Файл: Т.С. Жирнова Дифференциальные уравнения.pdf

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.

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

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.

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.

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