Find the current flowing through resistor

[dinospamoni]

Homework Statement

The power supply in the circuit shown has V(t) =
(120V)cos(ωt), where ω = 310 rad/s. Determine the current
flowing through the resistor at time t = 9.7 s, given R = 600 Ω,
C = 18 mF, and I(0) = 0 A. As a reminder, Kirkhoff’s voltage
law for this circuit (Eq. 8-1.3 in the book) reduces to:
dV/dt = R(dI/dt) + I/C

Homework Equations

The Attempt at a Solution

I've tried this about ten times and can't seem to get the right answer:

I found dV/dt = -37200 Sin(wt) (i'll call it v' from now on)

Rearranging the equation to make it in standard form:

dI/dt + (1/RC)I = v'/R

P= 1/RC = .0926

Q=v'/R = -62 Sin(wt)

F = ∫p dt

So e^F = e^.0926 t
and e^-F = e^-.0926 t

This equation was given in class for solving this type of DE:

I = (e^-I)∫Q*e^F dt + c1*e^-F

When plug this into mathematica, it gives me an imaginary answer

Any ideas?

 

[Simon Bridge]

Did you try using phasor diagrams?

 

[dinospamoni]

I have no idea what that is. We were told to use only differential equation methods

 

[Simon Bridge]

OK - I guess you'll learn about phasor analysis after you experience the pain of having to use the integral approach.
$$\renewcommand{\tder}[1]{\frac{d #1}{dt}}

\tder{i}+\frac{1}{RC}i= -\frac{\omega V}{R}\sin\omega t$$

it looks like you tried using an integrating factor?
then you mention an equation "given in class" ... but you don't seem to understand it, so there is confusion. You should back up to where you do understand what is going on.

Have you tried the method of undetermined coefficients?

You could try guessing - it's usually less painful.
You could also have used your understanding of the physics of RC circuits to guide you in guessing i(t). The effect of the capacitor is to change the phase relationship between voltage and current ... so you'd guess a sine wave, different amplitude and phase but same frequency, as v(t). The main trouble is that you need a table of trig identities and it ends up the same as undetermined coefficients.