Final answer:
To find a particular solution to the differential equation y'' - y' + 9y = 3sin(3x), we can use the method of undetermined coefficients. Assuming a particular solution of the form y_p = A sin(3x) + B cos(3x), we substitute it into the differential equation and solve for the coefficients A and B. Finally, we find that the particular solution is y_p = (1/6)sin(3x).
Step-by-step explanation:
To find a particular solution to the differential equation y'' - y' + 9y = 3sin(3x), we can use the method of undetermined coefficients. Since the right-hand side of the equation is a sinusoidal function, we assume a particular solution of the form:
y_p = A sin(3x) + B cos(3x)
Substituting this into the differential equation and solving for A and B will give us the particular solution.
Step 1: Compute the derivatives of y_p:
y_p' = 3A cos(3x) - 3B sin(3x)
y_p'' = -9A sin(3x) - 9B cos(3x)
Step 2: Substitute y_p, y_p', and y_p'' into the differential equation:
-9A sin(3x) - 9B cos(3x) - (3A cos(3x) - 3B sin(3x)) + 9(A sin(3x) + B cos(3x)) = 3sin(3x)
Step 3: Collect like terms and solve for A and B:
(9A - 3B + 9A)sin(3x) + (-9B - 3A + 9B)cos(3x) = 3sin(3x)
Simplifying:
18A sin(3x) - 6B cos(3x) = 3sin(3x)
Comparing coefficients:
18A = 3
-6B = 0
Solving for A and B:
A = 1/6
B = 0
Thus, the particular solution is:
y_p = (1/6)sin(3x)