Final answer:
The question involves solving a second-order linear homogeneous differential equation with constant coefficients. The characteristic equation yields a repeated root, resulting in a general solution that includes exponential and linear terms. The initial conditions are applied to find the specific coefficients for the particular solution.
Step-by-step explanation:
The question addresses a second-order linear homogeneous differential equation with constant coefficients. The standard form of such an equation is ay'' + by' + cy = 0, where a, b, and c are constants, and y'' and y' are the second and first derivatives of y with respect to t, respectively. In this case, the differential equation is y'' + 2y' + y = 0. To solve this initial value problem, we look for a characteristic equation associated with the differential equation, which in this case is r^2 + 2r + 1 = 0. The solutions to this quadratic equation will give us the roots necessary to determine the general solution to the differential equation.
Upon solving the characteristic equation, we find that it has a repeated root of r = -1. This implies that the general solution to the differential equation will be of the form y(t) = (C1 + C2*t)e^{-t}, where C1 and C2 are constants that we will determine using the given initial conditions, which are y(0) = 7 and y'(0) = 2.
Applying the initial conditions, we find that C1 = 7 and C2 = 9. Therefore, the particular solution that satisfies the initial conditions is y(t) = (7 + 9t)e^{-t}.