Final answer:
The solution involves finding the general solution to the associated homogeneous equation and a particular solution to the nonhomogeneous equation. Combining these yields the complete general solution for the given second-order nonhomogeneous differential equation.
Step-by-step explanation:
The student is seeking a solution to a second-order linear nonhomogeneous differential equation with constant coefficients. This is a problem in Mathematics, specifically in the area of differential equations. To solve this, we will find the general solution to the homogeneous equation y'' - 3y' + 4y = 0 and then find a particular solution to the nonhomogeneous equation y'' - 3y' + 4y = -16x2 + 24x - 8.
The characteristic equation of the homogeneous part is r2 - 3r + 4 = 0. Solving this gives us complex roots r = 3/2 ± (√(9-4*4))/2 = 3/2 ± i/2. The general solution of the homogeneous equation is yh(x) = e3/2x(C1cos(1/2x) + C2sin(1/2x)), where C1 and C2 are arbitrary constants.
Now, we guess a particular solution in the form of the right-hand side of the given nonhomogeneous equation. Assuming yp(x) = Ax2 + Bx + C and substituting it into the nonhomogeneous equation, we can solve for A, B, and C by equating coefficients.
Finally, the general solution to the differential equation will be y(x) = yh(x) + yp(x). Recall that to determine the particular constants (C1 and C2), initial conditions or additional constraints are necessary.