Math Problem Statement
Solution
The problem presents a system of two differential equations:
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 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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Jacobian Matrix
Equilibrium Points
Differential Equations
Dynamical Systems
Formulas
Jacobian Matrix: J = [[βfβ/βxβ, βfβ/βxβ], [βfβ/βxβ, βfβ/βxβ]]
Equilibrium Points: Set the derivatives to 0 and solve the system
Partial Derivatives: βfβ/βxβ = -1, βfβ/βxβ = 3, βfβ/βxβ = -1, βfβ/βxβ = -2xβ
Theorems
Jacobian matrix theorem
Equilibrium point definition
Suitable Grade Level
College level (Advanced Calculus / Differential Equations)
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