Heart Pacemaker A heart pacemaker, showninFIGURE2.7.10, consists of a switch, a battery

Chapter 2, Problem 47

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Heart Pacemaker A heart pacemaker, showninFIGURE2.7.10, 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 Sis at Q, the capacitor discharges, sending an electrical stimulus to the heart. In in Exercises 2.3, we saw that during the time the electrical stimulus is being applied to the heart, the voltage E across the heart satisfies the linear DE -= dE 1 --E. dt RC (a) Let us assume that over the time interval of length ti. (0, t1), the switch Sis at position P shown in Figure 2. 7 .10 and the capacitor is being charged. When the switch is moved to position Q at time t1 the capacitor discharges, sending an impulse to the heart over the time interval of length t2: [t1' t1 + t2). Thus, over the initial charging/discharging interval (0, t1 + t2) the voltage to the heart is actually modeled by the piecewise-defined differential equation dE { O, -- 1 dt - RCE, By moving S between P and Q, the charging and discharging over time intervals of lengths t1 and t2 is repeated indefinitely. Suppose t1 = 4 s, t2 = 2 s, E0 = 12 V, and E(O) = 0, E(4) = 12, E(6) = 0, E(lO) = 12, (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. Q switch p FIGURE 2.7.10 Model of a pacemaker in 47