help for # 34 and 37 Transcribed Image Text: 3.5 Problems In Problems 1 through 20, find a particular solution yp

Excerpt
32. y” + 3y’ + 2y = e*; y(0) = 0, y'(0) = 3 In Problems 47 through 56, use the method of variation of pa- rameters to find a particular solution of the given differential equation. = 2x; y(0) = 1, y'(0) = 2

help for # 34 and 37 Transcribed Image Text: 3.5 Problems
In Problems 1 through 20, find a particular solution yp of the
given equation. In all these problems, primes denote deriva-
tives with respect to x.
3. у” + 9у — sin 2x; y (0) — 1, у’ (0) — 0
34. y” + y = cos x; y(0) = 1, y'(0) = –1
35. у” — 2у’ + 2у %3D х + 1; у(0) — 3, у’ (0) —D 0
36. y(4) – 4y” = x²; y(0) = y’ (0) = 1, y”(0) = y(3) (0) =
37. y(3) – 2y” + y’ = 1 + xe*; y(0) = y'(0) = 0, y”(0) = 1
38. у” + 2y’ + 2у %3 sin 3x; y (0) 3 2, у’ (0) %3D 0
39. y(3) + y” = x +e¯*; y(0) = 1, y'(0) = 0, y”(0) = 1
40. y(4) – y = 5; y(0) = y'(0) = y”(0) = y(3) (0) = 0
41. Find a particular solution of the equation
= -1
1. у” + 16у — еЗx
3. у” — у’ — бу — 2 sin 3x
5. y” + y’ + y = sin? x
7. у” — 4у — sinh x
9. y” + 2y’ – 3y = 1 + xe*
10. y” + 9y = 2 cos 3x + 3 sin 3x
11. y(3) + 4y’ = 3x – 1
13. у” + 2у’ + 5у — е* sin x
15. у(5) + 5у(4) — у %3D 17
2. у” — у’ — 2у — Зх + 4
4. 4y” + 4y’ + у %3D 3хе*
6. 2y” + 4y’ + 7y = x²
8. y” – 4y = cosh 2x
y(4) – y(3) – y” – y’ – 2y = 8×5.
12. y(3) + y’ = 2 – sin x
14. у(4) — 2у” + у %— хе*
16. у” + 9у 3D 2х?езx + 5
42. Find the solution of the initial value problem consisting
of the differential equation of Problem 41 and the initial
conditions
17. y” + y = sin x + x cos x
18. y(4) – 5y” + 4y = e* – xe2x
19. y(5) + 2y(3) +2y” = 3x?
20. y(3) – y = e* +7
y (0) = y'(0) = y”(0) = y(3) (0)
= 0.
– 1
43. (а) Write
cos 3x + i sin 3x
= e31x
(cos x + i sinx)³
In Problems 21 through 30, set up the appropriate form of a
particular solution
coefficients.
by Euler’s formula, expand, and equate real and imag-
inary parts to derive the identities
Ур»
but do not determine the values of the
cos x =
cos x + cos 3x,
21. у” — 2у + 2y — е* sin x
22. y(5) – y(3) = e* +2x² – 5
23. у” + 4у %— 3x cos 2x
24. у(3) — у” — 12y’ — х — 2хе Зх
25. y” + 3y’ + 2y = x(e-* – e-2x)
26. у” — бу’ + 13у — хезx sin 2x
27. y(4) + 5y” + 4y
28. y(4) + 9y” = (x² + 1) sin 3x
29. (D — 1)3 (D2 — 4)у %3 хе* + е2х +е-2х
30. y(4) – 2y” + y = x² cos x
sin’ x =
sin x
i sin 3x.
=
(b) Use the result of part (a) to find a general solution of
y” + 4y = cos³ x.
Use trigonometric identities to find general solutions of the
equations in Problems 44 through 46.
= sin x + cos 2x
44. y” + y’ + y = sin x sin 3x
45. y” + 9y = sin* x
46. у” + у — х cos3 x
Solve the initial value problems in Problems 31 through 40.
31. у” + 4y
32. y” + 3y’ + 2y = e*; y(0) = 0, y'(0) = 3
In Problems 47 through 56, use the method of variation of pa-
rameters to find a particular solution of the given differential
equation.
= 2x; y(0) = 1, y'(0) = 2

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