Final answer:
To find the general solution of the given ODE, first find the characteristic equation and solve for the roots. Then, use the method of undetermined coefficients to find a particular solution.
Step-by-step explanation:
To find the general solution of the given ODE, we can first find the characteristic equation by setting the coefficients of the derivatives to zero. The characteristic equation is r^2 - r - 2 = 0. Solving this equation gives us the roots r1 = 2 and r2 = -1.
Therefore, the general solution of the ODE is y(t) = c1e^(2t) + c2e^(-t), where c1 and c2 are arbitrary constants.
To find a particular solution, we can use the method of undetermined coefficients. Since the right-hand side of the equation is 4sin(3t), we can assume a particular solution of the form yp(t) = Asin(3t) + Bcos(3t). Plugging this into the ODE and solving for A and B will give us the particular solution.