Heart Pacemaker A heart pacemaker, shown in Figure 3.1.12, consists of a switch, a b...

Heart Pacemaker A heart pacemaker, shown in Figure 3.1.12, 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 an electrical stimulus to the heart. In Problem 47 in Exercises 2.3 we saw that during this time the electrical stimulus is being applied to the heart, the voltage E across the heart satisfies the linear DE

(a) Let us assume that over the time interval of length t₁, 0 < t < t₁, the switch S is at position P shown in Figure 3.1.12 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 ≤ t < t₁ + t₂. Thus over the initial charging/discharging interval 0 < t , t₁ + t₂ the voltage to the heart is actually modeled by the piecewise-defined differential equation

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(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.

Reference: Problem 47 in Exercises 2.3

Heart Pacemaker A heart pacemaker consists of a switch, a battery of constant voltage E₀, 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 open, 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 equation