Inverse Laplace Transform

Find the inverse Laplace transform of the following in the attached file. Thanks.

Transfer Function

A battery of voltage Vi is connected in series with a resistor of resistance R, an inductor of inductance L and a capacitor of capacitance C. If, the output voltage across capacitor is Vo, derive the transfer function.

Second order to first order

Find the equivalent first-order system (that is, find the matrix A and the vector R of dv/dx = Av + R) for the second order equation (see attachment for equation)

Initial value problem

(See attached file for full problem description with proper symbols) --- 1. Solve the initial-value problem d2u/dt2 + w2u = (w2-2)cos(t), u(0)=2, du/dt (0) = 0 where w and  are constants. Show that the solution can be written in the form 2cos(t)cos(t) where =(-w)/2 and  = (+w)/2. Regarding  as small compared to either w or , roughly sketch the solution u as a function of t. This effect is an example of the beating of two nearly equal frequencies and  is the beat frequency ---

Linear Differential Equations

Please see attached file for homework specifics. Thank-you for your help. UPDATE: Maple would be satisfactory where HPGSolver is suggested.