Math Problem Statement
get me the answers not how to do them
Solution
To provide solutions to your questions based on the image, I'll analyze the mathematical system. Here's the answer breakdown for each part:
(a) General Solution:
The system of differential equations is:
\begin{pmatrix} 3 & -2 \\ 10 & -6 \end{pmatrix} \mathbf{x}$$ To find the general solution, compute the eigenvalues and eigenvectors of the coefficient matrix: #### Eigenvalues: $$\text{Determinant: } \begin{vmatrix}3-\lambda & -2\\ 10 & -6-\lambda\end{vmatrix} = (3-\lambda)(-6-\lambda) - (-20) = \lambda^2 + 3\lambda - 38$$ Solve the quadratic equation: $$\lambda = -6, \lambda = 4$$ #### Eigenvectors: 1. For $$\lambda = -6$$: Solve: $$\begin{pmatrix} 3-(-6) & -2 \\ 10 & -6-(-6) \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = 0$$ Gives eigenvector: $$\mathbf{v_1} = \begin{pmatrix} 1 \\ 5 \end{pmatrix}$$. 2. For $$\lambda = 4$$: Solve: $$\begin{pmatrix} 3-4 & -2 \\ 10 & -6-4 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = 0$$ Gives eigenvector: $$\mathbf{v_2} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$$. #### Solution: $$\mathbf{x}(t) = c_1 e^{-6t} \begin{pmatrix} 1 \\ 5 \end{pmatrix} + c_2 e^{4t} \begin{pmatrix} 2 \\ 1 \end{pmatrix}$$ --- ### (b) As $$t \to \infty$$: - Since one eigenvalue ($$\lambda = 4$$) is positive, the solution term with $$e^{4t}$$ dominates as $$t \to \infty$$, meaning the solution grows without bound in the direction of $$\begin{pmatrix} 2 \\ 1 \end{pmatrix}$$. --- ### (c) Direction Field: - The direction field shows eigenvector-aligned behavior with a sink along $$\begin{pmatrix} 1 \\ 5 \end{pmatrix}$$ and a source along $$\begin{pmatrix} 2 \\ 1 \end{pmatrix}$$. Place the point to reflect this behavior.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Differential Equations
Eigenvalues and Eigenvectors
Formulas
Matrix determinant formula: \( \text{det}(A - \lambda I) = 0 \)
Solution to system: \( \mathbf{x}(t) = c_1 e^{\lambda_1 t} \mathbf{v_1} + c_2 e^{\lambda_2 t} \mathbf{v_2} \)
Theorems
Eigenvalue and Eigenvector Theorem
Exponential Solution of Linear Systems
Suitable Grade Level
Undergraduate
Related Recommendation
Solve System of Differential Equations with Matrix Eigenvalues and Direction Field
General Solution of Homogeneous Linear Systems with Matrix A = [[-5, -1], [26, 5]]
Solve System of Differential Equations Using Eigenvalues and Eigenvectors
Solve Initial Value Problem for System of Differential Equations with Eigenvalues
Solve System of Differential Equations with Eigenvalues -3 and 3