Final answer:
To solve the given differential equation, use the method of undetermined coefficients to find a particular solution. Add the complementary solution to get the complete solution. Use the initial conditions to determine the values of the constants.
Step-by-step explanation:
To solve the given differential equation, we can use the method of undetermined coefficients. Since the right side of the equation contains both sine and cosine terms, we can assume a particular solution of the form y = Asin(2t) + Bcos(2t). We can then substitute this assumption into the differential equation and solve for A and B.
After solving the differential equation, we find that the particular solution is y = 4sin(2t) + 2cos(2t). To find the complete solution, we add the complementary solution which satisfies the homogeneous equation y'' + 2y' = 0. This complementary solution is given by y = C1e^(-t) + C2, where C1 and C2 are constants. Finally, we can use the initial conditions y(0) = 1 and y'(0) = 8 to determine the values of C1 and C2, and thus obtain the complete solution.