# RC Circuit Differential Equation

1. Apr 2, 2009

### sritter27

1. The problem statement, all variables and given/known data
Find the general solution of $$L \frac{d^2I}{dt^2} + R \frac{dI}{dt} + \frac{I}{C} = \frac{dV}{dt}$$ given $$L = 0$$ and $$V = V_0 cos(\omega t)$$.

2. Relevant equations

3. The attempt at a solution
So the equation basically turns into a first-order RC circuit equation $$R \frac{dI}{dt} + \frac{I}{C} = \frac{dV}{dt}$$, but I'm not sure how to approach it to find a general solution.

The answer the book gives is $$I = Ae^{-\frac{t}{RC}} - \frac{V_0 \omega C (sin(\omega t) - \omega R C cos(\omega t))}{1 + \omega^2 R^2 C^2}$$ and I'm not sure how they came to that conclusion, so any help or nudge in the right direction would be greatly appreciated.

2. Apr 2, 2009

### tiny-tim

Welcome to PF!

Hi sritter27! Welcome to PF!

To find the general solution of R dI/dt + I/C = -ωV0sinωt,

assume it's of the form Acosωt + Bsinωt, and you get the given result,

except that you've copied it wrong … it's $$V_0 \omega C\frac{ (sin(\omega t) - \omega R C cos(\omega t))}{1 + \omega^2 R^2 C^2}$$

3. Apr 3, 2009

### sritter27

Oh wow I should have been able to see that. My many thanks for the help!

Last edited: Apr 3, 2009