книги / Математический анализ динамических моделей
..pdfx(IV )(t) x(t) 0,
x(0) 2, x (0) 0, x (0) 2, x (0) 0.
x(IV )(t) 10x (t) 9x (t) sin(t),
x(0) 1, x (0) 1, x (0) 1, x (0) 1.
x(IV )(t) 5x (t) 4x(t) 0,
x(0) 6, x (0) 0, x (0) 12, x (0) 0.
x(IV )(t) 4x(t) 0,
x(0) 1, x (0) 2, x (0) 2, x (0) 0.
x(IV )(t) 6x (t) 9x (t) 0,
x(0) 0, x (0) 1, x (0) 6, x (0) 27.
x(IV )(t) 2x (t) x(t) 0,
x(0) 0, x (0) 1, x (0) 2, x (0) 8.
x(IV )(t) 8x (t) 16x (t) 0,
x(0) 1, x (0) 2, x (0) 4, x (0) 3.
x(IV )(t) 4x (t) 3x(t) 0,
x(0) 1, x (0) 1, x (0) 1, x (0) 1.
3x(IV )(t) x (t) 2, 36.
x(0) 1, x (0) 1, x (0) 2, x (0) 2.
Упражнение 4. Решить следующие задачи Коши. Найти передаточную матрицу-функцию и матрицу Коши, записать формулу Коши.
37. |
x (t) x(t) 2y(t), |
x(0) 0, |
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y (t) 2x(t) y(t) 1, |
y(0) 5. |
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38. |
x (t) 2y(t), |
x(0) 2, |
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y (t) 2x(t), |
y(0) 2. |
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39. |
x (t) 3x(t) 4y(t), |
x(0) 1, |
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y (t) 4x(t) 3y(t), |
y(0) 1. |
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40. |
x (t) y(t), |
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y(0) 1, |
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y (t) 2x(t) 2y(t), |
y(0) 1. |
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10e |
2t |
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x(0) 1, |
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41. |
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x (t) 2x(t) 2y(t) |
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2t |
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y(0) 3. |
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y (t) 2x(t) y(t) 7e |
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42. |
x (t) x(t) y(t), |
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x(0) 1, |
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y (t) x(t) y(t), |
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y(0) 1. |
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43. |
x (t) x(t) 3y(t),t |
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x(0) 0, |
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y (t) x(t) y(t) e |
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y(0) 0. |
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44. |
x (t) x(t) y(t), |
x(0) |
2, |
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y (t) x(t) y(t), |
y(0) |
0. |
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45. |
x (t) 2x(t) y(t), |
x(0) 1, |
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y (t) x(t) 2y(t), |
y(0) 3. |
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46. |
x (t) 4x(t) 6y(t), |
x(0) 0, |
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y (t) 2x(t) 3y(t) t, |
y(0) 0. |
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t |
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x(0) 1/8, |
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47. |
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x (t) y(t) |
3e |
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2e |
3t |
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y(0) 5/8. |
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y (t) x(t) |
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t |
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x(0) 0, |
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48. |
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x (t) y(t) e |
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t |
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y(0) 0. |
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y (t) x(t) e |
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x (t) x(t) y(t) et ,
y (t) x(t) y(t) et ,
50. x (t) y(t), |
x(0) 2, |
y (t) x(t), |
y(0) 5. |
x(0) 1, y(0) 1.
51. |
x (t) x(t) 2y(t), |
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x(0) 1, |
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y (t) x(t) 4y(t) 1, |
y(0) 1. |
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52. |
x (t) 2y(t) 3t, |
x(0) 2, |
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y (t) 2x(t) 4, |
y(0) 3. |
22
53. |
x (t) 3x(t) 8y(t), |
x(0) 6, |
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y (t) x(t) 3y(t), |
y(0) 2. |
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54. |
x (t) 9y(t), |
x(0) 3, |
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y (t) x(t), |
y(0) 1. |
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55. |
x (t) 3x(t) 4y(t), |
x(0) 1, |
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y (t) 2x(t) 5y(t), |
y(0) 4. |
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56. |
x (t) x(t) 5y(t), |
x(0) 2, |
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y (t) x(t) 3y(t), |
y(0) 1. |
x (t) 3x(t) 4y(t) 9e2t ,
y (t) 2x(t) 3y(t) 3e2t ,
58.x (t) 2x(t) 4y(t) cos(t),
y (t) x(t) 2y(t) sin(t),
x(0) 2, y(0) 0.
x(0) 0, y(0) 0.
