Final answer:
The question asks for solving an initial-value problem for an RL circuit, which involves integrating a first-order differential equation and applying given initial conditions to find the current over time and voltages across components.
Step-by-step explanation:
The initial-value problem presented corresponds to the behavior of an RL (resistor-inductor) circuit. To solve for the current i(t), we need to integrate the first-order differential equation given and apply the initial condition i(0) = i0. For an RL circuit, the time constant τ is given by the ratio of inductance L to resistance R. Solving the differential equation involves separating variables and integrating from an initial time t = 0 to any arbitrary time t. The solution will reflect how the current in the circuit evolves over time from the initial state influenced by the initial current i0 and the constant e.
To calculate the time constant of the circuit, we use the formula τ = L/R. The initial current through the resistor is the current at t = 0, which is the provided initial condition i0. The final current is determined by the steady-state behavior when the effect of the inductor is nullified, which is given by e/R. For specific moments in time, such as t equal to twice the time constant (2τ), calculations will involve using the exponential decay function. Lastly, the voltages across the inductor and the resistor can be found using Ohm's law and the inductor's voltage formula, VL = L di/dt.