e 2
.

HD gia’i:

34)

u+v
.
u − 2v

❈♦✐

x = x(y)

y +

1
=0
2x − y 2

✬❛
❧❛
✒ ❤❛
✒♠ ❝✉

y

t❛ ❝♦
✓✿

y =

1
x

✳ ✳
t❤❛② ✈❛
✒♦ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿

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1
1
+
= 0 ⇔ x + 2x = y 2 :
x
2x − y 2

9

✳ ✳
✓♥ t✏

♣❤✉ ♦ ♥❣ tr✏
✒♥❤ t✉②❫
❡
✓♥❤✳

✒
✓t✿
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
✳ ♠ t❫

x = Ce−2y
1
1
1
✒ ♥❣ s❫
✓♥ t❤✐❫
✓✿ C (y) = y 2 e2y ⇒ C(y) = y 2 e2y − ye2y + e2y + C
❇✐❫
❡
❡♥ ❤✕
❛
♦
2

2
4
1
1
1
✳
✳
−2y
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿ x = Ce
+ y2 − y +
❱❫
❛
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✳ ② ♥❣❤✐❫
✳ ♠ t❫
2
2
4

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
3
2

y 2 + 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
2

dy
y 2 + 2C1
2ydy
y 2 + 2C1

⇒
=
⇒ 2
= dx
2y
dx
2y
y + 2C1
⇒ y 2 = A1 ex + A2 .
x
x
2
x
✓ tr❛
❈❤✉
✓ ✓
② ✿ ❱❫
❡
✓ ✐ (yy ) = yy ⇔ yy = C1 e ⇔ ydy = C1 e dx ⇔ y = 2C1 e + C2
✳
❛
◆❤✉ ✈❫
✳ ②✿

39)

p=

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏

●✐❛
✒♥❤✿

HD gia’i: yx =

1
xy

yey = y (y 3 + 2xey )

✳
✈✓
♦✐

✳ ✳
✓♥ ❞
✒
❜✐❫
❡
✖♦
❫✬✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈❫
❡✿

◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t✿
✳ ♠ t❫

y(0) = −1

2
x − x = y 2 e−y
y

x = y 2 (C − e−y )

y(0) = −1 ⇒ C = e.
2
−y
❱❫
❛
✳ ② x = y (e − e )

40)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

HD gia’i:

✲ ❛
❉
✕
✳t

y = p;

◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t✿
✳ ♠ t❫

xy” = y + x

1
p − p=1
x
✓
s❫
♦ ✿ C = ln |x| + C1

✳ ✳
✬✳ t❤❛
✒♥❤✿
♣❤✉ ♦ ♥❣ tr✏
✒♥❤ tr♦

p = Cx

✒ ♥❣
✓♥ t❤✐❫
❜✐❫
❡
❡♥ ❤✕
❛

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10

⇒p=

dy
= (ln |x| + C1 )x ⇒ y =
dx

(ln |x| + C1 )xdx + C2

= C1 x2 +

41)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

x2
x2
ln |x| −
+ C2

2
4

y + xy = x3

✒
✓t
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✳ ♥ t❫
x2
✒ ♥❣ s❫
✓♥ t❤✐❫
✓✿ C(x) = (x2 − 2)e− 2 + ε
❜✐❫
❡
❡♥ ❤✕
❛
♦

HD gia’i:

❱❫
❛

❡
♦✬♥❣ q✉❛
✓ t✿
✳ ② ♥❣❤✐❫
✳ ♠ t❫

42)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

x2

y = Ce− 2

x2

y = εe− 2 + x2 − 2.
(x2 − y)dx + xdy = 0

✳ ✳
✳ ✳
2
✓t ❧❛
✒
✓t✿
P❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈✐❫

❡
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
✳ ✐✿ xy − y = −x ✱ ♣❤✉ ♦ ♥❣ tr✏
✒ ♥❣ s❫
✓♥ t❤✐❫
✓ s✉② r❛ C = −x + ε
❡♥ ❤✕
❛
✓ t✿ y = Cx ❜✐❫
❡
♦
❝♦
✓ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✳ ♠ t❫
2
✬
❱❫
❛
❡
♦ ♥❣ q✉❛
✓ t ✿ y = −x + εx
✳ ② ♥❣❤✐❫
✳ ♠ t❫

HD gia’i:

43)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

HD gia’i:

✳ ✳
✓♥ t✏
P❤✉ ♦ ♥❣ tr✏
✒♥❤ t✉②❫
❡
✓♥❤✿

1
y = εx2 − ;
x

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

HD gia’i:
✲ ❛
❉
✕
✳t

❳❡
✓t

✳
✈✓
♦✐

y(1) = 1

y = Cx2 ; C =

1
3
⇒C =− 3 +ε
4
x
x

y(1) = 1 ⇒ ε = 2

❱❫
❛
❡
♦✬♥❣ q✉❛
✓ t✿
✳ ② ♥❣❤✐❫
✳ ♠ t❫

44)

2
3
y − y= 2
x
x

xy − y = 0

y = 0,

y = 2x2 −

1
x

(x + 1)(y + y 2 ) = −y

1
.y = −y 2
x+1
1
✒
tr✏
✒♥❤ ✈❫
❡ z −
.z = 1.
x+1
✓t✿ z = C1 (x + 1) ❜✐❫
✓♥ t❤✐❫

♥❤❫
❛
❡
❡♥

✳ ✳
✒
✓♥ ❞
✒♥❤ ✈❫
❡ ❞❛
❜✐❫
❡
✖♦
❫✬✐ ♣❤✉ ♦ ♥❣ tr✏
✳ ♥❣

1
z
= z ⇒ y = − 2 = −y 2 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 = 0 ❝✉
✒ ♥❣❤✐❫
❡
✳ ♠✳
❱❫
❛
❡

♦✬♥❣ q✉❛
✓ t✿
✳ ② ♥❣❤✐❫
✳ ♠ t❫

45)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

HD gia’i:

y=

1
(x + 1)(ln |x + 1| + ε)

2xy + y =

✈❛
✒

y=0

♥❣❤✐❫
❡
✒ ❞✐
✳ ♠ ❦✏

✳✳

1
1−x

✲ ✉✳❛ ♣❤✉✳♦✳♥❣ tr✏
✒
❉
✒♥❤ ✈❫
❡ ❞❛
✳ ♥❣

y +

1
1
y =
2x
2x(1 − x)

✓♣ ✶
t✏
✓♥❤ ❝❫
❛

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✳ ✳
✓♥
♣❤✉ ♦ ♥❣ tr✏

✒♥❤ t✉②❫
❡


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◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t✿
✳ ♠ t❫

C
y=√ ✱
x

11

✒ ♥❣ s❫
✓♥ t❤✐❫
✓✿
❜✐❫
❡
❡♥ ❤✕
❛
♦

√

❱❫

❛
❡
♦✬♥❣
✳ ② ♥❣❤✐❫
✳ ♠ t❫

46)

√
1
x
x+1
C (x) =
|+ε
⇒ C = ln | √
2x(1 − x)
2
x−1
√
1 1
x+1
q✉❛
✓ t✿ y = √
ln | √
|+ε
x 2
x−1

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏

●✐❛
✒♥❤✿

HD gia’i: y −

xy − y = x2 sin x

y
= x sin x✱
x

✳ ✳
✓♥ t✏
♣❤✉ ♦ ♥❣ tr✏
✒♥❤ t✉②❫
❡
✓♥❤✳ ◆❚◗✿

y = Cx

✒ ♥❣
✓♥ t❤✐❫
❜✐❫
❡
❡♥ ❤✕
❛

✓✿
s❫
♦

◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t✿
✳ ♠ t❫

47)

y = (C − cos x)x

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

HD gia’i:

y cos2 x + y = tgx

✳ ✳
✓♥ t✏
✒♥❤ t✉②❫
❡
✓♥❤
P❤✉ ♦ ♥❣ tr✏

→

✬
t❤♦❛

◆❚◗

y(0) = 0

y = Ce−tgx ; y = tgx − 1

✭♠❫
♦
❡
✳ t ♥❣❤✐❫
✳♠

r✐❫
❡♥❣✮

⇒ ◆❚◗✿ y = Ce−tgx + tgx − 1
y(0) = 0 ⇒ C = 1✳ ❱❫
❛
❡
✳ ② ♥❣❤✐❫
✳♠

48)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

HD gia’i:

√

y(0) = 0 ⇒ C = 1 ⇒

1 − x2 + y = arcsin x

✬
t❤♦❛

y(0) = 0

✒
♥❣❤✐❫
❡
❡♥❣ ❝❫
❛♥ t✏
✒♠✿
✳ ♠ r✐❫

⑦♥ ❞
✒
✒
✬ ♠❛
t❤♦❛
✖✐❫
❡✉ ❦✐❫
❡
✖❫

❛✉
✳♥ ❞

❳❡♠

x

y = Ce−arcsinx

y = arcsinx − 1
+ arcsinx − 1

✳ ✳
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
❚✏
✒♠ ♥❣❤✐❫
❡
❡♥❣ ❝✉
✳ ♠ r✐❫

HD gia’i:

y = tgx − 1 + e−tgx .

