LRC-Circuit

[moy1989]

Hi, I'm having a whole lot of trouble with this one. Any help will be appreciated.

A series LRC-circuit has a resistor of 1 ohms, an inductor of .25 henry and a capacitor of 0.25 farads. The initial charge on the capacitor is 1 coulomb, and there is initially no current in the circuit.
Assume V(t) = 0 for t>0.

Determine the charge across the capacitor for t>0:

This is what I did:

(1/4)Q'' + Q' + 4Q = 0

characteristic equation:

(1/4)r^2 + r + 4 = 0

the roots are:

[-1 + sqrt(-3) ] / (1/2) and [-1 - sqrt(-3) ] / (1/2)
= -2 + 2*sqrt(-3) and -2 - 2*sqrt(-3)

so,

Q(t) = c1*exp(-2 + 2*sqrt(-3)) + c2*exp(-2 - 2*sqrt(-3))

from the prompt above, I know that:

Q(0) = 1 and Q'(0) = 0

so,

Q(0) = c1 + c2 and Q'(0) = (-2 + 2*sqrt(-3))*c1 + (-2 - 2*sqrt(-3))*c2

solve the set of equations:

c1 + c2 = 1
(-2 + 2*sqrt(-3))*c1 + (-2 - 2*sqrt(-3))*c2 = 1

I multiplied the first equation by -(-2 + 2*sqrt(-3)) and added to the second to attain:

(-2 - 2*sqrt(-3))*c2 - (-2 + 2*sqrt(-3))*c2 = -(-2 + 2*sqrt(-3))

so c2 =( 2 - 2*sqrt(-3) ) / (4*sqrt(-3))

and c1 = ( 2 + 2*sqrt(-3) ) / (4*sqrt(-3))

Q(t) = ( 2*sqrt(-3)+2 ) / (4*sqrt(-3)) * exp(-2+2sqrt(-3))t + ( (-2+(2sqrt(-3)) ) / (4sqrt(-3)) * exp(-2-2sqrt(-3))t)

 

[Deleted member 4993]

So now where is your problem?

 

[moy1989]

I gave this as my answer but it's wrong, supposedly.

 

[Deleted member 4993]

I gave this as my answer but it's wrong, supposedly.

Looks okay to me (may be some arithmatic mistake in there - I cannot catch it). May the answer was wanted in Sine/Cosine form. Anyway your method is correct.