Final answer:
To solve the initial value problem using Laplace transforms, apply the transform to the differential equation, incorporate initial conditions, solve for Y(s), and then find the inverse transform to obtain y(t).
Step-by-step explanation:
Solving the Initial Value Problem Using Laplace Transforms
To solve the initial value problem dy/dt + y = 5 sin(2t), y(0) = 0 using Laplace transforms, follow these steps:
Apply the Laplace transform to both sides of the differential equation to get L{dy/dt} + L{y} = L{5 sin(2t)}.
Utilize the initial condition y(0) = 0 and properties of the Laplace transform to simplify the equation to sY(s) - y(0) + Y(s) = 5/s^2 + 4, where Y(s) is the Laplace transform of y(t).
Solve for Y(s) to get Y(s) = 5/(s^2 + 4)(s + 1).
Find the inverse Laplace transform of Y(s) to determine y(t), which is the solution to the initial value problem.
After applying the inverse Laplace transform, you should obtain a function y(t) in terms of t, which will give you the time-domain solution to the given problem.