Answered: State the Eigenvalue Method for…

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

State the Eigenvalue Method for Homogeneous Systems?

Expert Solution

Step 1

consider a homogenous system

$x' = 4 x + 2 y y' = 3 x - y$

this can be written as

$[\begin{matrix} x' \\ y' \end{matrix}] = [\begin{matrix} 4 & 2 \\ 3 & - 1 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]$

step 1

for calculating the eigenvalue , subtract $λ$ from the diagonal elements of the matrix

$[\begin{matrix} 4 - λ & 2 \\ 3 & - 1 - λ \end{matrix}] . . . (i)$

Step 2

find the determinant of the matrix by putting it equal to zero.

$|\begin{matrix} 4 - λ & 2 \\ 3 & - 1 - λ \end{matrix}| = 0 \Rightarrow (4 - λ) (- 1 - λ) - 6 = 0 \Rightarrow - 4 - 4 λ + λ + λ^{2} - 6 = 0$

simplify the equation and find the roots of it

$\Rightarrow λ^{2} - 3 λ - 10 = 0 \Rightarrow λ^{2} - 5 λ + 2 λ - 10 = 0 \Rightarrow λ (λ - 5) + 2 (λ - 10) = 0 \Rightarrow (λ + 2) (λ - 5) = 0$

using the result that if $a b = 0$

implies either $a = 0 or b = 0$

$λ + 2 = 0 \Rightarrow λ = - 2 λ - 5 = 0 \Rightarrow λ = 5$

Step 3

thus, we get two eigenvalues of the system

$λ_{1} = - 2, λ_{2} = 5$

step 2

find the eigenvectors corresponding to eigenvalues

put $λ = - 2$ into $[\begin{matrix} 4 - λ & 2 \\ 3 & - 1 - λ \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}]$

$[\begin{matrix} 4 + 2 & 2 \\ 3 & - 1 + 2 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}] \Rightarrow [\begin{matrix} 6 & 2 \\ 3 & 1 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}] \Rightarrow 6 x + 2 y = 0 \Rightarrow y = - 3 x$

that is

$v_{1} = [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 1 \\ - 3 \end{matrix}] (x = 1)$

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