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Consider the equation [tex]y''+4y=2sin(2x)+e^{2x}+2[/tex]. According to the method of Undetermined Coefficients, the particular solution has the form:

C1 = 1st constant

C2 = 2nd constant

C3 = 3rd constant

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A) [tex]C1xsin(2x) + C3x + C4xe^{2x} + C5xe^{-2x}[/tex]

B) [tex]C1sin(2x) + (C2)2 + C3xe^{-2x}[/tex]

C) [tex]C1xcos(2x) + C2xsin(2x) + C3 + C4e^{2x}[/tex]

D) [tex]C1cos(2x) + C2xsin(2x) + C3 + C4xe^{2x}[/tex]

[tex]y^2 + 16 = 0[/tex]

[tex]y = -4, 4[/tex]

[tex]yh = c1cos(x) + c2sin(x)[/tex]

[tex]yp = Acos(2x) + Bsin(2x) + C + De^{2x}[/tex]

[tex]y'p = -2Asin(2x) + 2Bcos(2x) + 2De^{2x}[/tex]

[tex]y''p = -4Acos(2x) - 4Bsin(2x) + 4De^{2x}[/tex]

[tex]y'' + 4y = 2sin(2x) + 2 e^{2x}[/tex]

[tex][-4Acos(2x) - 4Bsin(2x) + 4De^{2x}] + 16[Acos(2x) + Bsin(2x) + C + De^{2x}] = 2sin(2x) + 2 + e^{2x}[/tex]

[tex]12Acos(2x) + 12Bsin(2x) + 20De^{2x} + 16C = 2sin(2x) + 2 + e^{2x}[/tex]

[tex]cos(2x)[/tex] ===> 12A = 0 ===> A = 0

[tex]sin(2x)[/tex] ===> 12B = 2 ===> B = 1/6

[tex]16[/tex] ===> 16C = 2===> C = 1/8

[tex]e^{2x}[/tex] ===> 20D = 1 ===> D = 1/20

So far what I got B) as a right answer, but I'm not too sure about A), C), and D)

C1 = 1st constant

C2 = 2nd constant

C3 = 3rd constant

.

.

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A) [tex]C1xsin(2x) + C3x + C4xe^{2x} + C5xe^{-2x}[/tex]

B) [tex]C1sin(2x) + (C2)2 + C3xe^{-2x}[/tex]

C) [tex]C1xcos(2x) + C2xsin(2x) + C3 + C4e^{2x}[/tex]

D) [tex]C1cos(2x) + C2xsin(2x) + C3 + C4xe^{2x}[/tex]

__Solution:__[tex]y^2 + 16 = 0[/tex]

[tex]y = -4, 4[/tex]

[tex]yh = c1cos(x) + c2sin(x)[/tex]

[tex]yp = Acos(2x) + Bsin(2x) + C + De^{2x}[/tex]

[tex]y'p = -2Asin(2x) + 2Bcos(2x) + 2De^{2x}[/tex]

[tex]y''p = -4Acos(2x) - 4Bsin(2x) + 4De^{2x}[/tex]

[tex]y'' + 4y = 2sin(2x) + 2 e^{2x}[/tex]

[tex][-4Acos(2x) - 4Bsin(2x) + 4De^{2x}] + 16[Acos(2x) + Bsin(2x) + C + De^{2x}] = 2sin(2x) + 2 + e^{2x}[/tex]

[tex]12Acos(2x) + 12Bsin(2x) + 20De^{2x} + 16C = 2sin(2x) + 2 + e^{2x}[/tex]

[tex]cos(2x)[/tex] ===> 12A = 0 ===> A = 0

[tex]sin(2x)[/tex] ===> 12B = 2 ===> B = 1/6

[tex]16[/tex] ===> 16C = 2===> C = 1/8

[tex]e^{2x}[/tex] ===> 20D = 1 ===> D = 1/20

So far what I got B) as a right answer, but I'm not too sure about A), C), and D)

**Can anybody help me out here, please?**
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