Final answer:
To solve the initial value problem in the homogeneous differential equation, determine the complementary function (CF) and the particular integral (PI) of the differential equation. Then, find the general solution by combining the CF and PI. Finally, use the initial conditions to determine the values of the constants and obtain the specific solution.
Step-by-step explanation:
To solve the initial value problem in the given homogeneous differential equation, first determine the complementary function (CF) of the differential equation by assuming y = e^(rx), where r is a constant. Substituting this assumption into the differential equation, we get a characteristic equation r^2 + 9 = 0. Solving this equation, we find r = ±3i. Therefore, the complementary function is y_cf = c1e^(3ix) + c2e^(-3ix), where c1 and c2 are constants.
The particular integral (PI) of the differential equation can be determined using the method of undetermined coefficients. Since the right-hand side of the equation contains terms of the form xe^(2x), we assume a particular solution of the form y_pi = Ax^2e^(2x), where A is a constant. Substituting this assumption into the differential equation, we can solve for A.
Finally, the general solution of the differential equation is given by y = y_cf + y_pi. Using the initial conditions y(0) = 0 and y'(0) = 0, we can find the values of the constants c1, c2, and A, and obtain the specific solution to the initial value problem.