Translate the given verbal statement into a differential equation using appropri

Question

Translate the given verbal statement into a differential equation using appropriate mathematical notation. Write the equation in a form so that the constant of proportionality is positive. Identify every letter that appears in the equation. A basic electrical circuit that is usually considered in elementary physics courses consists of a switch that can be opened or closed, a battery, a resistor, and a capacitor connected in series. When the switch is closed, an electrical current begins flowing through the circuit,; however, it immediately begins to decrease to zero. The rate of change of the current at a given moment is proportional to its value at that moment

Explanation / Answer

let i = current at any moment Vr = voltage drop across resistance Vc = voltage drop across capacitance Given that ''rate of change of the current at a given moment is proportional to its value at that moment" => di/dt = ki (k is a constant of proportionality) => di/i = kdt => ln(i) = kt +c ( c being integration constant) => i = e^(kt+c) Given that at t = infinity i =0 => k = -a ( a being a positive constant) => i = e^(-at) -----------( c = 0 as there is no current at t=0)

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