BaiTapPhuongTrinhViPhan

8/3/2019 BaiTapPhuongTrinhViPhan

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BI TP PHNG TRNH VI PHNCHNG I: PHNG TRNH VI PHN CP 1

Phng trnh bin s phn ly, phng trnh thun nht

Cu 1. Gii cc phng trnh bin s phn ly sau

1. (x2 yx2)y + y2 + xy2 = 0,2. y

cos(2y) sin y = 0,3. y

+ sin(x + y) = sin(x y),4. y

= cos(x y),5. y

= x2 + 2xy 1 + y2,

6. y

= 1xy + 1.

Cu 2. Gii cc phng trnh sau

1. xdy ydx =

x2 + y2dx,

2. xyy

+ x2 2y2 = 0,3. (3x2 + y2)y + (y2 x2)xy = 0,

4. 2(x + yy

)2

= y2

(1 + (y

)2

),5. x cos y

x(ydx + xdy) = y sin y

x(xdy ydx).


1. x y 1 + (y x + 2)y = 0,2. (x + 4y)y

= 2x + 3y 5,3. y

= 2( y+2

x+y1)2,

4. (y

+ 1) ln y+xx+3

= y+xx+3

,

5. (2x y + 4)dx + (4x 2y)dy = 0, y(0) = 1.Cu 4. Xt phng trnh y

= f(y), trong f(y) lin tc trn (a, b). Bit rng tnti nghim y0(x) ca phng trnh trn sao cho lim

xy0(x) = c (a, b). Chng minh

rng hm y(x) = c cng l nghim ca phng trnh.

Cu 5. a. Tm ng cong m i vi n giao im ca tip tuyn bt k vi trc honhcch u tip im v gc to .b. Tm ng cong sao cho khong cch gia tip tuyn bt k ca n ti gc to bnghonh tip im.


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Cu 6. a. Tm ng cong sao cho t s gia on thng trn trc Oy b ct bi tiptuyn v on thng trn trc Ox b ct bi php tuyn k t tip im l mt i lngkhng i.b. Tm ng cong m i vi n t s gia on thng b ct bi php tuyn trn trcOx v bn knh vct ti tip im l mt i lng khng i.

Cu 7. Chng minh rng bt k ng cong no nhn c t ng cong tch phn caphng trnh thun nht bng php bin i ng dng, tm ti gc to cng l ngcong tch phn.

Cu 8. Chng minh rng mi ng cong tch nghim ca phng trnh y

= 3

y2+1x4+1

c hai tim cn ngang.

Cu 9. Tm tt c cc hm f(x) sao cho

f(x + y) =f

(x

) +f

(y

)1 f(x)f(y)

Phng trnh tuyn tnh, phng trnh Bernoulli


1. 2x(x 1)y + (2x 1)y + 1 = 0,2. x(1 + x2)y

(x2

1)y + 2x = 0,

3. xy y = x2 arctanx,

4. y

+ tan y = xcos y

,

5. (x 2xy y2)y + y2 = 0,6. (x + y2)dy = ydx.


1. xy2 + x2(1 + x)yy

+ 3x 5 = 0,2. y

+ xy = x3y3,

3. y

+ y = ex

2

y, y(0) = 9

4,

4. ydx + (x + x2y)dy = 0,

5. y

(x2y3 + xy) = 1.

Cu 12. Gii cc phng trnh tch phn sau:

1. y(x) =x0

y(t)dt + 1 + x,


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2.x0

(x t)y(t)dt = 2x +x0

y(t)dt,

3. xx0

y(t)dt = (x + 1)x0

ty(t)dt.

Cu 13. Xt cc phng trnh

y

+ p(x)y = q(x) (1)

y

+ p(x)y = 0 (2)

Chng minh rng

1. Hiu hai nghim ca (1) l mt nghim ca (2),

2. Tng mt nghim ca (1) v mt nghim ca (2) l mt nghim ca (1),

3. Nghim tng qut ca (2) cng vi mt nghim ring ca (1) cho ta nghim tngqut ca (1).

Cu 14. Chng minh rng

1. phng trnh x(x2 + 1)y (2x2 + 3)y = 3 c mt nghim l tam thc bc hai.

Gii phng trnh y.

2. hm s y = xx

1 et2dt l nghim ca phng trnh xy

y = x2ex2. Tm nghim

ring ca phng trnh y tho mn iu kin y(1) = 1.

Cu 15. Cho y1, y2 l hai nghim khc nhau ca mt phng trnh tuyn tnh cp 1. Hybiu din nghim tng qut ca phng trnh qua hai nghim ny.

Cu 16. Da vo dng ca phng trnh Bernoulli hy chng t trc Ox l ng congduy nht c th cho ta nghim k d ca phng trnh .

Cu 17. Chng minh rng nghim ca phng trnh tuyn tnh

y

+ p(x)y = q(x)tho mn iu kin u y(x0) = y0 c th vit di dng

y = e

xx0

p(t)dt

[y0 +

xx0

q(t)e

tx0

p(s)ds

].

