Final answer:
The question involves solving an initial value inhomogeneous differential equation by finding a complementary solution and a particular solution, and then using the initial conditions to find the specific solution.
Step-by-step explanation:
The problem presented is an initial value inhomogeneous differential equation, which is a concept within the field of mathematics. This kind of problem involves finding the function y(x) that satisfies both the differential equation and the initial conditions provided. To tackle this problem, we will first find the complementary solution yc(x) to the homogeneous differential equation, 3y'' + 16y' + 16y = 0, by solving its characteristic equation.
Once we have the complementary solution, we need to find a particular solution yp(x) to the non-homogeneous equation. We do this by assuming a form for yp(x) that can satisfy the non-homogeneous part of the equation, which is typically a polynomial of the same degree as the inhomogeneous term (4x² + 9x + 10 in this case). We then substitute yp(x) into the differential equation to determine the coefficients.
The general solution to the differential equation is the sum of the complementary and particular solutions, y(x) = yc(x) + yp(x). Finally, to find the specific solution that satisfies the initial conditions y(0) = 0 and y'(0) = 0, we substitute these initial conditions into the general solution and solve for the constants.