Final answer:
To solve the given differential equation using Laplace transforms, we would typically transform the equation, solve for the Laplace transform of x(t), and then take the inverse transform. However, the provided expression seems to have a typo and more information or a correction is needed to provide a complete solution.
Step-by-step explanation:
To find the function x(t) from the given differential equation 2x + ∫ᵔ₀ xdx = 3, we can apply Laplace transforms. Given that x(0) = 0, we can ignore the constant of integration that arises from indefinite integration. Using the initial condition, we transform the differential equation with respect to t, and then we take the inverse Laplace transform of the resulting equation to find x(t).
The presence of an integral in the differential equation typically signifies a convolution in the transform domain. However, as the details for the Laplace transforms are not fully provided in the question, we cannot continue further with this example. Typically, the process would involve taking the Laplace transform of both sides of the equation, solving for the Laplace transform of x(t), and then finding the inverse Laplace transform to obtain x(t).
The provided expression x(t) = ¾/1² – 1³ does not seem to correlate directly to the differential equation and may be a result of a typo. To correctly answer the question, we would need more information or need to proceed with the solution steps as outlined for Laplace transforms.