Question

Using Laplace transform, determine the solution of the following ODE: x¨+3x˙+2x=0. Assume that x(0)=1 m and x˙(0)=2 m/s.

Answer 1

To solve the given ODE using Laplace transform, we write the equation in terms of the Laplace variable s, substitute the initial conditions, and simplify the equation. We then use partial fraction decomposition to find the inverse Laplace transform, which gives us the solution x(t) of the ODE.

Step-by-step explanation:

To determine the solution of the given ODE using Laplace transform, we first take the Laplace transform of the equation:

s^2X(s) + 3sX(s) + 2X(s) = x(0) + s * x(0) - x˙(0)

Substituting the initial conditions, we get:

s^2X(s) + 3sX(s) + 2X(s) = s + 1 - 2s

Simplifying the equation, we have:

X(s) = (s + 1)/(s^2 + 3s + 2)

Using partial fraction decomposition, we can write the above equation as:

X(s) = 1/(s + 1) - 1/(s + 2)

Taking the inverse Laplace transform, we get the solution of the ODE:

x(t) = e^(-t) - e^(-2t)