# Thread: Differential Equation - RC cricuit

1. ## 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

2. Originally Posted by QuestionSleep
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
$\displaystyle \frac{dE}{dt} = \frac{-1}{RC} E$

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

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

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

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

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

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

such that $\displaystyle K = e^{K^*}$