✓♥ t✏
✒
✓t✿
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤ t✉②❫

❡
✓♥❤ t❤✉❫
❛♥ ♥❤❫
❛
◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✳ ♠ t❫

⑦
✓② ♥❣❤✐❫
❉❫
❡ t❤❫
❛
❡
❡♥❣✿
✳ ♠ r✐❫
−arcsinx
⇒ ◆❚◗✿ y = Ce

49)

y

✒
r✐❫
❡♥❣ ❝❫
❛♥ t✏
✒♠✿

❧❛
✒ ❛
❫✬♥ ❤❛
✒♠✱ t❤❛②

y = e−arcsinx + arcsinx − 1

y =

1
2x − y 2

y(1) = 0✳

y =

1
✱
x

✳ ✳
✒♥❤ t❤❛
✒♥❤
♣❤✉ ♦ ♥❣ tr✏

1
1
=
⇐⇒ x − 2x = −y 2

2
x
2x − y
✳ ✳
✲ ❛
✓♥ t✏
✓♣ ♠❫
✓♥
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤ t✉②❫
❡
✓♥❤ ❝❫
❛
♦
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✒♥❤ t✉②❫
❡
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✳ t✱ ♥❣❤✐❫
✳ ♠ t❫
✳
✳
✳
✳
✳
−2y

✒
✓♥ t❤✐❫
✓ ❞
✒
✓t t✉ ♦ ♥❣ ✉
✓ ♥❣ ❧❛
✒ x = Ce
✳ ❇✐❫
❡
❡♥ ❤✕
❛ ♥❣ s❫
♦
✖✉ ♦
t✏
✓♥❤ t❤✉❫
❛♥ ♥❤❫
❛
✳ ❝ ◆❚◗✿

y2 y
− +
2
2
3
⑦♥ ❞
✒
✬ ♠❛
t❤♦❛
✖✐❫
❡✉ ❦✐❫

❡
✖✒
❛
❫✉ y(1) = 0 ❦❤✐ C =
✳
✳♥ ❞
4
3 −2y
⑦♥ ❞
✒
✬ ❛ ♠❛
❱❫
❛
❡
✖✐❫
❡✉ ❦✐❫
❡
✖✒
❛
❫✉✿ x =
e
+
✳ ② ♥❣❤✐❫
✳ ♠ t❤♦
✳♥ ❞
4
x = Ce−2y +

1
4

y2 y 1
− + ✳
2
2 4

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12

50)

✳ ✳
✒
✓t r✕
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤ s❛✉ ❞
✖❫
❛②✱ ❜✐❫
❡
❛ ♥❣ s❛✉ ❦❤✐ ❞
✖✕
❛
✳t

✳ ✳
✓♣ ❤❛✐ ❝♦
♠❫
♦
✒♥❤ ✈✐ ♣❤❫
❛♥ ❝❫
❛
✓ ♠❫
♦
❡
✳ t ♣❤✉ ♦ ♥❣ tr✏
✳ t ♥❣❤✐❫
✳♠

x2 y + 4xy + (x2 + 2)y = ex .

✿

z + z = ex ✱

z x2 − 4z x + 6z
z x − 2z
✳ ✳
;
y
=
✳ P❤✉ ♦ ♥❣ tr✏
✒♥❤
4
x3

x
x
e
∗
✒
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
r✐❫
❡♥❣ ❧❛
✒ y =
✱ ◆❚◗ ❝✉
✒♥❤ t❤✉❫
❛♥
2

y = zx2 =⇒ y =

✲ ❛
❉
✕
✳t

HD gia’i:

z
✳ ✳
✱ t❛ ♥❤❫
❛
✖✉ ♦
✳♥ ❞
✳❝

x2
1 x
∗
e ✿
r✐❫
❡♥❣ y =
2

y=

❝♦
✓ ♠❫
♦
❡
✳ t ♥❣❤✐❫
✳♠

z = C1 cos x + C2 sin x✳

t❤❛
✒♥❤
✓t✿
♥❤❫
❛

✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤ ❜❛♥ ❞
✖✒
❛
❫✉ ❧❛

✒✿
❱❫
❛
✳ ② ◆❚◗ ❝✉

ex
sin x
cos x
y = C1 2 + C2 2 + 2
x
x
2x

51)

✳ ✳
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
❚✏
✒♠ ♥❣❤✐❫
❡
❡♥❣ ❝✉
✒♥❤✿
✳ ♠ r✐❫
⑦♥ ❞
✒
✒
✬ ♠❛
t❤♦❛
✖✐❫
❡✉ ❦✐❫

❡
✖❫
❛✉
✳♥ ❞

HD gia’i:

x

❳❡♠

❧❛
✒ ❛
❫✬♥ ❤❛
✒♠✱ t❤❛②

yey = y (y 3 + 2xey )
y(0) = −1✳

1
✱
x

y =

✳ ✳
♣❤✉ ♦ ♥❣ tr✏
✒♥❤ t❤❛
✒♥❤

✳
✓♥ t✏
✒
✓t t✉✳♦✳♥❣ ✉
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤ t✉②❫
❡
✓♥❤ t❤✉❫
❛♥ ♥❤❫
❛
✓ ♥❣ ❧❛
✒
◆❚◗ ❝✉
✳ ✳
✓ ❞
s❫
♦
✖✉ ♦
✳❝
✳ ✳
❞
✖✉ ♦
✳❝

C(y) = −e−y + C ✳

C=

52)

1
✳
e

✳
◆❤✉ ✈❫
❛
✒
✳ ② ◆❚◗ ❧❛

✒
✬❛ ❞
t❤♦
✖✐❫
❡✉ ❦✐❫
❡
✳♥

✒ ♥❣
✓♥ t❤✐❫
❜✐❫
❡
❡♥ ❤✕
❛

✒
❚❤❛② ❞
✖✐❫
❡✉ ❦✐❫
❡

✖✒
❫
❛✉ ①❛
✓❝ ❞
✖✳✐♥❤
✳♥ ❞

y

y − y = cos x − sin x✳
x→∞

❜✐
❛
✳ ❝❤✕
✳ ♥ ❦❤✐

✓♥ t✏
✬ ✐ ♣❤✉✳♦✳♥❣ tr✏
●✐❛
✒♥❤ t✉②❫
❡
✓♥❤ r❛

✒
✬❛ ❞
t❤♦
✖✐❫
❡✉ ❦✐❫
❡

✳♥

53)

1
C
− y✳
y ye

C
❀
y

✳
✖✓
♦ ❑▲✳
❚✒
✉ ❞

✳ ✳
✬
❚✏
✒♠ ♥❣❤✐❫
❡
✒♥❤
✳ ♠ ❝✉❛ ♣❤✉ ♦ ♥❣ tr✏

HD gia’i:

x=

x=

2
x − x = y 2 e−y ✳
y

y

❜✐
❛
✳ ❝❤✕
✳ ♥ ❦❤✐

x→∞

y = Cex + sin x
C=0

❦❤✐

✳ ✳
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
❚✏
✒♠ ♥❣❤✐❫
❡
❡♥❣ ❝✉
✳ ♠ r✐❫
⑦♥ ❞

✒
✒
✬ ♠❛
t❤♦❛
✖✐❫
❡✉ ❦✐❫
❡
✖❫
❛✉
✳♥ ❞

y + sin y + x cos y + x = 0
π
y(0) = ✳
2

HD gia’i:
y + sin y + x cos y + x = 0 ⇐⇒ y + 2 sin

⇐⇒

❞
✖❛
✕
✳t

z = tan

z + z = −x✳

y
2

✬✐
●✐❛

⑦♥ ❞
✒
✬ ♠❛
t❤♦❛
✖✐❫
❡✉

y
y
y
cos + x.2 cos2 = 0
2
2
2

y

y
+ tan + x = 0
y
2
2 cos2
2

y

✳ ✳

✳ ✳

✓♥ t✏
✒♥❤ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ t✉②❫
❡
✓♥❤
y ✱ ♣❤✉ ♦ ♥❣ tr✏✒♥❤ t❤❛
2
−x
r❛✿ z = 1 − x + Ce
π
❦✐❫
❡
✖✒
❛
❫✉ y(0) =
❦❤✐ C = 0✳ ❱❫
❛
❡
❡♥❣ y = 2 arctan(1 − x)✳
✳♥ ❞
✳ ② ♥❣❤✐❫
✳ ♠ r✐❫
2

