Here is a problem similar to the one you provided:

Problem:

Show that $i(t) = I_0 e^{-t / \tau}$ is a solution to the following first-order differential equation:

$L \frac{di(t)}{dt} + Ri(t) = V_0 e^{-t/\tau}$

Where:

• $L$ is the inductance,
• $R$ is the resistance,
• $V_0$ is a constant voltage,
• $\tau$ is a time constant,
• $I_0$ is the initial current at $t = 0$.

Would you like to solve this problem or would you prefer additional details about the steps involved?

Here are five related questions that expand on the problem:

1. How does the solution change if the voltage term is constant instead of decaying over time?
2. What physical systems can be modeled by this type of first-order differential equation?
3. How can you determine the time constant $\tau$ in terms of $R$ and $L$?
4. What happens to the current $i(t)$ when $t \to \infty$?
5. How would the solution change if $R = 0$, representing a lossless circuit?

Tip: When solving differential equations, always check if the solution satisfies both the homogeneous and inhomogeneous parts of the equation (if applicable).