Math Problem Statement

Assume the following set of equations: π‘₯̇₁ = βˆ’π‘₯₁ + 3π‘₯β‚‚ π‘₯Μ‡β‚‚ = βˆ’π‘₯₁ βˆ’ π‘₯β‚‚Β². Select the correct pair of Jacobian Matrix and equilibrium points for the equation above.

Solution

The problem presents a system of two differential equations:

x1Λ™=βˆ’x1+3x2\dot{x_1} = -x_1 + 3x_2 x2Λ™=βˆ’x1βˆ’x22\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 JJ for a system of equations is given by the matrix of partial derivatives:

\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)$$, which corresponds to **Option L**. ### Final Answer: **Option B, L** --- Would you like further clarification on any part of the process? Here are five related questions you might consider: 1. How is the Jacobian matrix used to assess stability of equilibrium points? 2. What are the eigenvalues of the Jacobian at each equilibrium point? 3. How do nonlinear terms like $$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.

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Math Problem Analysis

Mathematical Concepts

Jacobian Matrix
Equilibrium Points
Differential Equations
Dynamical Systems

Formulas

Jacobian Matrix: J = [[βˆ‚f₁/βˆ‚x₁, βˆ‚f₁/βˆ‚xβ‚‚], [βˆ‚fβ‚‚/βˆ‚x₁, βˆ‚fβ‚‚/βˆ‚xβ‚‚]]
Equilibrium Points: Set the derivatives to 0 and solve the system
Partial Derivatives: βˆ‚f₁/βˆ‚x₁ = -1, βˆ‚f₁/βˆ‚xβ‚‚ = 3, βˆ‚fβ‚‚/βˆ‚x₁ = -1, βˆ‚fβ‚‚/βˆ‚xβ‚‚ = -2xβ‚‚

Theorems

Jacobian matrix theorem
Equilibrium point definition

Suitable Grade Level

College level (Advanced Calculus / Differential Equations)