Cu 18. Chng minh rng mt trong cc nghim ca phng trnh

y

+ ky = kq(x), (0 x < +)trong k l hng s, l biu thc y(x) = k

0

q(x t)ektdt (vi iu kin tch phnny tn ti).


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Cu 19. Chng minh rng, phng trnh

y

+ ay = P(x),

trong a = const, P(x) l a thc cp m ca x, c nghim ring dng y1 = Q(x),Q(x) l a thc cp m.

Cu 20. Chng minh rng bt k phng trnh tuyn tnh

y

+ p(x)y = q(x)

c nghim ring dng y1 = b, l phng trnh bin s phn ly.

Cu 21. Chng minh rng phng trnh tuyn tnh

y

= ky + f(x),

trong k = 0, f(x) l hm tun hon chu k T c mt nghim ring duy nht l hmtun hon chu k T. Hy tm nghim ring .

Phng trnh vi phn ton phn


1. (x + y + 1)dx + (x y2 + 3)dy = 0,2. (6xy2 + 4x3)dx + (3y2 + 6x2y)dy = 0,

3. 3x2(1 + ln y)dx + ( x3

y 2y)dy = 0,

4. [ y2

(xy)2 1x ]dx + [ 1y x2

(xy)2 ]dy = 0.

Cu 23. Gii cc phng trnh sau y, bit rng chng c tha s tch phn dng (x)hoc (y):

1. (x2 + y)dx = xdy,

2. (2xy2 y)dx + (y2 + x + y)dy = 0,3. (x

y+ 1)dx + ( x

y 1)dy = 0,

4. (x cosy y sin y)dy + (x sin y + y cos y)dx = 0.Cu 24. 1. Tm tha s tch phn dng (x + y) v gii phng trnh:

(2x3 + 3x2y + y2 y3)dx + (2y3 + 3xy2 + x2 x3)dy = 0.

2. Tm tha s tch phn dng (xy) v gii phng trnh:

xdy + ydx xy2 ln xdx = 0.


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3. Tm tha s tch phn dng (x2 y) v gii phng trnh:

(

x2 y + 2x)dx dy = 0.

Phng trnh cp mt cha gii ra o hm

Cu 25. Gii cc phng trnh sau:

1. yy2 (xy + 1)y + x = 0,

2. y3 y

4x= 0,

3. y2(1 + y2) = a2,

4. y = y2

2+ ln y

,

5. y3

+ y

3

3ayy

= 0,6. y

3 + (x + 2)ey = 0,

7. (xy

+ 3y)2 = 7x,

8. y

(2y y) = y2 sin2 x,9. y2 2xy = x2 4y,

10. x2y2 = xyy

+ 1,

11. 5y + y2 = x(x + y

),

12. y3 + y2 = xyy

,

13. y = xy x2y3,

14. 2y = xy2 2y3.

Bi ton qu o

Cu 26. Lp phng trnh vi phn nhn h ng cong sau lm h nghim

1. y = Cx3,

2. y = sin(x + C),

3. Cy sin(Cx) = 0,4. (x C)2 + y2 = 1,

5. x = y2 + 2y + C.


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Cu 27. a. Lp phng trnh vi phn ca h ng trn bn knh bng 1 v tm nm trnng thng y = 2x.b. Lp phng trnh vi phn ca nhng ng trn tip sc vi c hai trc to .c. Lp phng trnh vi phn ca nhng parabol c trc song song vi Oy v tip sc vicc ng thng y = 0 v y = x.

Cu 28. Hy vit phng trnh ca qu tch nhng im (x, y) l im cc i hocim cc tiu ca nghim phng trnh y

= f(x, y). Lm th no phn bit cim cc i v im cc tiu.

Cu 29. Tm qu o trc giao ca nhng h ng cong sau:

1. (x2 + y2)2 = 2xy,

2. x2

a2+ y

2

b2= , (a, b cho trc),

3. (x )2 + y2 = a2, (a cho trc),4. x(x2 + y2) = (x2 y2),5. x3 + (x 2)y2,6. r2 = ln tg + ,

7. r2n 2rnan cos(n) + a2n = b2n, (a, n cho trc).

CHNG II: PHNG TRNH VI PHN CP CAO

Phng trnh vi phn cp cao h cp c


1. x + sin y + 2y = 0,

2. x = ey

+ y,

3. y2 + y = xy,

4. (1 x2)y + xy = 2,5. xy = y + x sin y

x,

6. y = 2(y 1)cotgx.


1. y = 2yy ,

2. yy + 1 = y2,


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3. y(1 + y2) = ay,

4. y2 = (3y 2y)y,5. (y + 2y)y = y2,

6. 2yy = y2 + y2,

7. yy + y = y2.


1. xyy xy2 = yy,2. (1 + x2)(y2 yy ) = xyy,3. yy = y2 + 15y2

x,

4. x2yy = (y xy)2,5. y + y

x+ y

x2= y

2

y

6. x2yy = (y xy)2,Cu 33. Tm ng cong tch phn ca phng trnh yy + y2 = 1 i qua im (0, 1)v tip sc vi ng thng x + y = 1 ti im . C bao nhiu ng cong nh vy?.

