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Consider a simple RL circuit with a resistor \(R\), an inductor \(L\), and a constant voltage source \(E\). The current \(I(t)\) in the circuit is governed by the differential equation:$$ L\frac{dI}{dt} + RI = E $$The circuit is switched on at time \(t=0\), meaning there is initially no current: \(I(0) = 0\).
  1. Show that the equation can be written in the form \(I' = aI + b\). Identify the constants \(a\) and \(b\).
  2. Determine the general solution of this differential equation on \([0, +\infty)\).
  3. Using the initial condition \(I(0) = 0\), determine the unique function \(I(t)\) expressing the current in the circuit.

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