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RC Circuit Differential Equation

  1. Apr 2, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the general solution of [tex]L \frac{d^2I}{dt^2} + R \frac{dI}{dt} + \frac{I}{C} = \frac{dV}{dt}[/tex] given [tex]L = 0[/tex] and [tex]V = V_0 cos(\omega t)[/tex].


    2. Relevant equations



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

    The answer the book gives is [tex] 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}[/tex] 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. jcsd
  3. Apr 2, 2009 #2

    tiny-tim

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    Homework Helper

    Welcome to PF!

    Hi sritter27! Welcome to PF! :smile:

    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 [tex]V_0 \omega C\frac{ (sin(\omega t) - \omega R C cos(\omega t))}{1 + \omega^2 R^2 C^2}[/tex] :wink:
     
  4. Apr 3, 2009 #3
    Oh wow I should have been able to see that. My many thanks for the help!
     
    Last edited: Apr 3, 2009
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