https://file.aiquickdraw.com/mathbot/file/1726343000850ichpomic.jpg

The equation in your image is as follows:

\[
\frac{Vo(s)}{Vi(s)} = \frac{s^2 - 2s + 1}{s^3 + 9s}
\]

You are tasked with finding \( Vo(t) \), given \( Vi(t) = 4u(t) \), where \( u(t) \) is the unit step function.

### Approach:

1. **Identify the transfer function**:
   \[
   H(s) = \frac{Vo(s)}{Vi(s)} = \frac{s^2 - 2s + 1}{s^3 + 9s}
   \]
   
   This represents the system's transfer function in the Laplace domain.

2. **Find \( Vo(s) \)**:
   Given \( Vi(t) = 4u(t) \), its Laplace transform is:
   \[
   Vi(s) = \frac{4}{s}
   \]
   Therefore,
   \[
   Vo(s) = H(s) \cdot Vi(s) = \frac{s^2 - 2s + 1}{s^3 + 9s} \cdot \frac{4}{s}
   \]

   Simplify the expression:
   \[
   Vo(s) = \frac{4(s^2 - 2s + 1)}{s(s^3 + 9s)}
   \]

3. **Simplify and perform inverse Laplace transform**:
   We now break the expression into partial fractions and apply the inverse Laplace transform to get \( Vo(t) \) in the time domain.

I'll calculate the partial fraction expansion and then determine \( Vo(t) \).The expression for \( Vo(t) \), after applying the inverse Laplace transform, is:

\[
Vo(t) = \frac{4}{27} \left( 3t + 8\sin(3t) + 6\cos(3t) - 6 \right) u(t)
\]

Where \( u(t) \) is the unit step function \( Heaviside(t) \).

### Key points:
- This result describes the output voltage \( Vo(t) \) as a function of time.
- The expression contains sinusoidal terms (from system oscillations) and a linear term due to the step input.

Let me know if you'd like further details or have any questions!

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Here are five related questions:
1. What are the physical implications of the oscillations in \( Vo(t) \)?
2. How does changing \( Vi(t) \) affect the result for \( Vo(t) \)?
3. How is the Laplace transform used to model physical systems?
4. Can we use this transfer function to analyze stability?
5. What happens to \( Vo(t) \) as \( t \to \infty \)?

**Tip:** To check if your system is stable, you can examine the poles of the transfer function \( H(s) \).

Learn how to find the output voltage Vo(t) given the transfer function H(s) = (s^2 - 2s + 1) / (s^3 + 9s) and input Vi(t) = 4u(t). This solution involves applying the Laplace transform, partial fraction decomposition, and the inverse Laplace transform.

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