Final answer:
To solve the differential equation y′′+3y′=18sin(3t)+144cos(3t), you can use the method of undetermined coefficients and assume a particular solution of the form: y_p(t) = A sin(3t) + B cos(3t). By substituting this particular solution into the equation and comparing coefficients, you can find the values of A and B. The general solution to the differential equation is the sum of the homogeneous solution and the particular solution.
Step-by-step explanation:
To solve the differential equation y′′+3y′=18sin(3t)+144cos(3t), we can use the method of undetermined coefficients. Since the equation is non-homogeneous, we assume a particular solution of the form: y_p(t) = A sin(3t) + B cos(3t), where A and B are constants to be determined.
By substituting this particular solution into the equation and comparing coefficients, we find that A = -6 and B = 48. Therefore, the particular solution is y_p(t) = -6sin(3t) + 48cos(3t).
The general solution to the differential equation is the sum of the homogeneous solution and the particular solution: y(t) = y_h(t) + y_p(t). The homogeneous solution is y_h(t) = c1e^(-3t) + c2, where c1 and c2 are constants determined by initial conditions.