=⇒ z =

2 cos2

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54)

✳ ✳
✬ ❛ ❝❛
❚✏
✒♠ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ s❛✉✿
✳ ♠ t❫

HD gia’i:

✲ ❛
❉
✕
✳t

z = sin y,

13

y − x tan y =

✳ ✳
✬✳ t❤❛
❦❤✐ ❞
✖✓
♦ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤
✒♥❤ ❞
✖⑦
❛ ❝❤♦ tr♦

z − xz = x.
z = Ce − 1✳

✳ ✳
✓♥ t✏
✓♣ ✶ ✈❛
✒♥❤ t✉②❫
❡
✓♥❤ ❝❫
❛
✒ ❝♦
✓ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❧❛

✒
♣❤✉ ♦ ♥❣ tr✏
✳ ♠ t❫
x2
✳
✳
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
❱❫
❛
❡
✒♥❤ ❞
✖⑦
❛ ❝❤♦ ❧❛
✒ sin y = z = Ce 2
✳ ② ♥❣❤✐❫
✳ ♠ ❝✉

55)

x
cos y

✳ ✳
✬ ❛ ❝❛
❚✏
✒♠ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏

✒♥❤ s❛✉✿
✳ ♠ t❫

✲ ❛
❉
❫② ❧❛
✒

x2
2

−1
y − xy = x

HD gia’i:
✳ ✳
✲ ❛
✓♥ t✏
✓♣ ✶ ✈❛
✒♥❤ t✉②❫
❡
✓♥❤ ❝❫
❛
✒ ❝♦
✓ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❧❛
✒
❉

❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✳ ♠ t❫

56)

✳ ✳
✬ ❛ ❝❛
✒♥❤ s❛✉✿
❚✏
✒♠ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✳ ♠ t❫

HD gia’i:

y
√
= x y.
x

✳ ✳
✲ ❛
✒♥❤ ❇❡r♥♦✉❧❧✐ ✈❛
✒ ❝♦
✓ ♥❣❤✐❫
❡

♦✬♥❣ q✉❛
✓ t ❧❛
✒
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✳ ♠ t❫

√

57)

y +

C
1
y = √ + x2 .
x 5

✳ ✳
✬
✒♥❤ s❛✉✿
❚✏
✒♠ ♥❣❤✐❫
❡
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✳ ♠ ❝✉❛ ❝❛

y −

y
= x3
x

HD gia’i:
✳ ✳
✲ ❛
✓♥ t✏
✓♣ ✶ ✈❛
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ t✉②❫
❡
✓♥❤ ❝❫
❛
✒ ❝♦
✓ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❧❛
✒
✳ ♠ t❫

1
y = Cx + x4 .
3

58)

✳ ✳
✬
❚✏
✒♠ ♥❣❤✐❫
❡
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ s❛✉✿
✳ ♠ ❝✉❛ ❝❛

y − y = y2.

HD gia’i:
✳ ✳
✲ ❛
✒♥❤ ❇❡r♥♦✉❧❧✐ ✈❛
✒ ❝♦
✓ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❧❛
✒
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✳ ♠ t❫

y2 =

59)

1
Ce−2x

✳ ✳
✬
❚✏
✒♠ ♥❣❤✐❫
❡
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ s❛✉✿
✳ ♠ ❝✉❛ ❝❛

−1

.

y +

y
= sin x
x

HD gia’i:
✳ ✳
✲ ❛
✓♥ t✏
✓♣ ✶ ✈❛
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏

✒♥❤ t✉②❫
❡
✓♥❤ ❝❫
❛
✒ ❝♦
✓ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❧❛
✒
✳ ♠ t❫

y=

C sin x
+
− cos x.
x
x

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14

60)

✳ ✳
✬
❚✏
✒♠ ♥❣❤✐❫
❡
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ s❛✉✿
✳ ♠ ❝✉❛ ❝❛

√
y − y = x y.

HD gia’i:

✳ ✳
✲ ❛
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ❇❡r♥♦✉❧❧✐ ✈❛
✒ ❝♦
✓ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❧❛
✒

✳ ♠ t❫

√

61)

1

y = Ce 2 x − x − 2.

✳ ✳
✬ ❛ ❝❛
✒♥❤ s❛✉✿
❚✏
✒♠ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✳ ♠ t❫

y + 2xy = xe−x

2

HD gia’i:

✳ ✳
✲ ❛
✓♥ t✏

✓♣ ✶✳
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈✐ ♣❤❫
❛♥ t✉②❫
❡
✓♥❤ ❝❫
❛
x2 −x2
)e ✳
◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❧❛
✒ y = (C +
✳ ♠ t❫

2

62)

✳ ✳
✬ ❛ ❝❛
❚✏
✒♠ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏

✒♥❤ s❛✉✿
✳ ♠ t❫

HD gia’i:

y
√
= x y.
x

✳ ✳
✲ ❛
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ❇❡r♥♦✉❧❧✐ ✈❛
✒ ❝♦
✓ ♥❣❤✐❫
❡
✒
✳ ♠ ❧❛

√

63)

y −4

y=

1
ln x + Cx2 .
2

✳ ✳
✒
✬
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ s❛✉
❛✮ ❚✏
✒♠ ♠✐❫
❡♥ ♠❛
✒ tr♦♥❣ ❞
✖♦
✓ ♥❣❤✐❫
❡
✒✐ t♦❛
✓ ♥ ❈❛✉❝❤② ❝✉
✳ ♠ ❝✉❛ ❜❛
✒
✓t
❞
✖❫
❛② t❫
♦♥ t❛
✒ ❞✉② ♥❤❫
❛
✳ ✐ ✈❛

✬

❜✮ ❚✏
✒♠ ♥❣❤✐❫
❡
✒✐ t♦❛
✓ ♥ ❈❛✉❝❤② s❛✉ ❞
✖❫
❛②
✳ ♠ ❝✉❛ ❜❛

y = y + 3x.

1

y” − y = x
x
y(x = 1) = 1 va` y (x = 1) = 2.

HD gia’i:

✳ ✳
✲ ❛
✓♥ t✏
✓♣ ✶ t❤♦
✒
✒
✓t
✬❛ ❞
❛✮ ❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏

✒♥❤ t✉②❫
❡
✓♥❤ ❝❫
❛
✖✳✐♥❤ ❧②
✓ ❞
✖✐❫
❡✉ ❦✐❫
❡
♦♥ t❛
❛
✳ ♥ t❫
✳ ✐ ❞✉② ♥❤❫
2
♥❣❤✐❫
❡
❡♥ R .
✳ ♠ tr❫
✬ ✐ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤
❜✮ ●✐❛

y” −

y
= x✱
x

✳ ✳
t❛ ❞

✖✉ ♦
❡
♦✬♥❣ q✉❛
✓t
✳ ❝ ♥❣❤✐❫
✳ ♠ t❫

y = C1 + C2 x +

x2
.
2

✬ ❛ ❜❛
❱❫
❛
❡
✒✐ t♦❛
✓ ♥ ❈❛✉❝❤② ❧❛
✒
✳ ② ♥❣❤✐❫
✳ ♠ ❝✉

1
x2
y =− +x+ .
2
2

64)

✳ ✳
✬
✒♥❤ s❛✉✿
❚✏
✒♠ ♥❣❤✐❫
❡
✳ ♠ ❝✉❛ ♣❤✉ ♦ ♥❣ tr✏

y + ytgx = cos x

HD gia’i:

✳ ✳
✲ ❛
✓♥ t✏
✓♣ ✶✳
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈✐ ♣❤❫
❛♥ t✉②❫
❡
✓♥❤ ❝❫
❛
◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❧❛
✒✿

✳ ♠ t❫

y = (C + x) cos x.

