Find the Laplace Transform of the following functions:
(a) f(t) = cos(t) if 0 < t < 2; f(t) = 0 if t > 2.
Rewrite using the step function; then find the Laplace transform:
f(t) = 1 if 0 < t < 1/2; f(t) = -1 if 1/2 < t < 1; f(t) = 0 if t > 1.
Rewrite using step functions, then find the Laplace transform.
(c) f(t) = t^2 + 4(t - 3)^2.
Find the inverse Laplace Transform of the following functions:
F(s) = 4s / (s^2 + 9).
F(s) = 8s / (s^2 - 9).
Use the Laplace Transform to solve the following initial value problems:
1" + 4y' = 2u_1(t) - u_a(t); y(0) = 1, y'(0) = 0.
2" + 2y' + y = t + 6e^(-t); y(0) = 0, y'(0) = 2.
2" + 4y' + 5y = 0(t - 2) + 8(t - 2); y(0) = 0, y'(0) = 2.