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 = 50x₁ - 20x₂, x'₂ = 100x₁ - 60x₂

Expert Solution

Step 1

We can write the given system of linear differential equations in two variables in the matrix form as below:

$X' = (\begin{matrix} 50 & - 20 \\ 100 & - 60 \end{matrix}) X$ .

Here, $X = (\begin{matrix} x_{1} \\ x_{2} \end{matrix})$ and $X' = (\begin{matrix} x_{1}' \\ x_{2}' \end{matrix})$ . The coefficient matrix is $A = (\begin{matrix} 50 & - 20 \\ 100 & - 60 \end{matrix})$ . First, we have to find the

eigenvalues of the matrix.

Step 2

Consider the matrix $A - λ I = (\begin{matrix} 50 - λ & - 20 \\ 100 & - 60 - λ \end{matrix})$ . The determinant of this matrix is $λ^{2} + 10 λ - 1000$ .

Solving the equation $λ^{2} + 10 λ - 1000 = 0$ , we get, $λ_{1} = - 5 (1 + \sqrt{41})$ and $λ_{2} = 5 (- 1 + \sqrt{41})$ .

These are the eigenvalues.

For $λ = - 5 (1 + \sqrt{41})$ , we have, $A - λ I = (\begin{matrix} 5 (1 + \sqrt{41}) + 50 & - 20 \\ 100 & - 60 + 5 (1 + \sqrt{41}) \end{matrix})$ . The null space of this

matrix is $\{(\begin{matrix} - \frac{- 11 + \sqrt{41}}{20} \\ 1 \end{matrix})\}$ . This is an eigenvector corresponding to the eigenvalue $- 5 (1 + \sqrt{41})$ .

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