# A heart pacemaker, shown in the following figure, consists of a switch, a battery, a capacitor, and the heart as a resistor. When the switch S is at P,the capacitor charges; when S is at Q, the capacitor discharges, sending

dE/dt = - 1/RC E.

(a) Let us assume that over the time interval of length t_{1}, 0 < t < t_{1}, the switch S is at position P shown in the following figure and the capacitor is being charged. When the switch is moved to position Q at time t_{1} the capacitor discharges, sending an impulse to the heart over the time interval of length t_{2}: t_{1} ‰¤ t < t_{1} + t_{2}. Thus over the initial charging/ discharging interval 0 < t < t_{1} + t_{2} the voltage to the heart is actually modeled by the piece wise-linear differential equation

By moving S between P and Q, the charging and discharging over time intervals of lengths t_{1} and t_{2} is repeated indefinitely.

Suppose t_{1} = 4 s, t_{2} = 2 s, E_{0} = 12 V, and E(0) = 0, E(4) = 12, E(6) = 0, E(10) = 12, E(12) = 0, and so on. Solve for E(t) for 0 ‰¤ t ‰¤ 24.

(b) Suppose for the sake of illustration that R = C = 1. Use a graphing utility to graph the solution for the IVP in part (a) for 0 ‰¤ t ‰¤ 24.

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## a For 0 t 4 6 t 10 and 12 t 16 no voltage is applied to the heart and E t 0 At the other …View the full answer

**Related Book For**

## A First Course in Differential Equations with Modeling Applications

**ISBN:** 978-1305965720

11th edition

**Authors:** Dennis G. Zill

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