Final answer:
The complementary function of the differential equation 4y" - y = e^(x/2) is y_c(x) = C1e^(x/2) + C2e^(-x/2), obtained by solving the characteristic equation 4r^2 - 1 = 0.
Step-by-step explanation:
The question asks to find the complementary function of the differential equation 4y" - y = ex/2. In order to find the complementary function, we look at the associated homogeneous equation, which is 4y" - y = 0. This is a second-order linear homogeneous differential equation with constant coefficients. To find the complementary function, we first solve the characteristic equation related to the differential equation, which is 4r2 - 1 = 0. Solving this quadratic equation, we get two real distinct roots r1 = 1/2 and r2 = -1/2. Using these roots, we can construct the general solution to the homogeneous equation, which is the complementary function, expressed as yc(x) = C1ex/2 + C2e-x/2, where C1 and C2 are arbitrary constants determined by initial conditions.