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# 9.3.1: Undetermined Coefficients for Higher Order Equations (Exercises)

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## Q9.3.1

In Exercises 9.3.1-9.3.59 find a particular solution.

1. $$y'''-6y''+11y'-6y=-e^{-x}(4+76x-24x^2)$$

2. $$y'''-2y''-5y'+6y=e^{-3x}(32-23x+6x^2)$$

3. $$4y'''+8y''-y'-2y=-e^x(4+45x+9x^2)$$

4. $$y'''+3y''-y'-3y=e^{-2x}(2-17x+3x^2)$$

5. $$y'''+3y''-y'-3y=e^x(-1+2x+24x^2+16x^3)$$

6. $$y'''+y''-2y=e^x(14+34x+15x^2)$$

7. $$4y'''+8y''-y'-2y=-e^{-2x}(1-15x)$$

8. $$y'''-y''-y'+y=e^x(7+6x)$$

9. $$2y'''-7y''+4y'+4y=e^{2x}(17+30x)$$

10. $$y'''-5y''+3y'+9y=2e^{3x}(11-24x^2)$$

11. $$y'''-7y''+8y'+16y=2e^{4x}(13+15x)$$

12. $$8y'''-12y''+6y'-y=e^{x/2}(1+4x)$$

13. $$y^{(4)}+3y'''-3y''-7y'+6y=-e^{-x}(12+8x-8x^2)$$

14. $$y^{(4)}+3y'''+y''-3y'-2y=-3e^{2x}(11+12x)$$

15. $$y^{(4)}+8y'''+24y''+32y'=-16e^{-2x}(1+x+x^2-x^3)$$

16. $$4y^{(4)}-11y''-9y'-2y=-e^x(1-6x)$$

17. $$y^{(4)}-2y'''+3y'-y=e^x(3+4x+x^2)$$

18. $$y^{(4)}-4y'''+6y''-4y'+2y=e^{2x}(24+x+x^4)$$

19. $$2y^{(4)}+5y'''-5y'-2y=18e^x(5+2x)$$

20. $$y^{(4)}+y'''-2y''-6y'-4y=-e^{2x}(4+28x+15x^2)$$

21. $$2y^{(4)}+y'''-2y'-y=3e^{-x/2}(1-6x)$$

22. $$y^{(4)}-5y''+4y=e^x(3+x-3x^2)$$

23. $$y^{(4)}-2y'''-3y''+4y'+4y=e^{2x}(13+33x+18x^2)$$

24. $$y^{(4)}-3y'''+4y'=e^{2x}(15+26x+12x^2)$$

25. $$y^{(4)}-2y'''+2y'-y=e^x(1+x)$$

26. $$2y^{(4)}-5y'''+3y''+y'-y=e^x(11+12x)$$

27. $$y^{(4)}+3y'''+3y''+y'=e^{-x}(5-24x+10x^2)$$

28. $$y^{(4)}-7y'''+18y''-20y'+8y=e^{2x}(3-8x-5x^2)$$

29. $$y'''-y''-4y'+4y=e^{-x}\left[(16+10x)\cos x+(30-10x)\sin x\right]$$

30. $$y'''+y''-4y'-4y=e^{-x}\left[(1-22x)\cos 2x-(1+6x)\sin2x\right]$$

31. $$y'''-y''+2y'-2y=e^{2x}[(27+5x-x^2)\cos x+(2+13x+9x^2)\sin x]$$

32. $$y'''-2y''+y'-2y=-e^x[(9-5x+4x^2)\cos 2x-(6-5x-3x^2)\sin2x]$$

33. $$y'''+3y''+4y'+12y=8\cos2x-16\sin2x$$

34. $$y'''-y''+2y=e^x[(20+4x)\cos x-(12+12x)\sin x]$$

35. $$y'''-7y''+20y'-24y=-e^{2x}[(13-8x)\cos 2x-(8-4x)\sin2x]$$

