Final answer:
The maximum displacement of a mass attached to a spring in SHM is determined by solving the second-order linear differential equation given the initial conditions and then rounding the amplitude to three decimal places.
Step-by-step explanation:
The question revolves around determining the maximum vertical displacement of a mass attached to a spring that is in a state of simple harmonic motion (SHM) dictated by a second-order linear differential equation with given initial conditions. To solve this initial-value problem, we must find the solution to the differential equation 4x'' + x' + x = 0 that satisfies the initial conditions x(0) = 8 and x'(0) = 4. Using the theory of differential equations and characteristic equations, we can solve for the displacement function x(t), and subsequently find the amplitude of oscillation, which in this case represents the maximum vertical displacement of the mass.
After solving the equation, we use the amplitude to determine the maximum displacement since during SHM, the mass will oscillate symmetrically about the equilibrium position, reaching a maximum positive and negative displacement that is equal to the amplitude. The mass will have its maximum vertical displacement at these positions. To find the value of this displacement, one would solve the characteristic equation, apply the initial conditions to determine the constants, and finally arrive at the amplitude, which gives the maximum displacement when rounded to three decimal places.