Math Problem Statement
the eigan values are -2-2i with the eigan vectors [3-2i; 1 ] and the second one is -2+2i with the eigan vectors [3+2i ; 1]
Solution
Given the initial value problem:
\begin{pmatrix} 1 & -13 \\ 1 & -5 \end{pmatrix} \mathbf{x}, \quad \mathbf{x}(0) = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$$ and the eigenvalues $$-2 - 2i$$ and $$-2 + 2i$$ with corresponding eigenvectors \(\mathbf{v}_1 = \begin{pmatrix} 3 - 2i \\ 1 \end{pmatrix} \) and \(\mathbf{v}_2 = \begin{pmatrix} 3 + 2i \\ 1 \end{pmatrix} \), the solution can be expressed as follows: ### General Solution The general solution to the system is given by: $$\mathbf{x}(t) = c_1 e^{\lambda_1 t} \mathbf{v}_1 + c_2 e^{\lambda_2 t} \mathbf{v}_2$$ where: - $$\lambda_1 = -2 - 2i$$, $$\lambda_2 = -2 + 2i$$ - \(\mathbf{v}_1 = \begin{pmatrix} 3 - 2i \\ 1 \end{pmatrix}\), \(\mathbf{v}_2 = \begin{pmatrix} 3 + 2i \\ 1 \end{pmatrix}\) ### Solving for Constants Using Initial Conditions Substitute the initial condition \(\mathbf{x}(0) = \begin{pmatrix} 1 \\ 1 \end{pmatrix}\): At $$t = 0$$: $$\mathbf{x}(0) = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2$$ Substitute the eigenvectors: $$\begin{pmatrix} 1 \\ 1 \end{pmatrix} = c_1 \begin{pmatrix} 3 - 2i \\ 1 \end{pmatrix} + c_2 \begin{pmatrix} 3 + 2i \\ 1 \end{pmatrix}$$ This expands to the system: 1. $$1 = c_1 (3 - 2i) + c_2 (3 + 2i)$$ 2. $$1 = c_1 + c_2$$ ### Solve for $$c_1$$ and $$c_2$$ From the second equation: $$c_2 = 1 - c_1$$ Substitute into the first equation: $$1 = c_1 (3 - 2i) + (1 - c_1)(3 + 2i)$$ Simplify: $$1 = c_1 (3 - 2i) + (3 + 2i) - c_1 (3 + 2i)$$ $$1 = (3 + 2i) + c_1 (-4i)$$ Equating real and imaginary parts: - Real part: $$1 = 3 \implies c_1 = 0$$ - Imaginary part: solve next. Would you like the solutions updated for projection final line through a direct methodAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues and Eigenvectors
Differential Equations
Formulas
General solution: x(t) = c1 * e^(λ1 * t) * v1 + c2 * e^(λ2 * t) * v2
Eigenvalue equation: A * v = λ * v
Theorems
Fundamental theorem of linear differential equations
Eigenvalue decomposition theorem
Suitable Grade Level
Undergraduate (Mathematics or Engineering students)
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