59. x (t) 7x(t) y(t), |
x(0) 1, |
y (t) 2x(t) 5y(t), |
y(0) 1. |
x (t) x(t) y(t) 3t2 /2,
y (t) 4x(t) 2y(t) 4t 1,
61. |
x (t) y(t), |
x(0) |
1, |
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y (t) x(t), |
y(0) |
1. |
62. |
x (t) 2y(t) 3t, |
x(0) 0, |
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y (t) 2x(t) 4t, |
y(0) 0. |
x(0) 0, y(0) 0.
63. |
x (t) x(t) y(t), |
x(0) 2, |
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y (t) 4x(t) y(t), |
y(0) 0. |
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64. |
x (t) y(t), |
x(0) 1, |
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y (t) x(t), |
y(0) 1. |
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65. |
x (t) y(t) 1, |
x(0) 0, |
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y (t) x(t) 1, |
y(0) 1. |
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66. |
x (t) x(t) 5y(t), |
x(0) 0, |
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y (t) 2x(t) y(t), |
y(0) 3. |
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67. |
x(t) y(t) t, |
x(0) |
1, |
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y (t) x(t) t, |
y(0) |
1. |
68. |
x (t) 2x(t) y(t), |
x(0) 1, |
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y (t) x(t) 4x(t), |
y(0) 1. |
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69. |
x (t) x(t) 8y(t), |
x(0) 1, |
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y (t) x(t) y(t), |
y(0) 1. |
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70. |
x (t) x(t) y(t), |
x(0) 2, |
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y (t) x(t) y(t), |
y(0) 1. |
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71. |
x (t) 2x(t) y(t), |
x(0) 1, |
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y (t) 2x(t) 4y(t), |
y(0) 1. |
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72. |
x (t) x(t) y(t), |
x(0) 0, |
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y (t) 2(t) 4y(t), |
y(0) 1. |
Упражнение 5. Решить следующие задачи Коши, Найти передаточную матрицу-функцию и матрицу Коши, записать формулу Коши.
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x(0) 0, |
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x (t) 2x(t) y(t) z(t), |
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1. |
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y(0) 1, |
y (t) 2x(t) z(t), |
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z(0) 2. |
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z (t) 2x(t) y(t) 2z(t), |
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x(0) 2, |
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x (t) 2x(t) y(t) z(t), |
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2. |
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y(0) 1, |
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y (t) x(t) z(t), |
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z(0) 3. |
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z (t) 3x(t) y(t) 2z(t), |
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x(0) 2, |
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x (t) x(t) 2y(t) 2z(t), |
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3. |
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y(0) 2, |
y (t) x(t) 4y(t) 2z(t), |
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z(0) 3. |
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z (t) x(t) 5y(t) 3z(t), |
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2z(t), |
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x(0) 0, |
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x (t) x(t) 2y(t) |
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4. |
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2z(t), |
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y(0) 1, |
y (t) 2x(t) y(t) |
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z(0) 1. |
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z (t) 3x(t) 2y(t) 3z(t), |
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x(0) 0, |
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x (t) 3x(t) 2y(t) 2z(t), |
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5. |
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y(0) 1, |
y (t) 3x(t) y(t) z(t), |
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z(0) 1. |
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z (t) x(t) 2y(t), |
x (t) 3x(t)
6.y (t) 3x(t)
z (t) x(t)
3y(t) z(t), |
x(0) 1, |
2y(t) 2z(t), |
y(0) 1, |
2y(t), |
z(0) 1. |
x (t) 2x(t) y(t)
7.y (t) x(t) z(t),z (t) x(t) y(t),
z(t), x(0) 1, y(0) 1, z(0) 1.
