Final answer:
To solve the initial-value problem, we must find the solution to the corresponding homogeneous equation using the characteristic equation, guess a particular solution for the nonhomogeneous part, and apply the initial conditions to determine any constants.
Step-by-step explanation:
To solve the given initial-value problem y" + 4y' + 4y = (7x)e⁻²⁰, we first need to solve the homogeneous equation y" + 4y' + 4y = 0. This has a characteristic equation r² + 4r + 4 = 0, which factors to (r + 2)² = 0. The repeated root is r = -2, so the general solution of the homogeneous equation is yh(x) = (C1 + C2x)e⁻²⁰.
For the nonhomogeneous part, we use the method of undetermined coefficients. Since the right side of the equation is (7x)e⁻²⁰, we guess a particular solution yp(x) = x(Ax+B)e⁻²⁰ and find A and B by differentiating yp and substituting in the nonhomogeneous equation. After determining the coefficients A and B, we obtain the particular solution.
The complete solution is y(x) = yh(x) + yp(x). To find the constants C1 and C2, we use the initial conditions y(0) = 5 and y'(0) = 8. Substituting x = 0 in y(x) and y'(x) and solving the resulting equations will give us the values for C1 and C2.
The final step is to combine all the parts to form the solution to the initial-value problem that satisfies the given conditions.