Final answer:
To solve the given differential equation y'' - 4y = sin(3x), we use the method of undetermined coefficients. We find the complementary solution and a particular solution. The general solution is the sum of the complementary and particular solutions.
Step-by-step explanation:
To solve the given differential equation y'' - 4y = sin(3x), we can use the method of undetermined coefficients. First, we find the complementary solution by solving the homogeneous equation y'' - 4y = 0. The characteristic equation is r^2 - 4 = 0, which has roots r = 2 and r = -2. So, the complementary solution is y_c(x) = c_1e^(2x) + c_2e^(-2x), where c_1 and c_2 are constants.
Next, we find a particular solution to the non-homogeneous equation y'' - 4y = sin(3x). We can guess a solution of the form y_p(x) = A sin(3x) + B cos(3x), where A and B are constants to be determined. Plugging this into the differential equation, we get -9A sin(3x) - 9B cos(3x) - 4A sin(3x) - 4B cos(3x) = sin(3x). Equating coefficients of like terms, we get -9A - 4B = 1 and -9B + 4A = 0. Solving this system of equations, we find A = -1/10 and B = 9/40.
Therefore, the general solution to the given differential equation is y(x) = y_c(x) + y_p(x) = c_1e^(2x) + c_2e^(-2x) - (1/10)sin(3x) + (9/40)cos(3x).