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65)

✳ ✳
✬
✒♥❤ s❛✉✿
❚✏
✒♠ ♥❣❤✐❫
❡
✳ ♠ ❝✉❛ ♣❤✉ ♦ ♥❣ tr✏

y +

15

y
ex
= x( x
)y 2 .
x
e +1

HD gia’i:
✳ ✳
✲ ❛
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈✐ ♣❤❫
❛♥ ❇❡r♥♦✉❧❧✐ ✈❛
✒ ❝♦
✓ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✒♥❤ ❧❛
✒
✳ ♠ t❫

y=

66)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

HD gia’i:

✲ ❛

❉
✕
✳t

y = p✱

(x + 1)y” + x(y )2 = y

✳ ✳
✳ ✳
✳
✬✳ t❤❛
♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ tr♦
✒♥❤ ❇❡r♥♦✉✐❧✐ ✭✈✓
♦✐

p −
✲ ❛
❉
✕
✳t

z = p−1 = 0✱

✳
❞
✖✉ ❛

1
.
Cx − x ln(ex + 1)

(∗)

x 2
1
p=−
p
x+1
x+1

(∗)

✳ ✳
✒
✓♥ t✏
✓♣ ♠❫
✈❫
❡ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ t✉②❫
❡
✒♥❤ ❝❫
❛
♦
✳ t✿

z +

1
x
z=
1+x
x+1
C
x+1
x2 + C1
1
2(x + 1)
z=
⇒y = = 2
2(x + 1)
z
x + C1

✒
✓t✿
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
✳ ♠ t❫
✳ ✳
✓♥❣ s❫
✓♥ t❤✐❫

✓ ❝✉❫
✓✐ ❝✉
❇✐❫
❡
❡♥ ❤✕
❛
♦
♦
✒♥❣ ❞
✖✉ ♦
✳ ❝✿

z=

✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤✿
❙✉② r❛ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✳ ♠ t❫


x
2


ln |x2 + C1 | + √ arctg √ + C2
C1
C1√

1
x
−
−C


√ 1 | + C2
ln |
ln |x2 + C1 | + √
−C1
x + −C1
❈❤✉
✓ ✓
②

67)

y=C

nˆe´u C1 > 0
nˆe´u C1 < 0

❧❛
✒ ◆❑❉

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

x2 y = y(x + y)

1
1
= 2 y 2 : ♣❤✉✳♦✳♥❣ tr✏✒♥❤ ❇❡r♥♦✉✐❧❧✐
y
x
1
1
−1
✲ ❛
❉
✕
(y = 0) : −z − z = 2 .
✳t z = y
x
x
✒
✓t✿
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
z = Cx
◆❚◗ ❝✉
1
1
✒ ♥❣ s❫
✓♥ t❤✐❫
✓ ❈✿ C(x) = ε −

❜✐❫
❡
❡♥ ❤✕
❛
♦
. ❱❫
❛
)
✳ ② z = x(ε −
2
2x
2x2
2x
❱❫
❛
❡
♦✬♥❣ q✉❛
✓ t ❧❛
✒✿ y =
✳ ② ♥❣❤✐❫
✳ ♠ t❫
2
εx − 1
HD gia’i: x2 y = y(x + y) ⇔ y −

68)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛

✒♥❤✿

✬
t❤♦❛

yy” − (y )2 = y 3

1

y(0) = −
2
y (0) = 0

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16

HD gia’i:

✲ ❛
❉
✕
✳t

y = p(y);

y = p.py
py

✓♣✿
❞
✖❛
✕
❡
✳ t t✐❫

p(y) = y.z(y)

✳ ✳
t❤❛② ✈❛
✒♦ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤

dp
− p2 = y 3 ,
dy

✳
✳ ✳
✒
✒♥❤ ✈❫
❡
❞
✖✉ ❛ ♣❤✉ ♦ ♥❣ tr✏

1
dy
dz
= ⇒ z 2 = 2(y + C1 ) ⇔
=y
dy
z
dx

|2y + C|

1
✳
y(0) = − ; y (0) = 0 ⇒ C = 1✳ ❚✒
✉ ❞
✖✓
♦ s✉②
2
|2y + 1| − 1
dy
= x + C2 .
= y |2y + 1| ⇒ ln
dx
|2y + 1| + 1
1
❞♦ y(0) = −
⇒ C2 = 0.
2
|2y + 1| − 1

✒
✬ ✿ ln
= x.
❱❫
❛
❡
❡♥❣ ❝❫
❛♥ t✏
✒♠ t❤♦❛
✳ ② ♥❣❤✐❫
✳ ♠ r✐❫
|2y + 1| + 1
✒
❉♦ ❞
✖✐❫
❡✉ ❦✐❫
❡
✳♥

69)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

√
2y x
dy
ydx + 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| + = x
y
y
√
1
tgy + ln | cos y| = x
y

✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
❱❛
✒ ❚P❚◗ ❝✉
✒♥❤✿

y(0) = π ⇒ ε = 0

70)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏

✒♥❤✿
●✐❛

HD gia’i:
✒
✈❫
❡ ❞❛
✳ ♥❣✿

❉♦

y=0

1
y − y = y −1
x

❱❫
❛
✳ ② ❚P❚◗✿

71)

✈❫
❛
✳ ② ❚P❘ ✿

tgy +

xydy = (y 2 + x)dx

✓ ❝❤♦
✬ ✐ ❧❛
❦❤❫
♦♥❣ ♣❤❛
✒ ♥❣❤✐❫
❡
❡
✳ ♠✱ ❝❤✐❛ ❤❛✐ ✈❫
✲ ❛
❇❡r♥♦✉✐❧❧✐❀ ❉
✕
✳t

z = y2

(y +

√

✳ ✳
✓♥ ❞
❜✐❫
❡
✖♦
❫✬✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤

✳
✳ ✳

✒
❞
✖✉ ❛ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈❫
❡ ❞❛
✳ ♥❣✿

2
z − z = 2 → z = −2x + Cx2
x
2
2
y = −2x + Cx

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

xy

xy)dx = xdy

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HD gia’i:
✲ ❛

❉
✕
✳t

✲ ✉✳❛ ♣❤✉✳♦✳♥❣ tr✏
✒
✒♥❤ ✈❫
❡ ❞❛
❉
✳ ♥❣
1

1 1
1
y − y = √ .y 2 ; x = 0
x
x

1
1
z = √ ♣❤✉✳♦✳♥❣ tr✏✒♥❤
2x
x
2
✬
t❫
♦ ♥❣ q✉❛
✓ t✿ y = x(ln x + C)

z = y2 : z −

❱❫
❛
❡
✳ ② ♥❣❤✐❫
✳♠

72)

✓♥ t✏
✬ ✐ r❛
t✉②❫
❡
✓♥❤ ❣✐❛

z=

√

x(ln x + C)

√
xy − 2x2 y = 4y

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

HD gia’i:

17

✳ ✳
P❤✉ ♦ ♥❣ tr✏
✒♥❤ ❇❡r♥♦✉✐❧❧✐✱ ❞
✖❛
✕
✳t

z = y 1−α =

√

1
y⇒z = √
2 y

4
z − z = 2x → ◆❚◗ z = Cx4 − x2
x
y = (Cx2 − 1)2 x4 .

✳ ✳
✬✳ t❤❛
♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
✒♥❤ tr♦
❱❫
❛

❡
♦✬♥❣ q✉❛
✓ t✿
✳ ② ♥❣❤✐❫
✳ ♠ t❫

73)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

2x2 y = y 2 (2xy − y)

✓♥ y ✿ x y 3 − 2xy 2 = −2x2 ❇❡r♥♦✉✐❧❧✐
HD gia’i: ❳❡♠ x ❧❛
✒ ❤❛
✒♠ t❤❡♦ ❜✐❫
❡
2
2z
1
✳ ✳
✲ ❛
✬✳ t❤❛
✱ ♣❤✉ ♦ ♥❣ tr✏
= 3 → ❚P❚◗✿ y 2 = x ln Cy 2 ✱
✒♥❤✿ z +
❉

✕
✒♥❤ tr♦
✳t z =
x
y
y
❦②
✒ ❞✐
y
=
0.
✳

74)

✳ ✳
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
❚✏
✒♠ ♥❣❤✐❫
❡
❡♥❣ ❝✉
✳ ♠ r✐❫
⑦♥ ❞
✒
✒
✬ ♠❛
t❤♦❛
✖✐❫
❡✉ ❦✐❫

❡
✖❫
❛✉
✳♥ ❞

x2 y = y(x + y)
y(−2) = −4✳

HD gia’i: ❉♦ y(−2) = −4 ♥❫
❡♥ y ≡ 0✳
y2
✳
−1
✓♣ t✉
❡
✖❛
✕
❞
✖✉ ❛
y − 1y = 2 ✳ ❚✐❫
✳❝ ❞
✳t z = y
x

