Second Order Linear Differential Equations

i)An electrical circuit consists of an inductor of inductance *L* in series with a capacitor of capacitance *C*. An alternating voltage of Eocos(wt) is applied to the circuit at time *t*=0 with the initial charge *q* in the circuit being zero and the initial current *i* (=*dq/dt*) being zero. Using Kirchoff’s laws the charge in the circuit satisfies the equation[/FONT]

$\displaystyle

L \frac{d^2q}{dt^2} + \frac{q}{C} = Eocos(wt)

$

Show that the charge in the circuit at any time *t* is given by the equation

$\displaystyle

q = \frac{Eo}{L(w^2 - n^2)}[cos(nt) - cos(wt)]

$

where

$\displaystyle

n^2 = \frac{1}{CL}

$

I can get most of this. I dont get the cos(nt) bit and my denominator is

$\displaystyle

n^2 - w^2

$

ii)

What is the current in the circuit at any time *t* ?.

What is the charge in the circuit when *w *= *n ?*