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this question below Suppose a system of constant coefficient differential equations '= A has a 2 x 2 matrix A with 1 . Find the real part ₁ and the -5+6i] eigenvalues A1, A2 = 4±2i and A₁ = 4+2i has eigenvector imaginary part 2 and write the fundamental matrix [₁2] for the general solution: Submit Question 1 x -5 x 0x 6 x

this question below Suppose a system of constant coefficient differential equations '= A has a 2 x 2 matrix A with 1 . Find the real part ₁ and the -5+6i] eigenvalues A1, A2 = 4±2i and A₁ = 4+2i has eigenvector imaginary part 2 and write the fundamental matrix [₁2] for the general solution: Submit Question 1 x -5 x 0x 6 x

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Educational Website Content: Differential Equations and Eigenvalues** --- **Topic: System of Constant Coefficient Differential Equations** Suppose a system of constant coefficient differential equations \( \mathbf{x}' = A\mathbf{x} \) has a 2 x 2 matrix \( A \) with eigenvalues \( \lambda_1, \lambda_2 = 4 \pm 2i \) and \( \lambda_1 = 4 + 2i \) has eigenvector \( \begin{bmatrix} 1 \\ -5 + 6i \end{bmatrix} \). Find the real part \( \mathbf{x}_1 \) and the imaginary part \( \mathbf{x}_2 \), and write the fundamental matrix \( [\mathbf{x}_1 \ \mathbf{x}_2] \) for the general solution: **Input Fields:** - Real part input: Enter a 1 in the first box and 0 in the second box for \( \mathbf{x}_1 \). - Imaginary part input: Enter -5 in the first box and 6 in the second box for \( \mathbf{x}_2 \). [Submit Question Button] --- The form above provides a way to input the real and imaginary parts of the eigenvector corresponding to the given complex eigenvalues, helping students practice constructing the fundamental matrix for solving differential equations.
Transcribed Image Text:**Educational Website Content: Differential Equations and Eigenvalues** --- **Topic: System of Constant Coefficient Differential Equations** Suppose a system of constant coefficient differential equations \( \mathbf{x}' = A\mathbf{x} \) has a 2 x 2 matrix \( A \) with eigenvalues \( \lambda_1, \lambda_2 = 4 \pm 2i \) and \( \lambda_1 = 4 + 2i \) has eigenvector \( \begin{bmatrix} 1 \\ -5 + 6i \end{bmatrix} \). Find the real part \( \mathbf{x}_1 \) and the imaginary part \( \mathbf{x}_2 \), and write the fundamental matrix \( [\mathbf{x}_1 \ \mathbf{x}_2] \) for the general solution: **Input Fields:** - Real part input: Enter a 1 in the first box and 0 in the second box for \( \mathbf{x}_1 \). - Imaginary part input: Enter -5 in the first box and 6 in the second box for \( \mathbf{x}_2 \). [Submit Question Button] --- The form above provides a way to input the real and imaginary parts of the eigenvector corresponding to the given complex eigenvalues, helping students practice constructing the fundamental matrix for solving differential equations.
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