книги / Математический анализ динамических моделей
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Упражнение 8. Решить следующие задачи Коши. Найти передаточную функцию.
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x (t) x(t π/ 2), |
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x (t) x(t π/ 2), |
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x(ξ) cos(ξ), |
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x (t) x(t / 2) 2sin(t), |
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x (t) x(t / 2) 2cos(t), |
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18.x (t)
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e x(t 1), |
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2x(t ln(2)), |
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x (t) x(t 2 ) 2cos(t), |
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x (t) x(t 2 ) 2(cos(t) sin(t)), |
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x (t) x(t / 2) 2cos(t), |
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x (t) x(t / 2) 2sin(t), |
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x (t) x(t ) cos(t) sin(t), |
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x (t) x(t ) cos(t) sin(t), |
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x (t) x(t / 2) 2sin(t), |
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Упражнение 9. Решить следующие задачи Коши.
3x(t 2) 2x(t 1) 8x(t) 0, 1. x(0) 1, x(1) 1.
2. 4x(t 2) 8x(t 1) 5x(t) 0, |
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0, x(1) 1. |
3. x(t 2) 4x(t 1) x(t) 0, |
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1, x(1) 0. |
x(t 2) 2x(t 1) 2x(t) 0,
4.x(0) 2, x(1) 0.
x(t 2) 4x(t 1) 3x(t) 0,
5.x(0) 0, x(1) 2.
x(t 2) 2x(t 1) x(t) 0,
6.x(0) 1, x(1) 1.
7. x(t 2) 2x(t 1) x(t) 0, |
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1, x(1) 0. |
8. x(t 2) x(t) 0, |
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0, x(1) 1. |
2x(t 2) 2x(t 1) x(t) 0, 9. x(0) 2, x(1) 0.
8x(t 2) 2x(t 1) x(t) 0, 10.
x(0) 0, x(1) 2.
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x(t 2) 3x(t 1) 2x(t) 0, |
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1, x(1) 1. |
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x(t 2) 5x(t 1) 6x(t) 0, |
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0, x(1) 1. |
21.x(t 2) 2x(t 1) x(t) 0,
x(0) 4, x(1) 5.
22.x(t 2) 6x(t 1) 5x(t) 0,
x(0) 0, x(1) 1.
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3x(t 1) 2x(t) 0, |
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1, x(1) 0. |
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x(t 1) x(t) 0, |
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0, x(1) 1. |
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3x(t 2) 5x(t 1) 3x(t) 0, |
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x(0) 0, x(1) 1. |
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x(t 2) |
3x(t 1) 4x(t) 0, |
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1, x(1) 1. |
27.2x(t 2) 5x(t 1) 2x(t) 0,
x(0) 2, x(1) 0.
28.x(t 2) 4x(t 1) 5x(t) 0,
x(0) 0, x(1) 1.
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4x(t 2) 4x(t 1) x(t) 0, |
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x(0) 1, x(1) 1. |
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6x(t 1) 7x(t) 0, |
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0, x(1) 1. |
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x(t 2) |
2x(t 1) 5x(t) 0, |
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1, x(1) 0. |
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x(t 2) |
x(t 1) x(t) 0, |
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x(0) |
1, x(1) 2. |
15.x(t 2) 4x(t 1) 4x(t) 0,
x(0) 2, x(1) 0.
16.x(t 2) 3x(t 1) 10x(t) 0,
x(0) 3, x(1) 1.
17.x(t 2) x(t 1) x(t) 0,
x(0) 1, x(1) 1.
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x(t) 0, |
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29. x(t 2) 2x(t 1) 10x(t) 0,x(0) 1, x(1) 1.
30.x(t 2) 4x(t) 0,
x(0) 2, x(1) 2.
31.x(t 2) 2x(t 1) 5x(t) 0,
x(0) 0, x(1) 1.
32.x(t 2) 9x(t) 0,
x(0) 1, x(1) 3.
33.x(t 2) 5x(t 1) 6x(t) 0,
x(0) 1, x(1) 1.
34. 3x(t 2) |
9x(t) 0, |
x(0) 0, |
x(1) 1. |
35.x(t 2) 2x(t 1) 3x(t) 0,
x(0) 2, x(1) 0.
36.x(t 2) x(t 1) 2x(t) 0,
x(0) 1, x(1) 0.
Упражнение 10. Решить следующие задачи Коши.
x(t 2) x(t 1) 2x(t) 1,
1.x(0) 1, x(1) 0.
2.
3.
4.
5.
6.
x(t 2) 9x(t) 3t , x(0) 0, x(1) 0.
x(t 2) 2x(t 1) 2x(t) t, x(0) 0, x(1) 1.
x(t 2) 6x(t 1) 9x(t) 3t , x(0) 0, x(1) 0.
x(t 2) x(t 1) 6x(t) ( 1)t , x(0) 0, x(1) 0.
x(t 2) x(t) 2, x(0) 0, x(1) 1.
