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**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).

Show that i = I₀ e^(-t/τ) is a solution of the differential equation iR + L(di/dt) = 0.

This page demonstrates how to solve the differential equation iR + L(di/dt) = 0, using an exponential decay current i = I₀ e^(-t/τ). Learn how to apply this to analyze electrical circuits involving resistors and inductors.

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