Determine the Laplace transforms of the following functions:
1. 5e^{-2t} cos 3t
2. 1/2 t^4 e^{-3t}
Determine the inverse Laplace transforms of the following:
3. (s+1) / (s^2 + 2s + 10)
4. 2(s-3) / (s^2 - 6s + 13)
Use Laplace transforms to solve the given differential equations
5. 9 d^2y/dt^2 - 24 dy/dt + 16y = 0, given y(0) = 3 and y'(0) = 3
6. d^2x/dt^2 + 100x = 0, given x(0) = 2 and x'(0) = 0
7. d^2i/dt^2 + 1000 di/dt + 250000i = 0, given i(0) = 0 and i'(0) = 100
8. d^2x/dt^2 + 6 dx/dt + 8x = 0, given x(0) = 4 and x'(0) = 8
9. d^2y/dx^2 - 2 dy/dx + y = 3e^{4x}, given y(0) = -2/3 and y'(0) = 4 1/3
10. d^2y/dx^2 + 16y = 10 cos 4x, given y(0) = 3 and y'(0) = 4
11. d^2y/dx^2 + dy/dx - 2y = 3 cos 3x - 11 sin 3x, given y(0) = 0 and y'(0) = 6
12. d^2y/dx^2 - 2 dy/dx + 2y = 3e^x cos 2x, given y(0) = 2 and y'(0) = 5