Answered: In the following system Problem,…

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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Question

In the following system Problem, categorize the eigenvalues and eigenvectors of the coefficient matrix A and sketch the phase portrait of the system by hand. Then use a computer system or graphing calculator to check your answer.

x'₁ = 7x₁ - 5x₂, x'₂ = 4x₁ + 3x₂

Expert Solution

Step 1

It is given that $x_{1}' = 7 x_{1} - 5 x_{2}$ and $x_{2}' = 4 x_{1} + 3 x_{2}$ . We can write the system of differential equations in

the matrix form as $[\begin{matrix} x_{1}' \\ x_{2}' \end{matrix}] = [\begin{matrix} 7 & - 5 \\ 4 & 3 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}]$ .

Thus, the coefficient matrix is $A = [\begin{matrix} 7 & - 5 \\ 4 & 3 \end{matrix}]$ . We have to find the eigenvalues and the corresponding

eigenvectors of the matrix A.

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