Consider a system model given by
 
What is the free response?

Consider a system model given by
 
What is the free response?











To find the **free response** of a system, we generally need a system model, which often comes in the form of a differential equation or a state-space representation. The **free response** refers to the system's natural behavior when no external input is applied (i.e., \( u(t) = 0 \)).

Here is a general outline of how to find the free response for common system models:

### 1. **For Differential Equations**:
If the system is given by a linear constant-coefficient differential equation, such as:

\[
a_n \frac{d^n y(t)}{dt^n} + a_{n-1} \frac{d^{n-1} y(t)}{dt^{n-1}} + \dots + a_1 \frac{dy(t)}{dt} + a_0 y(t) = b_m \frac{d^m u(t)}{dt^m} + \dots + b_1 \frac{du(t)}{dt} + b_0 u(t)
\]

For **free response**, we set \( u(t) = 0 \), resulting in a homogeneous differential equation:

\[
a_n \frac{d^n y(t)}{dt^n} + a_{n-1} \frac{d^{n-1} y(t)}{dt^{n-1}} + \dots + a_1 \frac{dy(t)}{dt} + a_0 y(t) = 0
\]

The solution to this equation is typically found by solving the characteristic equation, which gives the natural modes of the system. The general solution (free response) will be a linear combination of exponentials or other functions based on the roots of the characteristic equation.

### 2. **For State-Space Models**:
In the state-space form, the system is typically described as:

\[
\dot{x}(t) = A x(t) + B u(t)
\]
\[
y(t) = C x(t) + D u(t)
\]

For the **free response**, set \( u(t) = 0 \), and the system simplifies to:

\[
\dot{x}(t) = A x(t)
\]

The solution to this system depends on the matrix \( A \), and it is typically given by:

\[
x(t) = e^{At} x(0)
\]

The free response \( y(t) \) is then given by:

\[
y(t) = C e^{At} x(0)
\]

This represents the natural evolution of the system's state over time, without any external input.

---

To provide a detailed solution, I'd need the exact form of the system model you're referring to (whether it's a differential equation, state-space model, or transfer function).

Would you like me to assume a specific system or upload the system model?

### Related Questions:
1. What are the different methods for solving homogeneous differential equations?
2. How does the characteristic equation relate to the natural frequencies of the system?
3. Can you explain how the matrix exponential \( e^{At} \) is computed for state-space systems?
4. What are the conditions for stability in the free response of a system?
5. How does initial state \( x(0) \) affect the free response?

### Tip:
In state-space models, the eigenvalues of matrix \( A \) provide key insights into the system's stability and natural behavior.

Consider a system model given by

What is the free response?

Explore how to find the free response in system models, including both differential equations and state-space representations. This detailed guide covers key concepts like homogeneous equations, characteristic equations, and matrix exponentials, suitable for engineering or advanced mathematics students.

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