Final Answer:
The solution to the given initial value problem (IVP) using Laplace Transform is y(t) = e^(-t) - e^(-2t) + 2e^(t).
Step-by-step explanation:
Given Differential Equation:
y'' + 3y' + 2y = g(t), y(0) = 0, y'(0) = -2.
Laplace Transform of the Differential Equation:
s^2Y(s) - sy(0) - y'(0) + 3(sY(s) - y(0)) + 2Y(s) = G(s), where Y(s) is the Laplace Transform of y(t) and G(s) is the Laplace Transform of g(t).
Given g(t):
g(t) = 2e^t.
Laplace Transform of g(t):
G(s) = 2/(s - 1).
Substitute into the Laplace Transform Equation and Solve for Y(s).
Inverse Laplace Transform:
y(t) = e^(-t) - e^(-2t) + 2e^(t).
Therefore, the solution to the IVP is y(t) = e^(-t) - e^(-2t) + 2e^(t).