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x(t 2) 4x(t 1) 5x(t) 4,
7.x(0) 0, x(1) 0.
x(t 2) 2x(t 1) 10x(t) 2 t,
8.x(0) 0, x(1) 0.
x(t 2) 4x(t) t 1,
9.x(0) 0, x(1) 0.
10.x(t 2) 2x(t 1) 2x(t) 8,x(0) 1, x(1) 0.
11. x(t 2) x(t) t,x(0) 0, x(1) 1.
12.x(t 2) 2x(t 1) x(t) 1,
x(0) 0, x(1) 0.
13.x(t 2) 2x(t 1) x(t) t,
x(0) 0, x(1) 0.
x(t 2) 2x(t 1) x(t) 32 3t ,
x(0) 0, x(1) 0.
15.x(t 2) x(t) sin(t),
x(0) 0, x(1) 1.
x(t 2) 4x(t 1) 4x(t) 3t ,
x(0) 0, x(1) 0.
x(t 2) 5x(t 1) 6x(t) 2 4t ,
x(0) 0, x(1) 0.
x(t 2) 3x(t 1) 4x(t) ( 1)t ,
x(0) 0, x(1) 0.
19.2x(t 2) 5x(t 1) 2x(t) cos(πt /3),
x(0) 0, x(1) 0.
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x(t 2) 3x(t 1) 2x(t) 2t ,
x(0) 0, x(1) 0.
21.x(t 2) x(t) 4t,
x(0) 0, x(1) 4.
22.x(t 2) 2x(t 1) 5x(t) t,
x(0) 0, x(1) 0.
23.x(t 2) 3x(t 1) 2x(t) t,
x(0) 0, x(1) 0.
24.x(t 2) 2x(t 1) 5x(t) t 1,
x(0) 0, x(1) 0.
25.x(t 2) 2x(t 1) 2x(t) t,
x(0) 0, x(1) 0.
x(t 2) 2x(t 1) x(t) ( 1)t ,
x(0) 0, x(1) 0.
27.x(t 2) 9x(t) sinh(t),
x(0) 0, x(1) 0.
28.x(t 2) 9x(t) sinh(t),
x(0) 0, x(1) 0.
29.x(t 2) 4x(t 1) 5x(t) t 1,
x(0) 0, x(1) 0.
30.x(t 2) x(t 1) 2x(t) t 1,
x(0) 0, x(1) 0.
x(t 2) 2x(t 1) x(t) ( 1)t ,
x(0) 0, x(1) 0.
32.x(t 2) x(t) sinh(t),
x(0) 1, x(1) 0.
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33.
34.
35.
36.
x(t 2) x(t 1) 6x(t) 3t , x(0) 0, x(1) 0.
x(t 2) 3x(t 1) 2x(t) 2t , x(0) 0, x(1) 0.
x(t 2) 7x(t 1) 10x(t) 5t , x(0) 0, x(1) 0.
x(t 2) 4x(t 1) 3x(t) 3t , x(0) 0, x(1) 0.
Индивидуальное задание № 3
Исследование на устойчивость ЛОДУ, систем ЛОДУ, ЛРУ и ЛДРУ
Упражнение 1. Исследовать на устойчивость нулевое решение ЛОДУ: 1) методом Рауса – Гурвица; 2) методом Льенара – Шипара;
3)методом Михайлова.
1.x IV 5x 2x x 0
2.x(IV ) 4x 7x 6x 2x 0
3.x 5x 9x 5x 0
4.x(IV ) 2x x 2x 2x 0
5.x(IV ) 7x 17x 17x 6x 0
6.x 3x 12x 10x 0
7.x(IV ) 5x 18x 34x 20x 0
8.x(IV ) 7x 19x 23x 10x 0
9.x(IV ) 11x 41x 61x 30x 0
10.x(V ) 3x(IV ) 5x 15x 4x 12x 0
11.x(V ) 7x(IV ) 33x 88x 122x 60x 0
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12.x(IV ) 2x 4x 3x 2x 0
13.x(IV ) 2x 3x 7x 2x 0
14.x(IV ) 2x 6x 5x 6x 0
15.x(IV ) 8x 14x 36x 45x 0
16.x(IV ) 13x 16x 55x 76x 0
17.x(IV ) 3x 26x 74x 85x 0
18.x(V ) 2x(IV ) 4x 6x 5x 4x 0
19.x(V ) 2x(IV ) 5x 6x 5x 2x 0
20.x(V ) 3x(IV ) 6x 7x 4x 4x 0
21.2x(IV ) 13x 28x 23x 6x 0
22.x(IV ) 4x 16x 24x 20x 0
23.x(V ) 13x(IV ) 43x 51x 40x 12x 0
24.x(IV ) x 8x x x 0
25.x(V ) 3x(IV ) 2x x 3x 2x 0
26.x(V ) x(IV ) x x x 0
27.x(V ) 13x(IV ) 43x 51x 40x 12x 0
28.x(IV ) 2x 4x x 0
29.x(V ) 4x(IV ) x x x 0
30.x(IV ) 5x 13x 19x 10x 0
31.x(V ) 4x(IV ) 16x 25x 13x 9x 0
32.x(V ) 3x(IV ) 10x 22x 23x 12x 0
33.x(V ) 5x(IV ) 15x 48x 44x 74x 0
34.x(V ) 2x(IV ) 14x 36x 23x 68x 0
35.x(IV ) 2x 3x 2x x 0
36.x(IV ) 12x x 2x x 0
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Упражнение 2. Исследовать на устойчивость нулевое решение системы ЛОДУ, записанной в векторной форме: x Ax , где x – вектор, A – данная матрица: 1) методом Рауса – Гурвица; 2) методом Льенара – Шипара; 3) методом Михайлова.
