Файл: Т.С. Жирнова Дифференциальные уравнения.pdf

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25

№11

1.

y′ =

y ln y

;

7.

y′′−12y′+36y = e3x (49cos 2x +50sin 2x);

1 − x2 arcsin x

2.

xy′ = y(3 + ln y −ln x),

y(1)= e−1 ;

8.

y IV

− y = xex + cos x;

3.

y′+ xy = (1 + x)e−x y 2 ,

y(0)= −1;

9.

y1 = e−2 x cos x,

y2 = e−2 x sin x;

y′′ =

x

;

y′′−6 y′+9 y =

2e3x

4.

10.

;

x + 4

1 − x

2

5.

′′

′

+1;

y tqx = y

x′ = 4x −5y, x(0)= 0,

6.

y′′+ y′−2 y = 6x2 ;

11.

y′ = x,

y(0)=1.

№12

1.

(xy2 − y 2 )dx = (x2 y + x2 )dy;

2.

(y2 − 2x2 )dy + 2xydx = 0, y(1)=1;

3.xy′ln x = y + ln x;

4.y′′ = sin 2x sin 5x;

7.

y′′+ y′− 2 y = 6x2 e−2 x ;

8.

y IV −2 y′′′+ y′′ = xex

+cos x;

9.

y1 = e−x ,

y2 = xe−x ,

y3 = x2e−x ;

10. y′′−10 y′

+ 25y =

e5x

;

x

−1

x(0)

= −

7

,

5.

xy′′ = y′ln

y

′

;

11.

x′

= x + y

+t,

9

5

x

y′ = x − 2 y + 2t,

y(0)

= −

9 .

6.

y′′− y′ = 7 sin x;

№13

1.

sin2 xcos2

ydx −cos2 xdy = 0,

y(0)=

π

;

7.

y′′+ 6y′

+13y = e−3x

cos 4x;

2.

(x + 2y)dx − xdy = 9;

4

8.

y′′′+8y = e−2 x ;

3.

y′sin x − y cos x = y3 ,

π

=1;

9. y1 = e−7 x

,

y2

= xe−7 x ,

y3

= e2 x ;

y

2

4.

y′′ = e4 x +

1

;

10.

y′′−6 y

′+8y =

4e2 x

;

x

2 x

1

e

+

5.

(1 + x2 )y′′+ y′2 +1 = 0;

119

′

= e

t

− y −5x,

x(0)=

,

900

6.

y′′− 2 y′+ 2y = xex ;

11.

x

′

2t

211

= e

+ x

−3y,

y

y(0)=

.

900

26

№14

1.

( xy −2

x )y′− y = 0;

7.

y′′−8y′+16y = e3x (31sin 4x −22cos 4x);

3y

2

dx

=

(x

2

+3xy

+ y

2

)dy,

1

IV −

′′′

+

′′

=

+

x

2.

y

−1;

8.

y

2 y

y

x

e

;

3

dy = (ytqx −1)dx,

y(0)= 3;

3.

9.

y1

= e−x cos x,

y2

= e−x sin x;

4.

y′′ = cos x cos 2x cos 3x;

5.

y′′+

2

y′2 = 0;

10.

y′′− 2 y′+ y =

ex

;

1

− y

x

6.

y′′− 4 y = 8x3 ;

11.

x′ = 3x

+8y,

x(0)= 6,

y(0)= −2.

№15

y′ = −3y − x,

1.

cos x cos ydx −sin x sin ydy = 0,

π

= 0;

7.

y′′−4y = 8x2 e2 x ;

y

(x2

+ y 2 )dy −2xydx = 0,

y(2)=1;

2

2.

8.

yV − y′′′ = x;

3.

(y2

+ 2

xy )− xyy′ = 0;

9.

y1

= cos

3x,

y2

= sin

3x, y3 =1;

4.

y′′

=

1 + x − x2

;

10. y′′+ 4 y = 8ctq2x;

(1 − x2 )3

5.

xy′′− y′+ x = 0;

11.

x +2 y′=17x +8y,

x(0)= 2, y(0)= −1.

13x′=53x +2 y.

6.

y′′− 2 y′+ 2 y = ex cos 2x;

№16

7.

y′′+ 4 y′+5y = 5x2

1.

sin 2 yctqxdx + cos2 xtqydy = 0;

−32x +5;

xy′− y = x cos

2

y

y(4)

=π;

− y′′′ = (x −3)ex +e−x ;

2.

