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 an electrical stimulus to the heart. 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

dE/dt = - 1/RC E.

(a) Let us assume that over the time interval of length t₁, 0 < t < t₁, 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 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 piece wise-linear 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.

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