13.26.y′′′ − 2y′′ − 3y′ = (8x −14)e− x .

13.27.y′′′ + 2y′′ − 3y′ = (8x + 6)ex .

13.28.y′′′ + 6y′′ + 9y′ = (16x + 24)ex .

13.29.y′′′ − y′′ − 9y′ + 9y = (12 −16x)ex .

13.30.y′′′ + 4y′′ + 3y′ = 4(1− x)e− x .

13.31.y′′′ + y′′ − 6y′ = (20x +14)e2 x .

Задача 14. Найти общее решение дифференциального уравнения.

14.1. y′′ + 2y′ = 4ex (sin x + cos x).	14.2. y′′ − 4y′ + 4y = −e2x sin6x.
14.3. y′′ + 2y′ = −2ex (sin x + cos x).	14.4. y′′ + y = 2cos7x + 3sin 7x.
14.5. y′′ + 2y′ + 5y = −sin 2x.	14.6. y′′ − 4y′ + 8y = ex (5sin x − 3cos x).
14.7. y′′ + 2y′ = ex (sin x + cos x).	14.8. y′′ − 4y′ + 4y = e2x sin3x.
14.9. y′′ + 6y′ +13y = e−3x cos4x.	14.10. y′′ + y = 2cos3x − 3sin3x.
14.11. y′′ + 2y′ + 5y = −2sin x.	14.12. y′′ − 4y′ + 8y = ex (−3sin x + 4cos x).
14.13. y′′ + 2y′ =10ex (sin x + cos x).	14.14. y′′ − 4y′ + 4y = e2x sin5x.
14.15. y′′ + y = 2cos5x + 3sin5x.	14.16. y′′ + 2y′ + 5y = −17sin 2x.
14.17. y′′ + 6y′ +13y = e−3x cos x. 14.18.	y′′ − 4y′ + 8y = ex (3sin x + 5cos x).
14.19. y′′ + 2y′ = 6ex (sin x + cos x).	14.20. y′′ − 4y′ + 4y = −e2x sin 4x.
14.21. y′′ + 6y′ +13y = −e3x cos5x.	14.22. y′′ + y = 2cos7x − 3sin 7x.
14.23. y′′ + 2y′ + 5y = −cos x.	14.24. y′′ − 4y′ + 8y = ex (2sin x − cos x).
14.25. y′′ + 2y′ = 3ex (sin x + cos x).	14.26. y′′ − 4y′ + 4y = e2x sin 4x.
14.27. y′′ + 6y′ +13y = e−3x cos8x.	14.28. y′′ + 2y′ + 5y = 10cos x.

14.29. y′′ + y = 2cos4x + 3sin 4x. 14.30. y′′ − 4y′ + 8y = ex (−sin x + 2cos x).

14.31. y′′ − 4y′ + 4y = e2x sin 6x.

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Задача 15. Найти общее решение дифференциального уравнения.

15.1.y′′ − 2y′ = 2ch 2x.

15.2.y′′ + y = 2sin x − 6cos x + 2ex .

15.3.y′′′ − y′ = 2ex + cos x.

15.4.y′′ − 3y′ = 2ch3x.

15.5.y′′ + 4y = −8sin 2x + 32cos2x + 4e2x .

15.6.y′′′ − y′ =10sin x + 6cos x + 4ex .

15.7.y′′ − 4y′ =16ch 4y.

15.8.y′′ + 9y = −18sin3x −18e3x .

15.9.y′′′ − 4y′ = 24e2x − 4cos2x + 8sin 2x.

15.10.y′′ − 5y′ = 50ch5x.

15.11.y′′ +16y =16cos4x −16e4x .

15.12.y′′′ − 9y′ = −9e3x +18sin3x − 9cos3x.

15.13.y′′ − y′ = 2ch x.

15.14.y′′ + 25y = 20cos5x −10sin5x + 50e5x .

15.15.y′′′ −16y′ = 48e4x + 64cos4x − 64sin 4x.

15.16.y′′ + 2y′ = 2sh 2x.

15.17.y′′ + 36y = 24sin6x −12cos6x + 36e6x .

15.18.y′′′ − 25y′ = 25(sin5x + cos5x) − 50e5x .

15.19.y′′ + 3y′ = 2sh3x.

15.20.y′′ + 49y =14sin7x + 7cos7x − 98e7 x .

15.21.y′′′ − 36y′ = 36e6 x − 72(cos6x + sin6x).

15.22.y′′ + 4y′ = 16sh 4x.

15.23.y′′ + 64y =16sin8x −16cos8x − 64e8x .