,

8.

yV

x

3.

2xy′+3y = (5x2 +3)y 2 ,

y(1)=1;

9.

y1

= ex ,

y2

= e0,5 x ,

y3 = e−0,5 x ;

4.

y′′ = ln 2x;

10.

y′′+12y′+36y =

12e−6 x

;

x5

xy′′′+ y′′ =1 + x;

5.

x′ = x +5y,

x(0)= −2,

6.

y′′− 2 y′+ 2 y = 2sin x;

y(0)=1.

11. y′ = −x −3y,

№17

1.

ln x sin3

ydx + x cos ydy = 0;

7. y′′+ y′−12 y = 54 cos 3x;

xdx = (

− y)dy, y(1)= 0;

27

2.

x2

+ y 2

8.

y IV

−2 y′′′+ y′′ = x2

+ 2 +sin x;

3.

(1 + x2 )y′−2xy = (1+ x2 )2 , y(3)= 40;

9.

y1 =1,

y2 = x,

y3

= e−5 x ;

4.

y′′ =

x2

−1

;

10.

y′′−12 y′+36 y =

2xe6 x

;

x

2

+1

x

2

+ 2

5.

y′′+ 2 yy′3

= 0;

x′+3x + y = 0,

x(0)=1,

6.

y′′− y′ = ex x;

11.

y(0)=1.

y′− x + y = 0,

(x + 2)(y2 +1)dx + (y 2

№18

1.

− x2 y2 )dy = 0;

7.

y′′+3y′+ 2 y = sin 2x + 2 cos 2x;

2.

xy′ = y + x

x

e

y

,

1

8.

y′′′+9 y′ = ex cos 3x;

y

= 0;

e

3.

3xy′+5y = (4x −5)y4 , y(1)=1;

9.

y1 =1, y2 = e7 x , y3 = e−3x ;

4.

yy′′ = y 2 y′+ y′2 ;

10.

y′′−

8y′+16 y

=

e4 x

;

x2

+1

y′′′ = sin x + cos 2x;

5.

′

2

x = 3x −0,5y −

3t

−0,5t +1,5,

6.

y′′+ 2y′+ y = ex ;

11.

y′ = 2 y −2t −1,

x(0)= 0, y(0)= 3.

1.

6ex cos2

ydx + (1 − 2ex )ctqydy = 0;

2.

(x2 + 2xy)dy −2 y2 dx = 0, y(2)=1;

3.

y 1 − y 2 dx + (x 1 − y 2 + y)dy = 0,

4.

y′′ =

1

;

1− x2

5.y′′ = ae y ;

6.y′′+5y′+ 6y = e−x + e−2 x ;

№19

7.y′′+ y′+ 2,5 y = 25 cos 2x;

8.y′′′+9 y′ = 3x cos 3x;

y(1)=1;

9. y1 = у−x ,

y2 = ex ,

y3 = e−5 x , y4 = e5x ;

y′′+10 y′+ 25y =

6e−5x

10.

;

x

4

x′ = 3

− 2 y,

x(0)

11.

=1,

y(0)= 0.

y′ = 2x − 2t,

1.

y′ = cos(y − x);

№20

y′′−5y′+ 6 y =13sin 3x;

7.

y

2

2 y

y(1)=1; 8.

IV

+ 2 y′′′+ y′′ = (x −7)e

−x

2.

4

−

dx +

dy = 0,

y

;

x

2

x

28

y′(1 − x

2

)= xy +1,

3

2π

3.

;

9.

y1 =1,

y2

= sin x,

y3 = cos x;

y

2

=

3

4.

y′′ = x sin x;

10.

y′′+ 2 y′+ 2 y = 2e−xtqx;

5.

y′′ =

a

;

11.

x′+ 7x − y = 0,

x(0)= 0,

y(0)=1.

y3

y′+5y + 2x = 0,

6.

y′′− 2 y = xe−x ;

№21

dy = x(2y + y 2 )dx;

y′′−3y′+ 2 y = x cos x;

1.

y(1)= 25;

7.

2.