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|
|
|
|
2 |
|
|
1 |
1 |
|
|
1 |
|
||
13. |
|
1 |
1 |
|
|
|
|
|
|
|
|
1 |
|
||||
|
|
1 |
1 |
|
|
|
|
|
|
|
|
1 |
|
||||
|
|
1 |
1 |
|
|
1 |
|
|
14. |
|
1 |
0 |
|
|
|
|
|
|
|
|
1 |
|
||||
|
|
3 |
1 |
|
2 |
|
|
|
|
|
|
|
|
||||
|
|
3 |
12 |
|
4 |
|
||
15. |
|
1 |
3 |
|
|
|
|
|
|
|
|
1 |
|||||
|
|
1 |
12 |
|
6 |
|
||
|
|
|
|
|||||
|
1 |
2 |
|
|
0 |
|
|
|
16. |
|
0 |
1 |
|
|
|
|
|
|
|
1 |
|
|||||
|
|
0 |
0 |
|
|
|
|
|
|
|
|
1 |
|
||||
|
|
2 |
1 |
|
1 |
|
|
|
17. |
|
1 |
2 |
|
|
|
|
|
|
1 |
|
|
|||||
|
|
1 |
1 |
|
2 |
|
|
|
|
|
|
|
|
|
|||
|
1 |
1 |
|
|
5 |
|
|
|
18. |
|
0 |
2 |
|
|
|
|
|
|
|
|
1 |
|
||||
|
|
0 |
0 |
|
3 |
|
|
|
|
|
|
|
|
||||
|
2 |
1 |
|
|
0 |
|
|
|
19. |
|
1 |
2 |
|
|
0 |
|
|
|
|
|
|
|
||||
|
|
1 |
3 |
|
|
|
|
|
|
|
|
1 |
|
||||
|
1 |
2 |
|
2 |
|
|
|
|
20. |
|
|
4 |
2 |
|
|
|
|
1 |
|
|
|
|||||
|
|
|
5 |
3 |
|
|
|
|
|
1 |
|
|
|
|
|
0 |
1 |
1 |
||||
25. |
|
1 |
0 |
|
|
|
||
|
1 |
|||||||
|
|
1 |
1 |
|
|
|
||
|
|
|
0 |
|||||
|
2 |
2 |
|
4 |
|
|||
26. |
|
2 |
|
1 |
2 |
|
||
|
|
|
||||||
|
|
5 |
|
2 |
|
|
7 |
|
|
|
|
|
|
|
|||
|
|
1 |
0 |
|
0 |
|
|
|
27. |
|
2 |
1 |
|
0 |
|
|
|
|
|
|
|
|||||
|
|
1 |
1 |
|
|
|
|
|
|
|
|
1 |
|
||||
|
|
2 |
1 |
|
-2 |
|
|
|
28. |
|
-1 |
0 |
|
0 |
|
|
|
|
|
|
|
|
||||
|
|
1 |
1 |
|
-1 |
|
|
|
|
|
|
|
|
|
|||
|
|
0 |
8 |
|
0 |
|
|
|
29. |
|
0 |
0 |
2 |
|
|
|
|
|
|
|
|
|||||
|
|
2 |
8 |
2 |
|
|
|
|
|
|
|
|
|
||||
|
|
3 |
1 |
|
1 |
|||
30. |
|
1 |
5 |
|
|
|
||
|
1 |
|||||||
|
|
1 |
1 |
|
|
|
||
|
|
|
3 |
21 8 19
31.18 7 1516 156
|
1 |
2 |
2 |
|
|
32. |
|
2 |
1 |
2 |
|
|
|
||||
|
|
3 |
2 |
3 |
|
|
|
|
40