Follow these steps ode initial conditions

Solved Step by Step With Explanation-LAPLACE transforms

Questions

Solved Step by Step With Explanation- LAPLACE transforms

Answer

L{x¨ - 4x} = L{0}

Using the linearity property of Laplace transforms, we can write this as:

Substitute this into the equation:

s²X(s) - s(1) - 2 - 4X(s) = 0

Now, divide both sides by (s² - 4):

X(s) = (s + 2) / (s² - 4)

Step 5: Take the inverse Laplace transform of X(s) to find x(t):

Using the inverse Laplace transform property:

x(0) = e^(2 * 0) = e^0 = 1 (matches the given initial condition)

x˙(0) = 2e^(2 * 0) = 2 (matches the given initial condition)