Math Problem Statement
Solution
To find the Jacobian matrix for the given system of differential equations at the equilibrium point , follow these steps:
1. Write the system of equations:
2. Calculate the partial derivatives:
The Jacobian matrix is computed as:
\frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{bmatrix}$$ Where: - $$f_1(x, y) = 10 - x^2 - y^2$$ - $$f_2(x, y) = 3x - y$$ #### Partial derivatives: 1. $$\frac{\partial f_1}{\partial x} = \frac{\partial}{\partial x} (10 - x^2 - y^2) = -2x$$ 2. $$\frac{\partial f_1}{\partial y} = \frac{\partial}{\partial y} (10 - x^2 - y^2) = -2y$$ 3. $$\frac{\partial f_2}{\partial x} = \frac{\partial}{\partial x} (3x - y) = 3$$ 4. $$\frac{\partial f_2}{\partial y} = \frac{\partial}{\partial y} (3x - y) = -1$$ ### 3. Evaluate the partial derivatives at $$(x, y) = (1, 3)$$: - $$\frac{\partial f_1}{\partial x} = -2(1) = -2$$ - $$\frac{\partial f_1}{\partial y} = -2(3) = -6$$ - $$\frac{\partial f_2}{\partial x} = 3$$ (independent of $$x$$ or $$y$$) - $$\frac{\partial f_2}{\partial y} = -1$$ (independent of $$x$$ or $$y$$) ### 4. Construct the Jacobian matrix: $$J = \begin{bmatrix} -2 & -6 \\ 3 & -1 \end{bmatrix}$$ ### Final Answer: $$J = \begin{bmatrix} -2 & -6 \\ 3 & -1 \end{bmatrix}$$ Let me know if you need further clarification or a detailed explanation of any step! --- ### Related Questions: 1. What is the significance of the Jacobian matrix in dynamical systems analysis? 2. How do equilibrium points affect the stability of a system? 3. Can you explain how eigenvalues of the Jacobian matrix are used to determine stability? 4. How does linearization relate to the Jacobian matrix in nonlinear systems? 5. What would the Jacobian matrix look like for a different equilibrium point? ### Tip: Always verify that the equilibrium point satisfies the original equations before calculating the Jacobian matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Jacobian Matrix
Partial Derivatives
Dynamical Systems
Equilibrium Points
Formulas
Jacobian matrix formula: J = [[∂f1/∂x, ∂f1/∂y], [∂f2/∂x, ∂f2/∂y]]
Theorems
Jacobian matrix for nonlinear systems
Stability analysis using eigenvalues
Suitable Grade Level
Grades 11-12
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