Part A: 1. Solve the homogeneous equation (t) + cdx(t) + kx(t) = 0 for the following cases: • • • Underdamped: c² < 4mk. Critically damped: c² - 4mk, Overdamped: c² > 4mk. 迎 2. For each case, describe the general solution and analyze the behavior of the displacement a(t) as t→ ∞. Part B: 1. Use the method of undetermined coefficients to solve the non-homogeneous equation dz(t) dx(t) m +c+ kr(t) = Focos(wt) for the particular solution. с 2. Determine the general solution (t) for the full non-homogeneous equation by adding the homogeneous solution and the particular solution. 3. Discuss the phenomenon of resonance that occurs when the external driving frequency w approaches the natural frequency of the system wh) - √ How does the behavior of (t) change in this case? Part C: Consider now that the damping coefficient c depends on time in the following way. c(t) - ce¯xt where C and X are constants. Modify the original differential equation to account for this time- dependent damping: m dx(t) dt² da(t) +kx(t) F₁ cos(wt). dt 1. Apply the method of variation of parameters to solve this modified non-homogeneous equation. 2. Analyze the long-term behavior of the system for different values of X, and describe how the solution is affected by the exponentially decaying damping. Part D: Now, consider a system of two coupled masses with displacements a₁ (t) and (t), connected by springs and subjected to external forces. The equations governing the motion of the masses are: dx mi dt² da₁ dt +k₁₁₁₂(x²₂₁) - F₁ cos(wit), m2 +C₂ dt² dx2 dt - +k2(221) F2 cos(w2t). 1. Find the general solution to this system of coupled differential equations using matrix methods or Laplace transforms. 2. Analyze how the natural frequencies of the system change when the spring constants k₁ and k₂ are varied. 3. Discuss the physical interpretation of the solution in terms of the oscillatory behavior of the two masses, including any beating patterns that may arise when w₁ =W₂-