36. $$y'''-6y''+18y'=-e^{3x}[(2-3x)\cos 3x-(3+3x)\sin3x]$$

37. $$y^{(4)}+2y'''-2y''-8y'-8y=e^x(8\cos x+16\sin x)$$

38. $$y^{(4)}-3y'''+2y''+2y'-4y=e^x(2\cos2x -\sin2x)$$

39. $$y^{(4)}-8y'''+24y''-32y'+15y=e^{2x}(15x\cos2x+32\sin2x)$$

40. $$y^{(4)}+6y'''+13y''+12y'+4y=e^{-x}[(4-x)\cos x-(5+x)\sin x]$$

41. $$y^{(4)}+3y'''+2y''-2y'-4y=-e^{-x} (\cos x-\sin x)$$

42. $$y^{(4)}-5y'''+13y''-19y'+10y=e^x (\cos2x+\sin2x)$$

43. $$y^{(4)}+8y'''+32y''+64y'+39y=e^{-2x}[(4-15x)\cos3x-(4+15x)\sin 3x]$$

44. $$y^{(4)}-5y'''+13y''-19y'+10y=e^x[(7+8x)\cos 2x+(8-4x)\sin2x]$$

45. $$y^{(4)}+4y'''+8y''+8y'+4y=-2e^{-x} (\cos x-2\sin x)$$

46. $$y^{(4)}-8y'''+32y''-64y'+64y=e^{2x} (\cos2x-\sin2x)$$

47. $$y^{(4)}-8y'''+26y''-40y'+25y=e^{2x}[3\cos x-(1+3x)\sin x]$$

48. $$y'''-4y''+5y'-2y=e^{2x}-4e^x-2\cos x+4\sin x$$

49. $$y'''-y''+y'-y=5e^{2x}+2e^x-4\cos x+4\sin x$$

50. $$y'''-y'=-2(1+x)+4e^x-6e^{-x}+96e^{3x}$$

51. $$y'''-4y''+9y'-10y=10e^{2x}+20e^x\sin2x-10$$

52. $$y'''+3y''+3y'+y=12e^{-x}+9\cos2x-13\sin2x$$

53. $$y'''+y''-y'-y=4e^{-x}(1-6x)-2x\cos x+2(1+x)\sin x$$

54. $$y^{(4)}-5y''+4y=-12e^x+6e^{-x}+10\cos x$$

55. $$y^{(4)}-4y'''+11y''-14y'+10y=-e^x(\sin x+2\cos2x)$$

56. $$y^{(4)}+2y'''-3y''-4y'+4y=2e^x(1+x)+e^{-2x}$$

57. $$y^{(4)}+4y=\sinh x\cos x-\cosh x\sin x$$

58. $$y^{(4)}+5y'''+9y''+7y'+2y=e^{-x}(30+24x)-e^{-2x}$$

59. $$y^{(4)}-4y'''+7y''-6y'+2y=e^x(12x-2\cos x+2\sin x)$$

## Q9.3.2

In Exercises 9.3.60-9.3.68 find the general solution.

60. $$y'''-y''-y'+y=e^{2x}(10+3x)$$

61. $$y'''+y''-2y=-e^{3x}(9+67x+17x^2)$$

62. $$y'''-6y''+11y'-6y=e^{2x}(5-4x-3x^2)$$

63. $$y'''+2y''+y'=-2e^{-x}(7-18x+6x^2)$$

64. $$y'''-3y''+3y'-y=e^x(1+x)$$

65. $$y^{(4)}-2y''+y=-e^{-x}(4-9x+3x^2)$$

66. $$y'''+2y''-y'-2y=e^{-2x}\left[(23-2x)\cos x+(8-9x)\sin x\right]$$

67. $$y^{(4)}-3y'''+4y''-2y'=e^x\left[(28+6x)\cos 2x+(11-12x)\sin2x\right]$$

68. $$y^{(4)}-4y'''+14y''-20y'+25y=e^x\left[(2+6x)\cos 2x+3\sin2x\right]$$

## Q9.3.3

In Exercises 9.3.69-9.3.74 solve the initial value problem and graph the solution.