✳ ✳
✲ ✉✳❛ ♣❤✉✳♦✳♥❣ tr✏
✒
❉
✒♥❤ ✈❫
❡ ♣❤✉ ♦ ♥❣ tr✏

✒♥❤ ❇❡r♥♦✉✐❧❧✐✿
✳ ✳
✒
✓♥ t✏
♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈❫
❡ P❚ t✉②❫
❡
✓♥❤

✳
✒
✓t t✉✳♦✳♥❣ ✉
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
◆❚◗ ❝✉
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
✓ ♥❣✿

z = Cx✱

1
✳
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
❛
❡
✒♥❤ ❜❛♥ ❞
✖✒
❛

❫✉
✳ ◆❤✉ ✈❫
✳ ② ♥❣❤✐❫
✳ ♠ ❝✉
2x
1
4x
✒
C = ✳ ❱❫
❛
❡
❡♥❣ ❝❫
❛♥ t✏
✒♠ ❧❛
✒ y =
✳ ② ♥❣❤✐❫
✳ ♠ r✐❫
2
x2 − 1

C(x) = Cx −
❞
✖✒
❛
❫✉ ❝❤♦

75)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏

●✐❛
✒♥❤✿

HD gia’i: P❤✉✳♦✳♥❣
y (1 + Ce−x ) = 1

tr✏
✒♥❤✿

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

HD gia’i: P❤✉✳♦✳♥❣
1
y=
✳
1 + Cx + ln x

77)

tr✏
✒♥❤

1
1
z + z = − 2✳
x
x

✳ ✳
✒
✓♥ t❤✐❫
✓ ❞
❜✐❫
❡
❡♥ ❤✕
❛ ♥❣ s❫
♦
✖✉ ♦
✳❝
❧❛
✒✿

y=

2x
✳
Cx2 − 1

✲ ✐❫
✒
❉
❡✉ ❦✐❫
❡
✳♥

y − xy = −xy 3

y − xy = −xy 3

2

76)

♥❣❤✐❫
❡
✳♠

✳ ✳
✳ ✳
✬ ✐ r❛ ❞
❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ❇❡r♥♦✉✐❧❧✐✱ ❣✐❛
✖✉ ♦
✳❝

xy + y = y 2 ln x.

xy + y = y 2 ln x

✳ ✳
✳ ✳
✬ ✐ r❛ ❞
❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ❇❡r♥♦✉✐❧❧✐✱ ❣✐❛
✖✉ ♦

✳❝

✳ ✳
✬ ❛ ❝❛
❚✏
✒♠ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ s❛✉✿
✳ ♠ t❫

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y −4

y
√
=x y
x


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18

✳ ✳
✲ ❛

✒ ♥❣ ❝❛
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ❇❡r♥♦✉❧❧✐✱ ❜✕
❛
✓ ❝❤ ❞
✖❛
✕
✳t

HD gia’i:
✒
tr✏
✒♥❤ ✈❫
❡ ❞❛
✳ ♥❣

x
2
z − z=
x
2

z =

√

y

✳
✳ ✳
t❛ ❞
✖✉ ❛ ♣❤✉ ♦ ♥❣

✈❛
✒ ❝♦
✓ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❧❛
✒
✳ ♠ t❫

1
z = x2 ( ln |x| + C).
2
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤ ❧❛
✒
❱❫
❛
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✳ ② ♥❣❤✐❫
✳ ♠ t❫

1
y = x4 ( ln |x| + C)2 .

2

78)

✳ ✳
✬ ❛ ❝❛
✒♥❤ s❛✉✿
❚✏
✒♠ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✳ ♠ t❫

HD gia’i:
y=

y +

y
= y 2 xtgx.
x

✳ ✳
✲ ❛
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ❇❡r♥♦✉❧❧✐ ✈❛

✒ ❝♦
✓ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❧❛
✒
✳ ♠ t❫

1
✳
Cx + x ln | cos x|

79)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

y 2 dx + (2xy + 3)dy = 0

∂P
∂Q
=
= 2y
∂y
∂x

HD gia’i: P (x, y) = y 2 , Q(x, y) = 2xy + 3;
(1) ⇔ d(xy 2 + 3y) = 0✳

80)

❱❫
❛
✳②

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

HD gia’i:

xy 2 + 3y = C

ex (2 + 2x − y 2 )dx − yex dy = 0

∂P
∂Q
=
= −2yex
∂y
∂x

✳ ✳
✳ ✳
✳ ✳
✳
✒♥❤ t✉ ♦ ♥❣ ❞

✖✉ ♦ ♥❣ ✈✓
♦ ✐✿
s✉② r❛ ♣❤✉ ♦ ♥❣ tr✏

d ex (2x − y 2 ) =

0.
❱❫
❛
✳②

ex (2x − y 2 ) = C.

81)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

3

(y 2 + 1) 2 dx + (y 2 + 3xy

3

HD gia’i: p = (y 2 + 1) 2 ; Q = y 2 + 3xy
✬❛
❙✉② r❛ ♥❣❤✐❫
❡

♦✬♥❣ q✉❛
✓ t ❝✉
✳ ♠ t❫

(∗)

1 + y2 ⇒

0

Q(x, y)dy = C
0

⇔

HD gia’i:

1 + y2

y

P (x, 0)dx +

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

∂P
∂Q

=
= 3y
∂y
∂x

❧❛
✒✿

x

82)

1 + y 2 )dy = 0

3
y3
+ x(1 + y 2 ) 2 = C
3

(y cos2 x − sin x)dy = y cos x(y sin x + 1)dx

∂P
∂Q
=
= y sin 2x + cos x
∂y
∂x

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◆❚◗✿
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 y
x
+ 2)dx −
dy = 0
sin y
2 sin2 y

∂Q
x cos y
∂P
=

=−
∂y
∂x
sin2 y

HD gia’i:
❚P❚◗✿

y

x

π
P (x, )dx +
2
0

85)

88)

(y + ex sin y)dx + (x + ex cos y)dy = 0

3x2 (1 + ln y)dx = (2y −

x2 + 2(x sin y − cos y) = C.

x3
)dy
y

✳ ✳
✒
P❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈✐ ♣❤❫
❛♥ t♦❛
✒♥ ♣❤❫
❛♥✿ ◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t✿
✳ ♠ t❫

✳ ✳
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
❚✏
✒♠ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✒♥❤ ✈✐ ♣❤❫
❛♥✿
✳ ♠ t❫

HD gia’i:

xy + ex sin y = C.

(x + sin y)dx + (x cos y + sin y)dy = 0

✳ ✳
✒
P❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈✐ ♣❤❫
❛♥ t♦❛
✒♥ ♣❤❫
❛♥✿ ◆❚◗

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

HD gia’i:

x2
(x2 + 1) 1
+ 2x −
(
− 1) = C
2
2
sin y

✳ ✳
✒
✒♥❤ ✈✐ ♣❤❫
❛♥ t♦❛
✒♥ ♣❤❫
❛♥✱ ♥❣❤✐❫

❡
♦✬♥❣ q✉❛
✓ t✿
P❤✉ ♦ ♥❣ tr✏
✳ ♠ t❫

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

HD gia’i:

87)

π
2

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

HD gia’i:

86)

Q(x, y)dy = C ⇔

x3 (1 + ln y) − y 2 = C

3x2 (1 + ln y)dx = (2y −

x3
)dy
y

✳ ✳
✲ ❛
✒
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈✐ ♣❤❫
❛♥ t♦❛
✒♥ ♣❤❫
❛♥ ❝♦
✓ t✏
✓❝❤ ♣❤❫
❛♥ t❫
♦✬♥❣ q✉❛
✓ t ❧❛
✒✿

x3 (1 + ln y) − y 2 = C

89)

✳ ✳
⑦ ② t✏

✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
❍❛
✒♠ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✒♥❤✿
✳ ♠ t❫

HD gia’i:

P❚❱P❚P ❝♦
✓ t✏
✓❝❤ ♣❤❫
❛♥ t❫
♦✬♥❣ q✉❛
✓ t✿

(x + sin y)dx + (x cos y + sin y)dy = 0

x2 + 2(x sin y − cos y) = C

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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 xydy = 0

HD gia’i:

92)

✳ ✳
✒
✒♥❤ ✈✐ ♣❤❫
❛♥ t♦❛
✒♥ ♣❤❫
❛♥ ❝♦
✓ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❧❛
✒
P❤✉ ♦ ♥❣ tr✏
✳ ♠ t❫

✳
✳ ✳
⑦ ② t✏
✓ t✏
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
❍❛
✒♠ t❤✒
✉ ❛ s❫
♦
✓ ❝❤ ♣❤❫
❛♥ ❝✉

