Question

using the Laplace transform solve: mx'' + cx' + kx = 0, x(0)=a, x'(0)=b, where m>0, c>0, k>0 and c² -4km>0 (system is overdamped)

Answer 1

To solve the given second-order linear differential equation representing an overdamped mass-spring system using the Laplace transform, apply the initial conditions to determine the solution in the Laplace domain and then find the inverse transform to obtain the time-domain solution.

Step-by-step explanation:

To solve the differential equation mx'' + cx' + kx = 0 with initial conditions x(0)=a and x'(0)=b, using the Laplace transform for an overdamped system where c² - 4km > 0, follow these steps:

1. Take the Laplace transform of each term in the differential equation.
2. Apply the initial conditions and solve for the Laplace transform of x(t), denoted as X(s).
3. Use partial fraction decomposition if necessary to simplify the transform expression.
4. Find the inverse Laplace transform of X(s) to get the solution x(t) in the time domain.

The key detail is that in an overdamped system, characterized by the equation c² - 4km > 0, the system returns to equilibrium without oscillating, which is in contrast to an underdamped system where the system would exhibit decaying oscillations.