Final answer:
The question involves solving a second-order linear differential equation using the Laplace transform, accounting for the given initial conditions. The process involves computing the Laplace transform of the given equation, solving algebraically for Y(s), and then finding the inverse Laplace transform to obtain y(t).
Step-by-step explanation:
The question asks to use Laplace transform to solve the initial value problem given by the differential equation y'' + 9y = g(t), with initial conditions y(0) = 1 and y'(0) = 0. First, take the Laplace transform of both sides of the equation, using the properties of the Laplace transform that relate to derivatives and initial conditions. After transforming the differential equation, solve for the Laplace transform of y(t), which we may represent as Y(s). Finally, find the inverse Laplace transform of Y(s) to determine the solution y(t) in the time domain. However, to fully solve the problem, we need the function g(t) since it appears in the transformed equation as G(s), which is the Laplace transform of g(t). Without the specific form of g(t), we cannot completely solve for Y(s) or provide the inverse Laplace transform to find y(t). Therefore, we can only provide the general approach to solving the problem using the Laplace transform, not the explicit solution.