(5 xy − y)dx + xdy = 0,

8.

y IV + y′′ = sin x;

3.

xy′− y = x3 sin x,

π

= 0;

9.

y1 =1,

y2

= sin 2x,

y3 = cos 2x;

y

2

3

4.

y′′ = tq 2 2x;

10.

y′′+ y =

;

(1− x2 )y′′− xy′ = 2;

sin5 x cos x

5.

x′ = −4(x + y),

x(0)=1,

6.

y′′+ 4 y′+5y = 5x2 −32x +5;

11.

y(0)= 0.

4 y′+ x′+ 4 y = 0,

1.

x2 y′−cos 2 y =1, y(∞)=

9π

;

4

2.

xy′ = x2 − y 2 + y;

3.

4xy′+ 2 y = 49x3 ln x, y(1)= 0;

4.y′′ = x cos x;

5.xy′′− y′+ x3 sin x = 0;

6.y′′+ 4 y′+ 4y = e2 x (148cos3x −

№22

7.

y′′− y′−6 y = e3x (45x2 − 2x +11),

′

y(0)= 0, y (0)= −2;

8.

y′′′−3y′+ 2 y = e−x (4x2

+ 4x −10);

9. y1

=1,

y2

= e−3x sin 2x,

y3 = e−3x cos 2x;

10.

y′′+

y′ =

1

;

e

x

+1

61sin 3x);

x′+ y = 0,

11. x′

− y′ = 3x + y.

№23

1.

y′3x2

+ x9−y

= 0, y(0)=1;

7.

y′′−4 y′+ 20y = e3x (36cos3x − 2sin 3x);

2.

( xy − y)dx + xdy = 0, y(1)=

1

;

8.

yV

− y′′′ = x2 −1;

4

3.

y′−6y = 2xe6 x ;

9.

y1

=1, y2 = x, y3 = e−2,5 x ;

4.

y′′ =

x cos x

;

10.

y′′+ 2 y′′+ 2y = 2e−x tqx;

sin 3 x

29

′

2

2x = 6x − y −

6t

−t +3,

5.

xy′′ =1 + x2 ;

11.

y′ = 2y −2t −1,

x(0)= 2, y(0)= 3.

6.

y′′−3y′+ 2y = ex ;

№24

(1 + x2 )y′ = xy − y 1+ x2 ;

y′′−6 y′+13y = e2 x (15x2 −2x −18);

1.

7.

2.

x(y′+ e y x )= y, y(1)= 0;

8.

y IV + y′′ = x;

3.

y′sin x − y cos x = x sin

3

x,

π

y

= 0;

1 −cos x

2

4.

y′′ =

;

9.

y1

= у−x ,

y2 = ex ,

y3 = e−5 x , y4 = e5x ;

1 + cos x

2ex

5.

x3 y′′+ x2 y′ =1;

10.

y′′

− y =

;

ex

−1

′

+ x − y

t

,

6.

y′′+3y′+ 2 y = sin 2x + 2 cos 2x;

11.

x

= e

′

t

− x + y

.

№25

y

= e

y′(x +

x )=

1.

1− y, y(0)=1;

7.

y′′−6 y′+ 25y = e4 x (17x2

+38x + 40),

(x

2

2

)dx + 2xydy = 0,

y(1)=1;

′

2.

−2y

y(0)=3,

y (0)=9;

3.

dy

=

y2

;

8.

y IV

+8y′′+16 y = cos x;

dx

x + ye− 1 y

4.

y′′ = 23x

+sin3 x;

9.

y1

=1,

y2

= e−2 x sin 3x,

y3

= e−2 x cos 3x;

5.

′′

=

2 y

′

2

;

10.

y

′′

− y =

ex −e−x

;

ex + e−x

y tqy

x(π)= −1,

6.

y′′−12y′+36y = e3x (49cos 2x +

50sin 2x);

11.

x′

= y,

y(π)= 0.

№26

x′− y′ = x + y,

ydx + (1 − x2 )dy = 0,

y(0)= 0;

1.

3x3

7.

y′′− 4 y′+ 4 y = xe2 x ;

2.

xy′′ = y cos ln

y

;

8.

y IV

+ y′′ = x2

+ cos x;

x

0,6 x

3.

2xy′−3y = −(20x2 +12)y3 ,

y(1)=

2

;

9.

y1 = e 0,6 x cos0,8x,

4

y2

= e

sin 0,8x;

4.

y′′ =

sin

4

x

;

10.

y′′+ y =

1

;

cos

2

x

cos 2x

cos 2x

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