This is a first-order linear differential equation with respect to $i(t)$, describing an RLC circuit. Let’s go through the problem:

Problem Setup

The differential equation is given by: $L \frac{di}{dt} + Ri(t) + \frac{1}{C} \int_0^t i(\tau) \, d\tau = E(t)$ where:

• $L = 0.1 \, \text{H}$
• $R = 3 \, \Omega$
• $C = 0.05 \, \text{F}$
• $E(t) = 80 \left[ u(t - 1) - u(t - 2) \right]$ with $u(t)$ as the unit step function.

The initial condition is: $i(0) = 0$

Solution Approach

The equation can be solved by using Laplace transforms, which simplify handling of integrals and step functions. The Laplace transform of the equation can be solved for $I(s)$ (the Laplace transform of $i(t)$), and then we can apply the inverse Laplace transform to find $i(t)$.

1. Transform the Equation:

   • Apply the Laplace transform to both sides.
   • Use properties of Laplace transforms for differentiation, integration, and the step function.

2. Solve for $I(s)$:

   • After transforming, solve the resulting algebraic equation in terms of $I(s)$.

3. Inverse Laplace Transform:

   • Take the inverse Laplace transform of $I(s)$ to find $i(t)$ in terms of time $t$.

Once solved, the solution form of $i(t)$ is expected to be expressed with step functions as: $i(t) = ( \text{expression in terms of } t ) u(t - 1) + ( \text{expression in terms of } t ) u(t - 2)$

I'll proceed with calculations if you'd like further steps in detail.

Would you like a detailed solution using Laplace transforms, or do you have specific questions on this setup?

Here are 5 questions for further exploration:

1. How does the presence of a step function $E(t)$ influence the solution $i(t)$?
2. What are the Laplace transform properties for handling integrals and step functions?
3. How does the initial condition $i(0) = 0$ simplify the Laplace transform solution?
4. What is the role of each component (R, L, C) in the behavior of an RLC circuit?
5. How would the solution differ if $E(t)$ had a different form, such as a sine wave?

Tip: For RLC circuit problems, Laplace transforms are especially useful to convert differential equations with step functions into algebraic forms.