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x(0) 1, |
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x (t) y(t) z(t), |
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8. |
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y(0) 1, |
y (t) x(t) z(t), |
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z(0) 2. |
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z (t) 2x(t) 2y(t) z(t), |
x (t)
9.y(t)z (t)
y(t) z(t), |
x(0) 1, |
x(t) y(t), |
y(0) 1, |
x(t) z(t), |
z(0) 1. |
x (t)
10.y (t)z (t)
x (t)
11.y (t)z (t)
x (t)
12.y (t)z (t)
x (t)
13.y (t)z (t)
2x(t) y(t) 2z(t), |
x(0) 1, |
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x(t) 2z(t), |
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y(0) 1, |
2x(t) 3z(t), |
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z(0) 1. |
y(t) z(t), |
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x(0) 1, |
x(t) z(t), |
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y(0) 1, |
2x(t) 2y(t) 3z(t), |
z(0) 0. |
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4x(t) 2y(t) 2z(t), |
x(0) 1, |
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x(t) 3y(t) z(t), |
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y(0) 1, |
3x(t) 3y(t) z(t), |
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z(0) 0. |
2x(t) z(t), |
x(0) 1, |
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x(t) y(t), |
y(0) 1, |
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3x(t) y(t) z(t), |
z(0) 2. |
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x (t)
14.y (t)z (t)
x (t)
15.y (t)z (t)
x (t)
16.y (t)z (t)
x (t)
17.y (t)z (t)
x(t) y(t) z(t), |
x(0) 1, |
x(t) y(t) z(t), |
y(0) 1, |
2x(t) y(t), |
z(0) 1. |
x(t) 2y(t) z(t), |
x(0) 1, |
x(t) y(t) z(t), |
y(0) 0, |
x(t) z(t), |
z(0) 1. |
2x(t) y(t) z(t), |
x(0) 0, |
x(t) 2y(t) z(t), |
y(0) 1, |
x(t) y(t) 2z(t), |
z(0) 1. |
3x(t) y(t) z(t), |
x(0) 1, |
x(t) y(t) z(t), |
y(0) 1, |
4x(t) y(t) 4z(t), |
z(0) 1. |
x (t)
18.y (t)z (t)
3x(t) 4y(t) 2z(t), |
x(0) 1, |
x(t) z(t), |
y(0) 1, |
6x(t) 6y(t) 5z(t), |
z(0) 0. |
x (t)
19.y (t)z (t)
x (t)
20.y (t)z (t)
x (t)
21.y (t)z (t)
x (t)
22.y (t)z (t)
x(t) y(t) z(t), |
x(0) 0, |
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x(t) y(t), |
y(0) 1, |
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3x(t) z(t), |
z(0) 1. |
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2x(t) y(t), |
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x(0) 1, |
x(t) 3y(t) z(t), |
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y(0) 0, |
x(t) 2y(t) 3z(t), |
z(0) 1. |
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2x(t) y(t) 2z(t), |
x(0) 0, |
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x(t) 2z(t), |
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y(0) 2, |
2x(t) y(t) z(t), |
z(0) 1. |
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4x(t) y(t) z(t), |
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x(0) 1, |
x(t) 2y(t) z(t), |
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y(0) 1, |
x(t) y(t) 2z(t), |
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z(0) 1. |
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x (t)
23.y (t)z (t)
2x(t) y(t) z(t), |
x(0) 1, |
3x(t) 2y(t) 3(t), |
y(0) 3, |
x(t) y(t) 2z(t), |
z(0) 1. |
x (t)
24.y (t)z (t)
2x(t) y(t) 2z(t), |
x(0) 1, |
x(t) 2y(t) 2z(t), |
y(0) 1, |
3x(t) 3y(t) 5z(t), |
z(0) 3. |
x (t)
25.y (t)z (t)
y(t) z(t), |
x(0) 1, |
x(t) y(t), |
y(0) 2, |
x(t) z(t), |
z(0) 3. |
x (t)
26.y (t)z (t)
x (t)
27.y (t)z (t)
x (t)
28.y (t)z (t)
x (t)
29.y (t)z (t)
x (t)
30.y (t)z (t)
x (t)
31.y (t)z (t)
4y(t) z(t), |
x(0) 5, |
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z(t), |
y(0) 0, |
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4y(t), |
z(0) 4. |
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4x(t) y(t), |
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x(0) 0, |
3x(t) y(t) z(t), |
y(0) 0, |
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x(t) z(t), |
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z(0) 1. |
3x(t) y(t) z(t), |
x(0) 1, |
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x(t) 5y(t) z(t), |
y(0) 2, |
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x(t) y(t) 3z(t), |
z(0) 1. |
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8y(t), |
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x(0) 4, |
2z(t), |
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y(0) 0, |
2x(t) 8y(t) 2z(t), |
z(0) 1. |
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x(t) y(t) z(t), |
x(0) 1, |
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x(t) y(t) z(t), |
y(0) 1, |
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x(t) y(t) z(t), |
z(0) 0. |
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2x(t) y(t) z(t), |
x(0) 1, |
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x(t) 2y(t) z(t), |
y(0) 1, |
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x(t) y(t) 2z(t), |
z(0) 1. |
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x (t)
32.y (t)z (t)
x (t)
33.y (t)z (t)
x (t)
34.y (t)z (t)
x (t)
35.y (t)z (t)
x (t)
36.y (t)z (t)
2x(t) y(t) z(t), |
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x(0) 0, |
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x(t) z(t), |
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y(0) 0, |
