Final answer:
The student is seeking the Laplace transform Y(s) for a differential equation with given initial conditions. Using the Laplace transform's linearity and properties for differential equations, we transform y'' and 9y, taking care of the initial conditions, and then solve for Y(s) to find the transform of the function y(t).
Step-by-step explanation:
The student is asking for the Laplace transform Y(s) of the solution to a second-order differential equation with piecewise function as non-homogeneous term, given the initial conditions y(0) = 2, and y'(0) = -1. We start by applying the Laplace transform to both sides of the differential equation y'' + 9y = f(t), where f(t) is the piecewise function defined in the problem. Considering the initial conditions and the properties of the Laplace transform, we transform y'' and 9y separately to the s-domain, using the linearity of Laplace transform. The transform of the piecewise function f(t) is treated separately where we use the Heaviside step function to account for the change in function at t=2.
To find the Laplace transform of the second-order term, we use L{y''} = s^2*Y(s) - s*y(0) - y'(0). For the initial conditions provided, this becomes s^2*Y(s) - 2s + 1. Next, we transform the term 9y straightforwardly to 9*Y(s). The Laplace transform of the piecewise function is handled by breaking it into two parts: one for 0 ≤ t < 2 and another for t ≥ 2, applying the Heaviside function for t=2.
After transforming all the terms to the s-domain and using the initial conditions, we can solve for Y(s) to find the Laplace transform of the function y(t).