6. Solve the following initial value problems using Laplace transforms.
a) \(x'' + 5x = H(t - 2)\), \(x(0) = 1\).
b) \(x'' + x = \sin 2t\), \(x(0) = 0\).
c) \(x'' - x' - 6x = 0\), \(x(0) = 2\), \(x'(0) = -1\)
d) \(x'' - 2x' + 2x = 0\), \(x(0) = 0\), \(x'(0) = 1\).
e) \(x'' - 2x' + 2x = e^{-t}\), \(x(0) = 0\), \(x'(0) = 1\).
f) \(x'' - x = 0\), \(x(0) = 1\), \(x'(0) = 0\).
g) \(x'' + 0.4x' + 2x = 1 - H(t - 5)\), \(x(0) = 0\), \(x'(0) = 0\).
h) \(x'' + 9x = \sin 3t\), \(x(0) = 0\), \(x'(0) = 0\).
i) \(x'' - 2x = 1\), \(x(0) = 1\), \(x'(0) = 0\).
j) \(x'' = 2x + H(t - 1)\), \(x(0) = 0\).