69. $$y'''-2y''-5y'+6y=2e^x(1-6x),\quad y(0)=2, \quad y'(0)=7,\quad y''(0)=9$$

70. $$y'''-y''-y'+y=-e^{-x}(4-8x),\quad y(0)=2, \quad y'(0)=0,\quad y''(0)=0$$

71. $$4y'''-3y'-y=e^{-x/2}(2-3x),\quad y(0)=-1, \quad y'(0)=15,\quad y''(0)=-17$$

72. $$y^{(4)}+2y'''+2y''+2y'+y=e^{-x}(20-12x),\, y(0)=3,\; y'(0)=-4,\; y''(0)=7,\; y'''(0)=-22$$

73. $$y'''+2y''+y'+2y=30\cos x-10\sin x, \quad y(0)=3,\quad y'(0)=-4,\quad y''(0)=16$$

74. $$y^{(4)}-3y'''+5y''-2y'=-2e^x(\cos x-\sin x),\; y(0)=2,\; y'(0)=0,\; y''(0)~=~-1, \; y'''(0)=-5$$

## Q9.3.4

75. Prove: A function $$y$$ is a solution of the constant coefficient nonhomogeneous equation

$a_0y^{(n)}+a_1y^{(n-1)}+\cdots+a_ny=e^{\alpha x}G(x) \tag{A}$

if and only if $$y=ue^{\alpha x}$$, where $$u$$ satisfies the differential equation

$a_0u^{(n)}+{p^{(n-1)}(\alpha)\over(n-1)!}u^{(n-1)}+ {p^{(n-2)}(\alpha)\over(n-2)!}u^{(n-2)}+\cdots+p(\alpha)u=G(x) \tag{B}$

and

$p(r)=a_0r^n+a_1r^{n-1} + \cdots + a_n\nonumber$

is the characteristic polynomial of the complementary equation

$a_0y^{(n)}+a_1y^{(n-1)}+\cdots+a_ny=0.\nonumber$

76. Prove:

1. The equation $\begin{array}{lll}{a_{0}u^{(n)}}&{+}&{\frac{p^{(n-1)}(\alpha )}{(n-1)!}u^{(n-1)}+\frac{p^{(n-2)}(\alpha )}{(n-2)!}u^{(n-2)}+\cdots + P(\alpha )u}\\{}&{=}&{(p_{0}+p_{1}x+\cdots +p_{k}x^{k})\cos\omega x} \\ {}&{+}&{(q_{0}+q_{1}x+\cdots +q_{k}x^{k})\sin\omega x} \end{array}\tag{A}$ has a particular solution of the form $u_p=x^m\left(u_0+u_1x+\cdots+u_kx^k\right)\cos\omega x+ \left(v_0+v_1x+\cdots+v_kx^k\right)\sin\omega x.\nonumber$
2. If $$\lambda+i\omega$$ is a zero of $$p$$ with multiplicity $$m\ge1$$, then (A) can be written as $a(u''+\omega^2 u)= \left(p_0+p_1x+\cdots+p_kx^k\right)\cos\omega x+ \left(q_0+q_1x+\cdots+q_kx^k\right)\sin\omega x,\nonumber$ which has a particular solution of the form $u_p=U(x)\cos\omega x+V(x)\sin\omega x,\nonumber$ where $U(x)=u_0x+u_1x^2+\cdots+u_kx^{k+1},\,V(x)=v_0x+v_1x^2+\cdots+v_kx^{k+1}\nonumber$ and $\begin{array}{rcl} a(U''(x)+2\omega V'(x))&=&p_0+p_1x+\cdots+p_kx^k\\[10pt] a(V''(x)-2\omega U'(x))&=&q_0+q_1x+\cdots+q_kx^k. \end{array}\nonumber$