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15.24.y′′′ − 49y′ =14e7 x − 49(cos7x + sin 7x).

15.25.y′′ + 5y′ = 50sh5x.

15.26.y′′ + 81y = 9sin9x + 3cos9x +162e9x .

15.27.y′′′ − 64y′ =128cos8x − 64e8x .

15.28.y′′ + y′ = 2sh x.

15.29.y′′ +100y = 20sin10x − 30cos10x − 200e10x .

15.30.y′′′ −81y′ =162e9x + 81sin9x.

15.31.y′′′ −100y′ = 20e10x +100cos10x.

Задача 16. Найти решение задачи Коши.

16.1. y′′ +π 2 y = π 2

cosπ x, y(

0) = 3,

y′(0) = 0.

16.2. y′′ + 3y′ = 9e3x

(1+ e3x ),

y(

0) = ln 4,

y′(0) = 3(1− ln 2).

16.3. y′′ + 4y = 8ctg2x,

y(π 4) = 5,

y′(π

4) = 4.

16.4. y′′ − 6y′ + 8y = 4 (1+ e−2x ),

y(0) =1+ 2ln 2, y′(0) = 6ln 2.

16.5. y′′ − 9y′ +18y = 9e3x

(1+ e−3x ),

y(0) = 0,

y′(0) = 0.

16.6. y′′ +π 2 y = π 2

sinπ x =1,

y(1 2),

y′(1 2) = π 2

2.

1

1

y(0) = 2,

y′(0) = 0.

16.7. y′′ +

y

=

,

π 2

π 2 cos(x π )

16.8. y′′ − 3y′ =

9e−3x

y(0) = 4ln 4,

y′(0) = 3(3ln 4 −1).

,

3 + e−3x

16.9. y′′ + y = 4ctg x, y(π

2) = 4,

y′(π

2) = 4.

16.10. y′′ − 6y′ + 8y = 4 (2 + e−2x ),

y(0) =1+ 3ln3,

y′(0) =10ln3.

16.11. y′′ + 6y′ + 8y = 4e−2x

(2 + e2x ),

y(0) = 0,

y′(0) = 0.

16.12. y′′ + 9y = 9 sin3x,

y(π

6) = 4,

y′(π 6) = 3π

2.

16.13. y′′ + 9y = 9 cos3x,

y(0) =1, y′(0) = 0.

126

16.14. y′′ − y′ = e−x (2 + e−x ),

y(0) = ln 27,

y′(0) = ln9 −1.

16.15. y′′ + 4y = 4ctg2x,

y(π

4) = 3,

y′(π 4) = 2.

16.16. y′′ − 3y′ + 2y =

1

,

y(

0) =1+ 8ln 2,

y′(

0) =14ln 2.

3 + e−x

16.17. y′′ − 6y′ + 8y = 4e2x

(1+ e−2x ),

y(0) = 0,

y′(0) = 0.

16.18. y′′ +16y =16 sin 4x,

y(π

8) = 3,

y′(π

8) = 2π.

16.19. y′′ +16y =16 cos4x,

y(0) = 3,

y′(0) = 0.

16.20. y′′ − 2y′ = 4e−2x (1+ e−2x ),

y(0) = ln 4,

y′(0) = ln 4 − 2.

16.21. y′′ +

y

=

1

ctg(x 2),

y(π ) = 2,

y′(π ) =1 2.

4

4

16.22. y′′ − 3y′ + 2y =1 (2 + e−x ),

y(0) =1+ 3ln3,

y′(0) = 5ln3.

16.23. y′′ + 3y′ + 2y = e−x (2 + ex ), y(

0) = 0,

y′(0) = 0.

16.24. y′′ + 4y = 4 sin 2x,

y(π 4) = 2,

y′(π

4) = π.

16.25. y′′ + 4y = 4 cos2x,

y(

0) = 2,

y′(0) = 0.

16.26. y′′ + y′ = ex (2 + ex ),

y(0) = ln 27,

y′(0) =1− ln9.

16.27. y′′ + y = 2ctg x,

y(π

2) =1,

y′(π

2) = 2.

16.28. y′′ − 3y′ + 2y =1 (1+ e−x ),

y(

0) =1+ 2ln 2,

y′(0) = 3ln 2.

16.29. y′′ − 3y′ + 2y = ex

(1+ e−x ),

y(0) = 0,

y′(

0) = 0.

16.30. y′′ + y =1 sin x,

y(π

2) =1,

y′(π

2) = π

2.

16.31. y′′ + y =1 cos x,

y(

0) =1,

y′(0) = 0.

127