✒♥❤✿

x sin(xy) = C ✳

(x + y 2 )dx − 2xydy = 0

✳ ✳
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
s✉② r❛ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✒♥❤✳
✳ ♠ t❫

HD gia’i:

✳
✓ t✏
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
❚❤✒
✉ ❛ s❫
♦
✓❝❤ ♣❤❫
❛♥ ❝✉
✒♥❤ ❧❛
✒

✳ ✳
✳

✓ t✏
✒
✬ ✐ r❛
♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ❝❤♦ t❤✒
✉ ❛ s❫
♦
✓❝❤ ♣❤❫
❛♥ r❫
♦✐ ❣✐❛

93)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

HD gia’i:

2xy ln ydx + (x2 + y 2

µ(x) =

y2

x = Ce x

✓ ❝✉
✬❛

◆❤❫
❛♥ ❤❛✐ ✈❫
❡

✳

y 2 + 1)dy = 0

✳ ✳
✳
✲ ❛
✒
✓ t✏
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈✐ ♣❤❫
❛♥ t♦❛
✒♥ ♣❤❫
❛♥✱ t❤✒
✉ ❛ s❫
♦
✓❝❤ ♣❤❫
❛♥✿

✳
✳ ✳
✓ t✏
✓ ❝✉
✒

✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✬ ✐ r❛ ❞
t❤✒
✉ ❛ s❫
♦
✓❝❤ ♣❤❫
❛♥ ✈❛
✒♦ ❤❛✐ ✈❫
❡
✒♥❤ r❫
♦✐ ❣✐❛
✖✉ ♦
✳ ❝✿

94)

1
✳
x2

µ(y) =

1
y

♥❤❫
❛♥

1
3

x2 ln y + (y 2 +1) 2 = 0
3

✳ ✳
✬
✒♥❤
❚✏
✒♠ ♥❣❤✐❫
❡
✳ ♠ ❝✉❛ ♣❤✉ ♦ ♥❣ tr✏
✒
✬❛ ❞
t❤♦
✖✐❫
❡✉

HD gia’i:

(x3 + xy 2 )dx + (x2 y + y 3 )dy = 0✳
❦✐❫
❡
✳ ♥ y(0) = 1✳

✳ ✳
✲ ❛
✒
❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈✐ ♣❤❫

❛♥ t♦❛
✒♥ ♣❤❫
❛♥ ◆❚◗ ❧❛
✒✿

x4 + 2x2 y 2 + y 4 = C
✳
✒
✬❛ ❞
t❤♦
✖✐❫
❡✉ ❦✐❫
❡
✳♥

95)

y(0) = 1

❦❤✐

C = 1✳

✳ ✳
✬ ❛ ❝❛
❚✏
✒♠ t✏
✓ ❝❤ ♣❤❫
❛♥ t❫
♦✬♥❣ q✉❛

✓ t ❝✉
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ s❛✉✿

HD gia’i:

✳ ✳
✳
✓ t✏
❚❛ t✏
✒♠ ❞
✖✉ ♦
✉ ❛ s❫
♦
✓❝❤ ♣❤❫
❛♥
✳ ❝ t❤✒

1 ✲ ✳
✳ ✳
✳ ❉✉ ❛ ♣❤✉ ♦ ♥❣
2
x
2
2
❧❛
✒ x − y = Cx.

µ(x) =

✒
❞❛
❛♥ t♦❛
✒♥ ♣❤❫
❛♥✳ ❑❤✐ ❞
✖✓
♦ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓t
✳ ♥❣ ✈✐ ♣❤❫
✳ ♠ t❫

a) − 2xydy + (y 2 + x2 )dx = 0

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✒
tr✏
✒♥❤ ❞
✖⑦
❛ ❝❤♦ ✈❫
❡


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96)

21

✳
✳
2x −x
✒
✓♥ t✏
❛✮ ❈❤✓
✉ ♥❣ ♠✐♥❤ r✕
❛ ♥❣ ❤❫
❡
✓ ❝ ✈❡❝t♦
❧❛
✒ ❤❫
❡
✖❫
♦
❛
❡
✓ ♥❤✳
✳ ❝❛
✳ ❞
✳ ❝ ❧❫
✳ ♣ t✉②❫
✳
✬
✓ ♥❣✳
❚✏
✓ ♥❤ ❞
✖✐
✉ ❝ ❲r♦♥s❦✐ ❝✉❛ ❝❤✉

✳♥❤ t❤✓

{e , e , cos x}

✳ ✳
✬ ❛ ❝❛
❜✮ ❚✏
✒♠ t✏
✓ ❝❤ ♣❤❫
❛♥ t❫
♦✬♥❣ q✉❛
✓ t ❝✉
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ s❛✉✿

x2 − ydy − 2x(1 +

x2 − y)dx = 0.

HD gia’i:
⑦❛ ❦✐❫
✓♥ t✏
❛✮ ❉✉
✒♥❣ ❞
✖✳✐♥❤ ♥❣❤✏
❡✬♠ tr❛ ❤❫
❡
✖♦
❫
❛

❡
✓♥❤✳
✳ ❞
✳ ❝ ❧❫
✳ ♣ t✉②❫
✳
x
✲
❉✳✐♥❤ t❤✓
✉ ❝ ❲r♦♥s❦✐ W [y1 , y2 , y3 ](x) = 3e (3 cos x − sin x).
✳ ✳
✲ ❛
✒
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
❜✮ ❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ✈✐ ♣❤❫
❛♥ t♦❛
✒♥ ♣❤❫
❛♥✳ ❚✏
✓❝❤ ♣❤❫
❛♥ t❫
♦✬♥❣ q✉❛
✓ t ❝✉
✒♥❤
❧❛
✒

3

2
x2 + (x2 − y) 2 = C
3

97)

✳ ✳
✬ ❛ ❝❛
❚✏
✒♠ t✏
✓ ❝❤ ♣❤❫
❛♥ t❫
♦✬♥❣ q✉❛
✓ t ❝✉
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ s❛✉✿

HD gia’i:

✳ ✳
✳
✓ t✏
♦
✓❝❤ ♣❤❫
❛♥
❚❛ t✏
✒♠ ❞
✖✉ ♦
✉ ❛ s❫
✳ ❝ t❤✒

µ(x) =

✒
❞❛
❛♥ t♦❛
✒♥ ♣❤❫
❛♥✳ ❑❤✐ ❞
✖✓
♦ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❧❛
✒
✳ ♥❣ ✈✐ ♣❤❫
✳ ♠ t❫

x2
− y 2 )dy − 2xdx = 0.
y

1 ✲ ✳
✳ ✳
✳ ❉✉ ❛ ♣❤✉ ♦ ♥❣
y
2x2 + y 3 = Cy.

✳
✳
x 2x

2
✒
❛✮ ❈❤✓
✉ ♥❣ ♠✐♥❤ r✕
❛ ♥❣ ❤❫
❡
✓ ❝ ✈❡❝t♦
✳ ❝❛
✳
✳
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ s❛✉✿
❜✮ ❚✏
✒♠ t✏
✓ ❝❤ ♣❤❫
❛♥ t❫
♦✬♥❣ q✉❛
✓ t ❝✉

{e , e , x }

98)

(

✒
tr✏
✒♥❤ ❞
✖⑦
❛ ❝❤♦ ✈❫

❡

✓♥ t✏
❧❛
✒ ❤❫
❡
✖❫
♦
❛
❡
✓ ♥❤✳
✳ ❞
✳ ❝ ❧❫
✳ ♣ t✉②❫

(x − y)dy + (x + y)dx = 0.

