Question

Given the differential equation x'+3x=sin(2t) with the initial condition x(0) = 0), use Laplace transforms to find x(t). Calculate the partial fraction coefficients with the method of residuals. What is the correct expression for x(t)?

Answer 1

To solve the differential equation x'(t) + 3x(t) = sin(2t) with x(0) = 0 using Laplace transforms, one should take the Laplace transform of both sides, use partial fraction decomposition with the method of residues, and then find the inverse Laplace transform to obtain x(t).

Step-by-step explanation:

The differential equation given is x'(t) + 3x(t) = sin(2t) with the initial condition x(0) = 0. To solve this using Laplace transforms, we first take the Laplace transform of both sides of the equation:

1. Let's denote L{ x(t) } as X(s) and use the property L{ x'(t) } = sX(s) - x(0). Since x(0) = 0, we have sX(s).
2. The Laplace transform of the right side is L{ sin(2t) } which equals 2/(s^2 + 4).
3. Thus, we have the equation sX(s) + 3X(s) = 2/(s^2 + 4) or X(s) = 2/(s^2 + 4)/(s + 3).
4. Now apply partial fractions to find the expression for X(s). Use the method of residues to calculate the coefficients.
5. Finally, take the inverse Laplace transform of X(s) to find x(t).

The exact expression for x(t) would involve the inverse Laplace transform of the terms obtained after partial fraction expansion.