Final answer:
The task involves solving a second-order linear nonhomogeneous differential equation with particular initial conditions. It requires finding the complementary function, determining the particular integral, and applying initial conditions to find the constants for the complete solution.
Step-by-step explanation:
The task is to compute the solution for a second-order linear nonhomogeneous differential equation with given initial conditions: y(0) = y'(0) = 0. The differential equation is d^2y/dt^2 + 4y = 6 + t^2 + e^t. This involves several steps which encompass finding the complementary function (solution to the homogeneous equation), determining the particular integral using methods such as undetermined coefficients or variation of parameters, and then applying the initial conditions to find the constants involved.
First, we find the complementary function by solving the homogeneous equation d^2y/dt^2 + 4y = 0. The characteristic equation is λ^2 + 4 = 0, which yields complex roots. The general solution of the complementary function is of the form y_c(t) = A cos(2t) + B sin(2t). Next, a particular solution y_p(t) of the nonhomogeneous equation is required. After finding y_p(t), the general solution can be expressed as y(t) = y_c(t) + y_p(t). Lastly, by applying the initial conditions y(0) = y'(0) = 0, we can determine the constants A and B.