State the form for the particular solution of the differential equation y''-3y'+2y=sin(3x)

Question

State the form for the particular solution of the differential equation y''-3y'+2y=sin(3x). Do not solve for any of the variables.

Answer 1

y = A sin(3x) + B cos(3x)

y' = 3A cos(3x) - 3B sin 3x

y'' = -9A sin3x - 9B cos 3x

coefficient table

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sin cos

y'' -9A -9B

-9B 9A

2A 2B

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(-7A - 9B) sin 3x + (9A - 7B) cos 3x = sin(3x)

-7A - 9B = 1

9A -7B = 0

7A + 9B = -1

9A - 7B = 0

49A + 63B = -7

91A - 63B = 0

130A = -7

A = -7/130

which forces B = 9/130

solutiion is

y = (-7/130) sin(3x) + (9/130) cos(3x)