HD gia’i:

✳ ✳
✓♥ t✏
✒♥❤ ❧❛
✒ ❞
✖♦
❫
❛
❡
✓♥❤ ✳
❛✮ ❑✐❫
❡✬♠ tr❛ ❤❫

❡
✳ ❝ ❧❫
✳ ♣ t✉②❫
✳ ♣❤✉ ♦ ♥❣ tr✏
✳ ✳
✲ ❛
✒
✒♥❤ ✈✐ ♣❤❫
❛♥ t♦❛
✒♥ ♣❤❫
❛♥ ♥❫
❡♥ t❛ ❝♦
✓
❜✮ ❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
❱❫
❛
✓❝❤ ♣❤❫
❛♥ t❫
♦✬♥❣ q✉❛
✓ t ❧❛
✒
✳ ② t✏

x2 − y 2 + 2xy = C.

d(xy −

y 2 x2

+ ) = 0✳
2
2

✳
✳
x
✒
✓♥ t✏
❛✮ ❈❤✓
✉ ♥❣ ♠✐♥❤ r✕
❛ ♥❣ ❤❫
❡
✓ ❝ ✈❡❝t♦
❧❛
✒ ❤❫
❡
✖❫
♦
❛
❡
✓ ♥❤✳
✳ ❝❛
✳ ❞
✳ ❝ ❧❫
✳ ♣ t✉②❫
✳ ✳
2
✬
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏

✒♥❤ s❛✉✿
❜✮ ❚✏
✒♠ t✏
✓ ❝❤ ♣❤❫
❛♥ t❫
♦ ♥❣ q✉❛
✓ t ❝✉

{1, x, e }

99)

(x − y)dx + xdy = 0

HD gia’i:
⑦❛ ❦✐❫
✓♥ t✏
❛✮ ❉✉
✒♥❣ ❞
✖✳✐♥❤ ♥❣❤✏
❡✬♠ tr❛ ❤❫
❡
✖♦
❫
❛
❡
✓♥❤ ✳
✳ ❞
✳ ❝ ❧❫
✳ ♣ t✉②❫

✳
✳ ✳
✓ t✏
❜✮ ❚✏
✒♠ t❤✒
✉ ❛ s❫
♦
✓❝❤ ♣❤❫
❛♥✱ t❛ ❞
✖✉ ♦
✳❝
✳ ✳
✒
❞❛
✒♥❤ ✈✐ ♣❤❫
❛♥ t♦❛
✒♥ ♣❤❫
❛♥
✳ ♥❣ ♣❤✉ ♦ ♥❣ tr✏

(1 −
✳ ✳
✬ ✐ ♣❤✉✳♦✳♥❣ tr✏
●✐❛
✒♥❤ ♥❛
✒② t❛ ❞
✖✉ ♦
✳❝

µ(x) =

1
✳
x2

✳ ✳
✳
✳ ✳
✒
✒♥❤ ❞
✖⑦
❛ ❝❤♦ ❞
✖✉ ❛ ❞
✖✉ ♦
❡
P❤✉ ♦ ♥❣ tr✏
✳ ❝ ✈❫

y
1
)dx + dy = 0.
2
x
x

y = Cx − x2 .

✳
✳
2x x

✒
❛✮ ❈❤✓
✉ ♥❣ ♠✐♥❤ r✕
❛ ♥❣ ❤❫
❡
✓ ❝ ✈❡❝t♦
✳ ❝❛
✳ ✳
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
❜✮ ❚✏
✒♠ t✏
✓ ❝❤ ♣❤❫
❛♥ t❫
♦✬♥❣ q✉❛
✓ t ❝✉
✒♥❤ s❛✉✿

100)

{e , e , x}

✓♥ t✏
❧❛
✒ ❤❫
❡
✖❫
♦
❛
❡
✓ ♥❤✳

✳ ❞
✳ ❝ ❧❫
✳ ♣ t✉②❫

(x − y)dx − (x + y)dy = 0.

HD gia’i:

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22

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✳ ✳
✓♥ t✏
❛✮ ❑✐❫
❡✬♠ tr❛ ❤❫
❡
✒♥❤ ❧❛
✒ ❞
✖♦
❫
❛
❡
✓♥❤✳
✳ ♣❤✉ ♦ ♥❣ tr✏
✳ ❝ ❧❫
✳ ♣ t✉②❫

✳
✳
✲ ❛
✒
✒♥❤ ✈✐ ♣❤❫
❛♥ t♦❛
✒♥ ♣❤❫
❛♥✳ ❙✉② r❛ t✏
✓❝❤ ♣❤❫
❛♥ t❫
♦✬♥❣ q✉❛
✓ t ❝♦
✓ ❞❛
❜✮ ❉
❫② ❧❛
✒ ♣❤✉ ♦ ♥❣ tr✏
✳ ♥❣✿

x2 + y 2 − 2xy = C.

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1

` TA
ˆ. P PHU.O.NG TR`INH VI PHAN

ˆ (tiˆ
BAI
e´p theo)
101)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

y” + y = x + e−x

✳ ✳
✳
2
P❤✉ ♦ ♥❣ tr✏
✒♥❤ ❞
✖✕
❛
✳ ❝ tr✉ ♥❣ λ + λ = 0 ⇔ λ1 = 0; λ2 = −1
✳
✳
✒
✓t✿ y = C1 + C2 e−x
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✒♥❤ t❤✉❫

❛♥ ♥❤❫
❛
✳ ♠ t❫
✳ ✳
✳ ✳
✳
✖✓
♦ y1 , y2 ❧❛
✒ ❝❛
✓ ❝ ♥❣❤✐❫
❡
❚✏
✒♠ ♥❣❤✐❫
❡
❡♥❣ ❞✉ ♦
✓ ✐ ❞❛
✓ ♥❣
✳ ♠ t✉ ♦ ♥❣ ✉
✳ ♠ r✐❫
✳ ♥❣ y = y1 + y2 ✱ tr♦♥❣ ❞
✳
✳
−x
✬ ❛ ❝❛
✒♥❤✿ y” + y = x ✈❛
✒ y” + y = e
❝✉
✓ ❝ ♣❤✉ ♦ ♥❣ tr✏
✳ ✳
✳

✬
• ❱✏✒ λ1 = 0 ❧❛
✒ ♥❣❤✐❫
❡
✒♥❤ ❞
✖✕
❛
❡♥ y1 = x(Ax + B)
✳ ♠ ❝✉ ❛ ♣❤✉ ♦ ♥❣ tr✏
✳ ❝ tr✉ ♥❣ ♥❫

HD gia’i:

✳ ✳
✒ ♥❣ ♣❤✉✳♦✳♥❣ ♣❤❛
✓ ❜❫
✓t ❞
✓ ♣ ❤❫
❡
♦
❛
✖✳✐♥❤ ❞
✖✉ ♦
❇✕
❛
✳ s❫
✳ ❝✿

1
y1 = x2 − x

2

✳
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
❧❛
✒ ♥❣❤✐❫
❡
✒♥❤ ❞
✖✕
❛
❡♥✿
✳ ♠ ❝✉
✳ ❝ tr✉ ♥❣ ♥❫
−x
✓ ❜❫
✓t ❞
❚❤❛② ✈❛
✒♦ ✈❛
✒ ❞✉
✒♥❣ ❤❫
❡
♦
❛
✖✳✐♥❤ s✉② r❛✿ y2 = −xe
✳ s❫

• λ2 = −1

✓✐ ❝✉
❈✉❫

♦
✒♥❣ ◆❚◗✿

102)

1
y = C1 + C2 e−x + x2 − x − xe−x
2

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

HD gia’i:

y2 = Axe−x

2y” + 5y = 29x sin x

5
2λ2 + 5λ = 0 ⇔ λ1 = 0, λ2 = −
2
5x
−
✒
✓t y = C1 + C2 e 2
tr✏
✒♥❤ t❤✉❫
❛♥ ♥❤❫

❛

✳ ✳
✳
✒♥❤ ❞
✖✕
❛
P❤✉ ♦ ♥❣ tr✏
✳ ❝ tr✉ ♥❣✿

✬ ❛ ♣❤✉✳♦✳♥❣
◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✳ ♠ t❫
✳
✬ ✐ ❧❛
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤ ❞
✖❛
✕
❡♥ t✏
✒♠ ♥❣❤✐❫
❡
❡♥❣ ❞❛
❱✏
✒ ±i ❦❤❫
♦♥❣ ♣❤❛
✒ ♥❣❤✐❫

❡
✳ ❝ tr✉ ♥❣ ♥❫
✳ ♠ r✐❫
✳ ♥❣✿
✳ ♠ ❝✉

y = (Ax + B) sin x + (Cx + D) cos x
✳ ✳
✳ ✳
❚❤❛② ✈❛
✒♦ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ ❞
✖✉ ♦
✳ ❝✿

103)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

A = −2; B =

16
185
; C = −5; D = −
29
29

y” − 2y + 5y = x sin 3x

✳ ✳
✳
2
✒♥❤ ❞
✖✕
❛
P❤✉ ♦ ♥❣ tr✏
✳ ❝ tr✉ ♥❣✿ λ − 2λ + 5 = 0 ⇔ λ1 = 1 − 2i; λ2 = 1 + 2i
✳
✳
✒
✓t✿ y = ex (C1 cos 2x + C2 sin 2x)
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
◆❚◗ ❝✉
✳
✬ ✐ ❧❛
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✬ ❛ ✭✷✮
✒♥❤ ❞
✖❛
✕
❡♥ ♥❣❤✐❫
❡
❡♥❣ ❝✉
❉♦ ±3i ❦❤❫

