Without solving for the undetermined coefficients the correc

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation y\" + 9y = sin(3pi) is

Solution

Find the particular integral by comparing coefficients:
y = Asin(3x) + Bcos(3x)
y\' = 3Acos(3x) - 3Bsin(3x)
y\'\' = -9Asin(3x) - 9Bcos(3x)
y\'\' - 9y = sin(3x)
-9Asin(3x) - 9Bcos(3x) - 9[Asin(3x) + Bcos(3x)] = sin(3x)
-9Asin(3x) - 9Bcos(3x) - 9Asin(3x) - 9Bcos(3x) = sin(3x)
-18Asin(3x) - 18Bcos(3x) = sin(3x)
18A = -1
A = -1 / 18
-18B = 0
B = 0
y = -sin(3x) / 18