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Math Help - Differential Equation - RC cricuit

  1. #1
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    Differential Equation - RC cricuit

    A pacemaker consists of a switch, a battery of constant voltage E sub 0, a capacitor with constant capacitance C, and the heart as a resistor with constant resistance R. When the switch is closed the capacitor charges; when the switch is opne, the capacitor discharges, sending an electrical stimulus to the heart. During the time the heart is being stimulated the voltage E across the heart satisfies the linear differential eq

    dE/dt = - 1/ (RC) * E

    Solve the DE subject to E(4) = E sub 0
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  2. #2
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    Quote Originally Posted by QuestionSleep View Post
    A pacemaker consists of a switch, a battery of constant voltage E sub 0, a capacitor with constant capacitance C, and the heart as a resistor with constant resistance R. When the switch is closed the capacitor charges; when the switch is opne, the capacitor discharges, sending an electrical stimulus to the heart. During the time the heart is being stimulated the voltage E across the heart satisfies the linear differential eq

    dE/dt = - 1/ (RC) * E

    Solve the DE subject to E(4) = E sub 0
     \frac{dE}{dt} = \frac{-1}{RC} E

     \frac{dE}{E} = \frac{-1}{RC} dt

    \int \frac{dE}{E} = \int \frac{-1}{RC} dt

    \ln|E| = \frac{-t}{RC} +K^*

    E = e^{\frac{-t}{RC} +K^*}

    E = e^{\frac{-t}{RC}}e^{K^*}

    E = Ke^{\frac{-t}{RC}}

    such that  K = e^{K^*}
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