https://file.aiquickdraw.com/mathbot/file/1726346075787tbie2pzw.jpg

The equation in the image is:

\[
\frac{Vo(s)}{Vi(s)} = \frac{s^2 - 2s + 1}{s^3 + 9s}
\]
It also asks to find \( Vo(t) \) given that \( Vi(t) = 1 \cdot u(t) \), where \( u(t) \) is the unit step function.

### Solution Outline:

1. **Laplace Transform of \( Vi(t) \):**  
   Since \( Vi(t) = u(t) \), we know from Laplace transform tables that:
   \[
   Vi(s) = \frac{1}{s}
   \]

2. **Find \( Vo(s) \):**  
   Using the transfer function, we have:
   \[
   \frac{Vo(s)}{Vi(s)} = \frac{s^2 - 2s + 1}{s^3 + 9s}
   \]
   Multiplying both sides by \( Vi(s) = \frac{1}{s} \):
   \[
   Vo(s) = \frac{s^2 - 2s + 1}{s(s^3 + 9s)}
   \]
   Simplify:
   \[
   Vo(s) = \frac{s^2 - 2s + 1}{s^4 + 9s^2}
   \]

3. **Inverse Laplace Transform:**  
   The next step is to perform a partial fraction decomposition on the above equation to find the inverse Laplace transform, which will give us \( Vo(t) \). 

I will compute the partial fraction decomposition and the inverse Laplace transform next.The partial fraction decomposition of \( Vo(s) \) is:

\[
Vo(s) = \frac{2(s + 4)}{9(s^2 + 9)} - \frac{2}{9s} + \frac{1}{9s^2}
\]

Taking the inverse Laplace transform of this expression, the result for \( Vo(t) \) is:

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

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

Would you like further details or clarification on any part of the process? 

Here are 5 related questions to consider:
1. What are the key steps in performing partial fraction decomposition?
2. How do we compute the inverse Laplace transform of trigonometric terms like \( \sin(3t) \) and \( \cos(3t) \)?
3. What role does the unit step function \( u(t) \) play in the context of these problems?
4. How can we interpret the response \( Vo(t) \) in terms of system behavior?
5. What happens to \( Vo(t) \) as \( t \to \infty \)?

**Tip:** Understanding partial fraction decomposition is crucial when simplifying complex transfer functions for inverse Laplace transforms.

Find Vo(t) given Vi(t) = u(t) using the transfer function Vo(s)/Vi(s) = (s^2 - 2s + 1)/(s^3 + 9s)

Learn how to find Vo(t) using Laplace transforms, partial fraction decomposition, and inverse Laplace transformations. This problem involves analyzing a system with a transfer function Vo(s)/Vi(s) = (s^2 - 2s + 1)/(s^3 + 9s) and using Vi(t) = u(t). Suitable for undergraduate students studying control systems or signals and systems.

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