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Part b of a differential equation problem

  1. Mar 19, 2008 #1
    1. The problem statement, all variables and given/known data

    Find the charge on the capacitor in an L-R-C circuit at time t = 0.001

    L = 0.05H, R = 2 ohms, C = 0.01F

    q(0) = 5
    i(0) = 0
    E(t) = 0

    2. Relevant equations

    3. The attempt at a solution

    [tex] L \frac {di(t)}{dt} + R \frac {dq(t)}{dt} + \frac {q}{C} = 0 [/tex]

    [tex] \frac {dq^2(t)}{dt^2} + 40 \frac {dq(t)}{dt} + 2000q = 0 [/tex]

    [tex] m^2 = 40m + 2000 = 0 [/tex]

    [tex] q(t) = e^{-20t} (c1 * cos(40t) + c2 * sin(40t)) [/tex]

    [tex] q'(t) = i(t) = -20e^{-20t} (c1 * cos(40t) + c2 * sin(40t) ) + e^{-20t}(-200 sin (40t) + 40*c2*cos(40t))[/tex]

    Do you agree with my q(t) =

    [tex] q(t) = e^{-20t} (5 cos(40t) + \frac{5} {2} sin(40t)) [/tex]


    There is a second part to this problem -

    Find the first time q is equal to 0.

    Thanks to Kreizhn I have

    [tex] 0 = e^{-20t} (5cos(40t) + \frac{5} {2}sin(40t)) [/tex]

    [tex] 0 = (5cos(40t) + \frac{5} {2}sin(40t)) [/tex]

    [tex] cos(40t) = -\frac{1} {2}sin(40t)) [/tex]

    [tex] 40t = -1.1.07 [/tex]

    [tex] t = -0.0276 [/tex]

    [tex] 40t = -1.1.07 + pi [/tex]

    [tex] t = 0.0508 [/tex]

    The book gets t = 0.0669.


    Last edited: Mar 19, 2008
  2. jcsd
  3. Mar 20, 2008 #2


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    Homework Helper

    I am also getting this result. If you use the circuit parameters to compute the time constant for this LRC circuit, you also get a frequency of [tex]\omega' = 40[/tex] ,
    so this appears to jibe.

    The book's solution does not work in the equation q(t) = 0. The equation you used reduces to solving for 40t = arctan(-2). If you back-figure using the given solution, you find it is a result for 40t = arctan(-1/2); the simplest explanation is that the solver for this problem botched the algebra. (Yeah, like that's never happened in a textbook...)
  4. Mar 21, 2008 #3

    I'll go with it.

    (I'm reviewing my Diff E. from way back for the fun of it.
    This is not officially for a class so I won't get the solutions.)

    Thanks again
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