3x(t) y(t) 2z(t), |
z(0) 1. |
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6x(t) 12y(t) z(t), |
x(0) 1, |
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x(t) 3y(t) z(t), |
y(0) 1, |
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4x(t) 12y(t) 3z(t), |
z(0) 1. |
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x(t) z(t), |
x(0) |
1, |
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x(t), |
y(0) |
1, |
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x(t) y(t), |
z(0) |
0. |
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x(t) y(t) z(t), |
x(0) 1, |
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x(t) y(t) z(t), |
y(0) 1, |
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2x(t) y(t), |
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z(0) 1. |
15x(t) 6y(t) 16z(t), |
x(0) 2, |
15x(t) 7y(t) 18z(t), |
y(0) 2, |
19x(t) 8y(t) 21z(t), |
z(0) 1. |
Упражнение 6. Решить следующие интегральные уравнения. Найти передаточную функцию и резольвентное ядро. Записать обратный оператор.
1. |
x(t) sin(t) t |
(t s)x(s)ds |
19. |
x(t) t 2t |
sinh[2(t s)]x(s)ds |
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2. |
x(t) t 1/2t |
(t s)2 x(s)ds |
20. |
x(t) t3 |
t |
sin(t s)x(s)ds |
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3. |
x(t) t t |
sin(t s)x(s)ds |
21. |
x(t) e 4t 6t |
cosh[4(t s)]x(s)ds |
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4. |
x(t) cos(t) t |
et s x(s)ds |
22. |
x(t) et |
t 1 t |
x(s)ds |
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0 |
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0 |
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5. |
x(t) 1 t t |
cos(t s)x(s)ds |
23. |
x(t) t2 |
/2 t |
(t s)x(s)ds |
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6. |
x(t) t2 /2 t |
(t s)es t x(s)ds |
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0 |
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7. |
x(t) e t 1/2t |
(t s)2x(s)ds |
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0 |
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8. |
x(t) e t t |
es t |
sin(t s)x(s)ds |
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0 |
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9. |
x(t) sin(t) 2t |
cos(t s)x(s)ds |
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0 |
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10. x(t) 1 t |
e2(t s)x(s)ds |
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11. x(t) 1 1/2t |
sin[2(t s)]x(s)ds |
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0 |
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12. x(t) et 2t cos(t s)x(s)ds
0
13. x(t) 1 1/6t (t s)3x(s)ds
0
14. x(t) t t sinh(t s)x(s)ds
0
15. x(t) sinh(t) t cosh(t s)x(s)ds
0
16. x(t) cos(3t) t et s x(s)ds
0
17. x(t) e t /2 t (1 es t )x(s)ds
0
18. x(t) 2 1/6t (t s)3x(s)ds
0
24. x(t) te2t t e2(t s)x(s)ds
0
25. x(t) sin(t) t cos(t s)x(s)ds
0
26. x(t) et t sin(t s)x(s)ds
0
27. x(t) sin(t) t sinh(t s)x(s)ds
0
28. x(t) t2 /2 1/2t (t s)2x(s)ds
0
29. x(t) e2t t (t s)et s x(s)ds
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0 |
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30. |
x(t) e t |
t |
sin(t s)x(s)ds |
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0 |
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31. |
x(t) 1 t cos(t) sin(t) |
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t |
(t s)sin(t s)x(s)ds |
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0 |
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32. |
x(t) 1 2t 4t2 |
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t |
[3 6(t s) 4(t s)2]x(s)ds |
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0 |
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33. |
x(t) 1 |
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t |
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e |
(t s)/2 |
cos |
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3(t s)/2 x(s)ds |
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0 |
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34. |
x(t) t 2t (t s) |
0
sin(t s) x(s)ds
35. x(t) sin(2t)
8/3t sinh[3(t s)]x(s)ds
0
36. x(t) cos(5t)
7/4t sinh[4(t s)]x(s)ds
0
29
Упражнение 7. Решить следующие задачи Коши. Найти передаточную функцию и функцию Коши, записать формулу Коши.
1. x (t) x(t 2) 1, |
t 0, |
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x( ) 0, |
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0. |
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2. x (t) x(t 2), |
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x( ) 1, |
0. |
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3. x (t) x(t 2) 1, |
t 0, |
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x(ξ) 0, |
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ξ 0. |
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4. x (t) x(t 2), |
t 0, |
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x(ξ) 1, |
ξ 0. |
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t 0, |
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x (t) x(t 2) 1, |
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5. |
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ξ 0, |
x(ξ) |
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1, |