♦♥❣ ♣❤❛
✒ ♥❣❤✐❫
❡
✳ ❝ tr✉ ♥❣ ♥❫
✳ ♠ r✐❫
✳ ♠ ❝✉
✳ ✳
✳ ✳
❞
✖✉ ♦
✒♠ ❞✉ ♦
✓ ✐ ❞❛
✳ ❝ t✏
✳ ♥❣✿ y = (Ax + B) cos 3x + (Cx + D) sin 3x

HD gia’i:

✳ ✳
❚❤❛② ✈❛
✒♦ ✭✷✮ t❛ ❞
✖✉ ♦
✳ ❝✿

104)

A=

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿

●✐❛

3
57
1
41
; B= ; C=− ; D=
26
26
13
13

y” − 2y − 3y = xe4x + x2

✳ ✳
✳
2
P❤✉ ♦ ♥❣ tr✏
✒♥❤ ❞
✖✕
❛
✳ ❝ tr✉ ♥❣✿ λ − 2λ − 3 = 0 ⇔ λ1
✳
✳
✒
✓t✿ y = C1 e−x + C2 e3x
✬ ❛ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛

◆❚◗ ❝✉
✳
✬❛
❚✏
✒♠ ♥❣❤✐❫
❡
❡♥❣ ❞❛
♦ ✐ y1 ❧❛
✒ ♥❣❤✐❫
❡
✳ ♠ r✐❫
✳ ♥❣ y = y1 + y2 ✈✓
✳ ♠ ❝✉

HD gia’i:

y1 = e4x (Ax + B) = e4x
❝♦
✒♥

y2

✬❛
❧❛
✒ ♥❣❤✐❫
❡
❡♥❣ ❝✉
✳ ♠ r✐❫

y” − 2y − 3y = x2

= −1; λ2 = 3.
y” − 2y − 3y = xe4x

x
6
−
5 25

❝♦
✓ ❞❛
✳ ♥❣✿

2
4
14
y2 = A1 x2 + B1 x + C1 = − x2 + x − .
3
9
27
❱❫
❛
❡
♦✬♥❣ q✉❛
✓ t✿
✳ ② ♥❣❤✐❫
✳ ♠ t❫

y = C1 e−x + C2 e3x +

e4x
6
1
4
14
(x − ) − (x2 − x + )
5
5
3
3
9

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2

105)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

x2 y” − 2y = x3 cos x
y1 = x2

✳ ✳
✓t ♠❫
✒
✓t ❧❛
✬
❜✐❫
❡
♦
❡
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
✒
✳ t ♥❣❤✐❫
✳ ♠ ❝✉❛ ♣❤✉ ♦ ♥❣ tr✏

HD gia’i:

✓ ❝❤♦
❈❤✐❛ ✷ ✈❫
❡

x2 (x = 0)✿

y” −

2
y = x cos x.
x2

✳
✒
✓t ❞❛
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
❚✏
✒♠ ♥❣❤✐❫
❡
❡♥❣ t❤✓
✉ ❤❛✐ ❝✉
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
✳ ♠ r✐❫
✳ ♥❣✿

p(x) = 0; q(x) = −

2
✳
x2

1 −
e
y12

y2 = y1

p(x)dx

dx = x2

1
dx
=−
4
x
3x

✒
✓t ❧❛
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
❱❫
❛
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
✒✿
✳ ② ♥❣❤✐❫
✳ ♠ t❫
❈♦✐

C1 , C2

✬❛
❧❛
✒ ❤❛
✒♠ ❝✉

x✱

y = C1 x2 − C2 .

✳ ✳
✒ ♥❣ s❫
✓ ❜✐❫
✓♥ t❤✐❫
✓
❛ ♣ ❞✉
✓ ♣ ❤✕
❛
♦
❡
❡♥✿
✳ ♥❣ ♣❤✉ ♦ ♥❣ ♣❤❛

1
3x



C1 x2 + C2 (− 1 ) = 0
3x
1

C1 2x + C2 (
) = 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
✳
, q(x) = 1, f (x) =
✳ ❚✏
✒♠ ♥❣❤✐❫
❡
❡♥❣ t❤✓
✉ ❤❛✐✿
✳ ♠ 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 =
dx + K1
sin x
sin x
dx
x
− sin xdx + K1 = ln |tg | + cos x + K1
sin x
2

C2 = cos x → C2 = sin x + K2
❱❫
❛
❡
♦✬♥❣ q✉❛
✓ t✿ y = · · ·
✳ ② ♥❣❤✐❫
✳ ♠ t❫

107)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

y” − 2y + y = 1 +

ex
x

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3

✳ ✳
✳
2
P❤✉ ♦ ♥❣ tr✏
✒♥❤ ❞
✖✕
❛
✳ ❝ tr✉ ♥❣✿ λ − 2λ + 1 = 0 ⇔ λ = 1
✒
✓t✿ y = ex (C1 x + C2 )
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
◆❚◗ ❝✉
✳ ✳
✒ ♥❣ s❫
✓♥ t❤✐❫
✓ t✏

❉✉
✒♥❣ ♣❤✉ ♦ ♥❣ ♣❤❛
✓ ♣ ❜✐❫
❡
❡♥ ❤✕
❛
♦
✒♠ ♥❣❤✐❫
❡
❡♥❣ ❞❛
✳ ♠ r✐❫
✳ ♥❣✿
x
x
y = α1 (x).xe + α2 (x).e .

HD gia’i:


α1 (x).xex + α2 (x).ex = 0
α1 (x)(ex + xex ) + α2 (x).ex = 1 +

ex
x


1

α1 = e−x +
⇔

x
α = −(xex + 1)
2

❱❫
❛
✳②

α1 = −e−x + ln |x|
α2 = xe−x + e−x − x
✳
−x
❛
❡
❡♥❣✿ y = (ln |x| − e
)xex + (xe−x + e−x − x)ex
◆❤✉ ✈❫
✳ ② ♥❣❤✐❫
✳ ♠ r✐❫
x
x
x
❱❛
✒ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t✿ y = e (C1 x + C2 ) + xe ln |x| − xe + 1
✳ ♠ t❫

108)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

y” + y = xe−x

✳ ✳
✳
2
P❤✉ ♦ ♥❣ tr✏
✒♥❤ ❞
✖✕
❛
✳ ❝ tr✉ ♥❣✿ λ + λ = 0 ⇔
✒
✓t✿
✬ ❛ ♣❤✉✳♦✳♥❣ tr✏
✒♥❤ t❤✉❫
❛♥ ♥❤❫
❛
◆❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t ❝✉
✳ ♠ t❫
−x
❚✏
✒♠ ♥❣❤✐❫

❡
❡♥❣ ❞❛
(Ax + B)
✳ ♠ r✐❫
✳ ♥❣✿ y = xe
2
x
✓t q✉❛
✬ ✿ y = C1 + C2 e−x − (
❑❫
❡
+ x)e−x

HD gia’i:

λ1 = 0; λ2 = −1
y = C1 + C2 e−x

2

109)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

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 ; y2 =

110)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
✒♥❤✿
●✐❛

y” + 4y + 4y = 1 + e−2x ln x

✳
2
HD gia’i: P❤✉✳♦✳♥❣ tr✏✒♥❤ ❞✖✕

❛
✳ ❝ tr✉ ♥❣✿ λ + 4λ + 4 = 0 ⇔ λ = −2
−2x
◆❚◗ ✿ y = e
(C1 x + C2 )
−2x
❚✏
✒♠ ♥❣❤✐❫
❡
❡♥❣ ❞❛
+ α2 e−2x .
✳ ♠ r✐❫
✳ ♥❣✿ y = α1 (x).xe
α1 (x).xe−2x + α2 e−2x = 0
α (e−2x − 2xe−2x ) + α2 (−2e−2x ) = 1 + e−2x ln x
 1
1

α1 = e−2x + ln x → α1 = e−2x + x ln |x| − x
2
x2 1 2x x2
1

−2x
α2 = −x(e
− xe −
ln x
+ ln x) → α2 = e2x +
4
4

2
2
⇒ ♥❣❤✐❫
❡
❡♥❣ ⇒ ♥❣❤✐❫
❡
♦✬♥❣ q✉❛
✓ t✿
✳ ♠ r✐❫
✳ ♠ t❫
2
3x
x2
−2x
−2x 1 2x
y = e (C1 x + C2 ) + e ( e −
+
ln x)
4
4
2

111)

✳ ✳
✬ ✐ ♣❤✉ ♦ ♥❣ tr✏
●✐❛
✒♥❤✿

y” + y = e−x (sin x − cos x)

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