Final answer:
To find the Laplace Transform and solve for Y(s), perform the Laplace Transform of each term in the given differential equation, including the delta function, the step function, and the t*u(t) term, and then solve for Y(s).
Step-by-step explanation:
The student is asking to find the Laplace Transform of the differential equation 4d² y/dt² + 10dy/dt + 2y = 10δ(t) + 25u(t) + 5tu(t) and to solve for the response Y(s) in its simplest form. To solve this, we take the Laplace Transform of each term in the equation. The Laplace Transform of 4d² y/dt² is 4s² Y(s) - 4sy(0) - 4y'(0), the Transform of 10dy/dt is 10sY(s) - 10y(0), and the Transform of 2y is 2Y(s). Since the equation involves a delta function δ(t) and a step function u(t), their Transforms are respectively 10 and 25/s. Additionally, the term 5tu(t) has a Transform of 5/s². Summing these up and solving for Y(s) gives the simplest form of the response function.