RLC circuit with very large Resistance

[gonzalesdp]

Homework Statement

Consider an RLC circuit with very large Resistance.
a) when t = 0, q(0) = 0
b) When t = T_L q(T_L) = 2Q(1+e)^-1cosh(T_L/T_C)
Where T_L and T_C are the inductive and capacitive time constants, respectively.

Show that the charge as a function of time is given by:

q(t) = Q(1+e)^-1e^-t/TC + Q(1+e^-1)^-1e^-t/TLe^t/TC

Homework Equations

There is a hint that you might need to use the binomial theorem.

The Attempt at a Solution

I'm not sure where to begin with this one.

 

[vela]

Start by writing down the differential equation that the charge q satisfies, or if you already have it, the general solution to that equation.

 

[gonzalesdp]

Thanks, I have gotten that far since I posted.

I have the diff eq as

L dq^2/dt +R dq/dt +q/c = 0

I'm solving using the general form ay" + by' +cy = 0

the auxiliary equation is

am^2 + bm +c = 0

From what I understand is there are three cases to this solution:
1: Distinct real roots when b^2 - 4ac > 0
Solution: y =c_1exp(xm_1) +c_2exp(xm_2)
2: Reapeated Real roots when b^2 - 4ac = 0
Solution: y = c_1exp(xm) +xc_2exp(xm)
3: conjugate complex roots when b^2 -4ac < 0
Solution: y = exp(alphax)[c_1cos(Bx) + c_2sin(Bx)
alpha = -b/2a
B = (1/2a)(4ac-b^2)^1/2
Usually the third case is the one to use, since circuits are only useful with small resistances.
However since one of the conditions is a "very large resistance". I've been trying to use the first two.

 

[vela]

What you want to do is use the quadratic equation to write down what the roots are, and then you want to use the fact that R is large to approximate the radical. That's where the hint comes in.