Thread: electrical circuit problem

Consider an electrical circuit comprising an inductor and a capacitor, with no external applied voltage, and no electrical resistance. The current, i, through the inductor is related to the charge, q, on the capacitor by the differential equation: where the inductance, L, and capacitance C, are positive constants.


L(di/dt)+q/c=0


(a)Given that the charge and the current are related through i=dq/dt derive a second order differential equation for q(t).


i know that the second order differential is


L(d2q/dt2)+q/c=0


please help me solve the problem


b)assuming that the initial charge q(0)=q0 and the initial current i(0)=i0, show that


q(t)=q0cos(wt)+(i0/w)sin(wt)


where w^2=1/LC

c) hence show that q(t) can be written as

q(t)=Acos(wt+fi)

where A and fi are to be determined


thanks for the help

Hi, math254.

Your DE is a second order linear homogeneous equation. One route you could take is the following:

1. Set up what's called the "characteristic equation." This is done by replacing your DE with the polynomial



(if you haven't seen this before and need more details let me know).

2. Solve the above for Note: The two solutions for will be a complex numbers.

3. The general solution to your DE is then



where is the real part of and is the imaginary part of . See Equation 11 of http://www.stewartcalculus.com/data/...arEqns_Stu.pdf

4. Now use the initial conditions to determine and .

5. To prove part c) I would suggest visiting http://www.mathcentre.ac.uk/resource...a-alphaetc.pdf. There is a nice outline showing how to express the sum of sine and cosine as one cosine function.

Does this help sort out the confusion?

Good luck!

thank you