Phng trnh tuyn tnh

Cu 34. Cho h hm y1(x), y2(x), . . . , yk(x) lin tc trn on [a, b]. Chng minhrng h hm trn ph thuc tuyn tnh trn on khi v ch khi

ba

y21(x)dxb

a

y1(x)y2(x)d x . . .b

a

y1(x)yk(x)dx

b

ay2(x)y1(x)dx

b

ay22(x)dx . . .

b

ay2(x)yk(x)dx

. . . . . . . . . . . .ba

yk(x)y1(x)dxb

a

yk(x)y2(x)d x . . .b

a

y2k(x)dx

= 0.

Cu 35. Chng minh rng phng trnh vi phn tuyn tnh khng thun nht cp n vicc h s lin tc trn (a, b) c ng n + 1 nghim c lp tuyn tnh trn (a, b).

Cu 36. Bit rng nh thc Wronsky ca cc hm s y1, y2, . . . yn bng 0 ti im x0v khc 0 ti im x1. C th ni g v s ph thuc tuyn tnh hoc c lp tuyn tnhca h hm trn khng?.

Cu 37. Chng minh rng hai nghim y1(x), y2(x) ca phng trnh y

+ p(x)y

+q(x)y = 0 (p(x), q(x) lin tc trn (a, b)) cng t cc tiu ti mt im th chngph thuc tuyn tnh.


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Cu 38. Hai nghim y1(x), y2(x) ca phng trnh y

+ p(x)y

+ q(x)y = 0(p(x), q(x) lin tc trn (a, b)) c th ct nhau c khng, c th tip sc vi nhauc khng?.

Cu 39. Tm phng trnh vi phn tuyn tnh nhn h hm sau y lm h nghim cbn:

1. 1, x , x2;

2. cosxx

, sinxx

;

3. cos2 x, sin2 x.

Cu 40. Gii cc phng trnh sau nu bit mt nghim ring ca phng trnh thunnht tng ng

1. y +2

x y + y = 0, y1 =sinx

x .

2. y sin2 x 2y = 0, y1 = cotgx.3. x2y 2xy + 2y = 2x3, y1 = x.4. y + x

1x y 1

1x y = x 1, y1 = ex.Cu 41. Tm nghim tng qut ca phng trnh

x3y

3x2y + 6xy

6y = 0,

bit hai nghim ring ca n l y1 = x, y2 = x2.

Phng trnh tuyn tnh h s hng


1. y 7y + 6y = sin x,2. y + 9y = 6e3x,

3. y 3y = 2 6x,4. y 2y + 3y = ex cosx,5. y + 4y = 2 sin(2x),

6. y + 2y + y = 4ex,

7. y + 2y + 5y = 2xex cos(2x),

8. y + y = x2 cos2 x,

9. y 3y = e3x 18x,10. y 2y + (1 + 2)y = (1 + 42) cos(x), y(0) = 1, y(0) = 0,


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11. y (m + 1)y + my = ex x 1.Cu 43. Chng minh rng nghim y(x) ca phng trnh

y + 2y = f(x)

vi iu kin u y(0) = y

(0) = 0 c dng

y(x) =

x0

sin (x t)f(t)dt.

Cu 44. Cho phng trnhy + ay + by = 0.

Tm iu kin ca cc hng s a, b tho mn

1. Mi nghim ca phng trnh u b chn trn [0, +).2. Mi nghim ca phng trnh u dn n 0 khi x +.

Cu 45. Tm mi gi tr ca p, q cho mi nghim ca phng trnh

y + py + qy = 0

l nhng hm tun hon ca x.

Cu 46. Vi gi tr no ca k v th phng trnhy + k2y = sin(x)

c t nht mt nghim tun hon.

CHNG III: H PHNG TRNH VI PHN

Cu 47. Gii cc h phng trnh vi phn sau

1.

{y

= 4y 2zz

= y + z

3.

{y

= z yz

= 3z y

5.

dx

dt= x + y + z

dydt

= x y + zdz

dt= x + y z

2.

{y

= 4y 3zz

= 3y + 4z

4.

{y

= y + 8z

z

= 2y + z

6.

dx

dt= x + y

dydt

= x + 2y + zdz

dt= x + z


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Cu 48. Gii cc h phng trnh vi phn sau

1.

y

= 1 1z

z

=1

y x

3. y

=

y2

z

z

=y

2

2.

y

=y

2y + 3z

z

=z

2y + 3z

4.

y

=

x

yz

z

=x

y2

Cu 49. Gii cc h phng trnh vi phn sau bng cc phng php khc nhau

1.

{y

= 3y + 2z + 4e5x

z

= y + 2z

3.{ y

= y + 2z + 16 + ex

z

= 2y 2z

2.

{y

= 2y 4z + 4e2xz

= 2y 2z

4.{ y

= 4y 3z + sin xz

= 2y z 2 cos x

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