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x(0) |
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t 0, |
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x (t) x(t 2), |
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6. |
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1, |
ξ 0, |
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x(ξ) |
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0. |
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x(0) |
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t 0, |
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x (t) x(t 2) 1, |
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7. |
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0, |
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ξ 0, |
x(ξ) |
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1. |
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x(0) |
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t 0, |
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x (t) x(t 2), |
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8. |
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1, |
ξ 0, |
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x(ξ) |
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0. |
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x(0) |
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9. x (t) x(t 2), |
t 0, |
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x(ξ) ξ, |
ξ 0. |
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10. x (t) x(t 2), |
t 0, |
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x(ξ) ξ, |
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ξ 0. |
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t 0, |
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x (t) x(t 2) 1, |
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ξ 0, |
11. x(ξ) ξ, |
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x(0) 1. |
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19. |
x (t) x(t 1) 1, |
t 0, |
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x(ξ) ξ, |
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ξ 0. |
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20. |
x (t) x(t 1) 1, |
t 0, |
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x(ξ) 1, |
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ξ 0. |
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21. |
x (t) x(t 1) t, |
t 0, |
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x(ξ) 0, |
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ξ 0. |
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22. |
x (t) x(t 1) t, |
t 0, |
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x(ξ) 1, |
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ξ 0. |
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t 0, |
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x (t) x(t 1) t, |
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23. |
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0, |
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ξ 0, |
x(ξ) |
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1. |
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x(0) |
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t |
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x (t) x(t 1), |
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24. |
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1, |
ξ 0, |
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x(ξ) |
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0. |
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x(0) |
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t 0, |
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x (t) x(t 1) 1, |
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25. |
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1, |
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ξ 0, |
x(ξ) |
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1. |
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x(0) |
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t 0, |
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x (t) x(t 1) 1, |
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26. |
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ξ, |
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ξ 0. |
x(ξ) |
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1. |
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x(0) |
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t 0, |
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x (t) x(t 1) t, |
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27. |
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1, |
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ξ 0, |
x(ξ) |
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0. |
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x(0) |
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28. |
x (t) x(t 1) t, |
t 0, |
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x(ξ) ξ, |
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ξ 0. |
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29. |
x (t) x(t 1), |
t 0, |
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x(ξ) 1, |
ξ 0. |
30