Final answer:
The student is asked to solve a second-order non-homogeneous differential equation using the method of variation of parameters, given specific initial conditions. The process involves finding the general solution composed of a complementary solution and a particular solution, fitting the initial conditions to solve for constants.
Step-by-step explanation:
The subject of the question is solving a second-order non-homogeneous differential equation using the method of variation of parameters. With the given initial conditions y(0) = 1, y'(0) = 0, and the differential equation 9y'' − y = xex/3, the steps to solve it involve finding a complementary solution (yc), determining two linearly independent solutions to the associated homogeneous equation, using variation of parameters to find a particular solution (yp), and then applying the initial conditions to find the constants involved in the general solution.
The general solution to a second-order linear differential equation is of the form y(x) = yc(x) + yp(x), where yc is the complementary solution and yp is the particular solution found using variation of parameters. To apply this method, we first solve the homogeneous equation (i.e., without the non-homogeneous part), find the Wronskian of the solutions, and then apply the variation of parameters formulas to find yp. Finally, by applying the initial conditions, we can solve for any constants in the complementary solution and combine it with the particular solution to find the final answer.