Kolozova_Operacionnoe_ischislenie

10.

x

′′

+ x

′

= cos t,

x(0) = 2,

′

x (0) = 0.

2

x(0) = 1,

′

11.

x′′ + 2x′ + x = t ,

x (0) = 0.

12.

x

′′′

+ x

′′

= sin

t,

′

′′

x(0) = x

(0) = 1, x

(0) = 0.

13.

x

′′′

+ x

′′

= t,

x(0) = −3,

′

′′

x (0) = 1,

x (0) = 0.

14.

x

′′

+ 2x

′

+ 5x

= 3,

x(0) = 1,

′

x (0) = 0.

15.

x

′′

+ 2x

′

+ 2x

=1,

′

x(0) = x

(0) = 0.

16.

x

′′

+ x = 1,

x(0) = −1,

′

x (0) = 0.

17.

x

′′

+ 4x = t,

x(0) = 1,

′

x (0) = 0.

18.

x

′′

− 2x

′

+ 5x

= 1 − t,

′

x(0) = x

(0) = 0.

19.

x

′′

+ 3x

′

+ 2x

− 3 = 0,

x(0) = 1,

′

x (0) = 4.

20.

x

′′

+ 6x

′

+ 8x

−1 = 0,

x(0) = 1,

′

x (0) = 2.

21.

x

′′

+ 6x

′

+ 5x

− 2 = 0,

x(0) = 1,

′

x (0) = 3.

22.

x

′′

+ 7x

′

+ 10x − 4 = 0,

x(0) = 1,

′

x (0) = 5.

23.

x

′′

+ 8x

′

+ 12x − 5 = 0,

x(0) = 1,

′

x (0) = 2.

24.

x

′′

+ 9x

′

+ 20x − 1 = 0,

x(0) = 1,

′

x (0) = 3.

25.

x

′′

+ 4x

′

+ 4x

−1 = 0,

x(0) = 1,

′

x (0) = 6.

26.

x

′′

+ 6x

′

+ 9x

− 2 = 0,

x(0) = 1,

′

x (0) = 4.

27.

x

′′

+ 8x

′

+ 16x −1 = 0,

x(0) = 1,

′

x (0) = 7.

28.

x

′′

+ 10x

′

+ 25x − 2 = 0,

x(0) = 1,

′

x (0) = 3.

29.

x

′′

+ 12x

′

+ 36x − 4 = 0,

x(0) = 1,

′

x (0) = 5.

30.

x

′′

+ 14x

′

+ 49x − 2 = 0,

x(0) = 1,

′

x (0) = 8.

&

dx

означает

.

скобках указан один из корней характеристического уравнения. x

dt

x = y + 2 e ,

&

= y − 5cos t,

&

t

2.

x

1.

&

2

&

= 2x + y.

= x + t .

y

y

x(0) = −2, y(0) = 1.

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

&

= 3x + 2y +

4 e

5t

,

&

= 2x − 4y + 4 e

−2 t

,

3.

x

4.

x

&

= x + 2y.

&

= 2x − 2y.

y

y

x(0) = 3, y(0) = 4.

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

&

2 t

x = 2y − x + 1,

x = 4x + y − e ,

6.

&

5.

&

&

= y − 2x.

y = 3y − 2x.

y

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

x(0) = 1, y(0) = 1.

&

= 5x − 3y +

2e

3t

,

&

t

,

7.

x

8.

x

= 2x + y + e

−t

&

= −2x + 2t.

y = x + y + 5e .

&

y

x(0) = 1, y(0) = 0.

x(0) = 3, y(0) = −3.

9.

x&

= x + 2y,

10.

x& = 2x − 4y,

= x − 5 sin t.

t

&

y = x − 3y + 3e .

y

&

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

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

&

= 2x − y,

t

&

= x + 2y + 16 t e ,

x

11.

&

= y − 2x + 18t.

12.

x

y

&

y = 2x − 2y.

x(0) = − 2, y(0) = 3.

x(0) = 1 , y(0) = 1.

&

&

= 2x − 3y,

13.

x = 2x + 4y − 8,

14.

x

&

&

y = 3x + 6y.

y = x − 2 y + 2 sin t.

x(0) = 9, y(0) = 0.

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

x& = 2x − y,

x&

= x + z − y,

15.

t

&

= x + y − z,

&

= x + 2 e .

16.

y

y

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

= 2x − y

z&

x(0) = 2, y(0) = −7, z(0) = −10.

(λ1 = 1) .

&

= x − 2y − z,

&

= 2x − y + z,

x

x

17.

&

= y − x + z,

18.

&

= x + 2y − z,

y

y

&

&

= x − y + 2z

z = x − z

z

x(0) = 4, y(0) = 0, z(0) = −2.

x(0) = 5, y(0) = 3, z(0) = 6.

(λ1 = 1).

&

= 3x − y + z,

&

= 4y − 2z − 3x,

x

x

19.

&

= x + y + z,

20.

&

= z + x,

y

y

&

&

= 6x − 6y + 5z

z = 4x − y + 4z

z

x(0) = 6, y(0) = 0, z(0) = 2.

x(0) = 3, y(0) = 0, z(0) = −4.

(λ1 = 1).

(λ1 = 1).

&

= x − y − z,

&

= 2x + y,

x

x

21.

&

= x + y,

22.

&

= x + 3y − z,

y

y

&

= 3x + z

&

= 2y + 3z − x

z

z

x(0) = 2, y(0) = 1, z(0) = −5.

x(0) = 4, y(0) = 3, z(0) = 3.

(λ1 = 1).

(λ1 = 2).

&

= 2x + 2z − y,

&

= 4x − y − z,

x

x

23.

&

= x + 2z,

24.

&

= x + 2y − z

y

y

&

= y − 2x − z

&

= x − y + 2z

z

z

x(0) = 2, y(0) = 4, z(0) = 4.

x(0) = 6, y(0) = 3, z(0) = 4 .

(λ1 = 1).

(λ1 = 2).

&

= 2x − y − z,

&

= y − 2x − 2z,

x

x

25.

&

= 3x − 2y − 3z,

26.

&

= x − 2y + 2z,

y

y

&

= 2z − x + y

&

= 3x − 3y + 5z

z

z

x(0) = 3, y(0) = 8, z(0) = −3.

x(0) = 5, y(0) = 1, z(0) = −8.

(λ1 = 3) .

&

= 3x − 2y

− z,

&

= x − y + z,

x

x

&

= 3x − 4y

− 3z,

28.

&

= x + y − z,

27. y

y

&

= 2x − 4y

&

= 2z − y

z

z

x(0) =3, y(0) = 6, z(0) = −2.

x(0) = 4, y(0) = −3, z(0) = 2 .

(λ1 = −5) .

(λ1 = 2).

&

= y − 2z

− x,

&

= 2x + y,

x

x

29.

&

= 4x

+ y,

30.

&

= 2y + 4z,

y

y

&

+ y

− z

&

= x − z

z = 2x

z

x(0) = 2, y(0) = −1, z(0) = 0.

x(0) = 13, y(0) = 12, z(0) = 2.

(λ1 = 1) .