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The problem presents a system of two differential equations:

\[
\dot{x_1} = -x_1 + 3x_2
\]
\[
\dot{x_2} = -x_1 - x_2^2
\]

The task is to find the Jacobian matrix for this system and identify the correct equilibrium points.

### Step 1: Calculate the Jacobian matrix
The Jacobian matrix \( J \) for a system of equations is given by the matrix of partial derivatives:

\[
J = \begin{bmatrix}
\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} \\
\frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2}
\end{bmatrix}
\]

Where:
- \( f_1(x_1, x_2) = -x_1 + 3x_2 \)
- \( f_2(x_1, x_2) = -x_1 - x_2^2 \)

### Partial derivatives:
1. \( \frac{\partial f_1}{\partial x_1} = -1 \)
2. \( \frac{\partial f_1}{\partial x_2} = 3 \)
3. \( \frac{\partial f_2}{\partial x_1} = -1 \)
4. \( \frac{\partial f_2}{\partial x_2} = -2x_2 \)

Thus, the Jacobian matrix is:

\[
J = \begin{bmatrix}
-1 & 3 \\
-1 & -2x_2
\end{bmatrix}
\]

### Step 2: Find the equilibrium points
Equilibrium points occur where \( \dot{x_1} = 0 \) and \( \dot{x_2} = 0 \). Set the system of equations to zero and solve:

\[
-x_1 + 3x_2 = 0
\]
\[
-x_1 - x_2^2 = 0
\]

From the first equation: 
\[
x_1 = 3x_2
\]

Substitute this into the second equation:
\[
-(3x_2) - x_2^2 = 0 \quad \Rightarrow \quad -3x_2 - x_2^2 = 0 \quad \Rightarrow \quad x_2(-x_2 - 3) = 0
\]

Thus, \( x_2 = 0 \) or \( x_2 = -3 \).

1. If \( x_2 = 0 \), then \( x_1 = 0 \).
2. If \( x_2 = -3 \), then \( x_1 = -9 \).

The equilibrium points are \( (0, 0) \) and \( (-9, -3) \).

### Step 3: Matching the Jacobian and Equilibrium Points
Now, let's match the Jacobian and equilibrium points with the provided choices.

- For \( (0, 0) \), the Jacobian becomes:

\[
J = \begin{bmatrix}
-1 & 3 \\
-1 & 0
\end{bmatrix}
\]

- For \( (-9, -3) \), the Jacobian becomes:

\[
J = \begin{bmatrix}
-1 & 3 \\
-1 & 6
\end{bmatrix}
\]

From the options given, the correct Jacobian is **B** and the equilibrium points are \( (0, 0) \) and \( (-9, -3) \), which corresponds to **Option L**.

### Final Answer:
**Option B, L**

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Would you like further clarification on any part of the process? Here are five related questions you might consider:

1. How is the Jacobian matrix used to assess stability of equilibrium points?
2. What are the eigenvalues of the Jacobian at each equilibrium point?
3. How do nonlinear terms like \( x_2^2 \) affect the stability of the system?
4. What is the significance of equilibrium points in a dynamical system?
5. How can phase portraits help visualize the behavior near equilibrium points?

**Tip:** Always verify if an equilibrium point is stable by examining the eigenvalues of the Jacobian at that point.

Assume the following set of equations: 
 𝑥̇₁ = −𝑥₁ + 3𝑥₂ 
 𝑥̇₂ = −𝑥₁ − 𝑥₂². 
 Select the correct pair of Jacobian Matrix and equilibrium points for the equation above.

Learn how to calculate the Jacobian matrix and equilibrium points for a system of differential equations. This step-by-step guide includes formulas, partial derivatives, and methods for identifying equilibrium points.

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