Answered: In the following problem apply the…

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 problem apply the eigenvalue method to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. use a computer system or graphingcalculator to construct a direction field and typical solutioncurves for the given system.

x'₁ = 5x₁ - 9x₂, x'₂ = 2x₁ - x2

Expert Solution

Step 1

The given system is--

$\begin{array}{rcl} x_{1}^{'} & = & 5 x_{1} - 9 x_{2} \\ x_{2}^{'} & = & 2 x_{1} - x_{2} \end{array}$

So,

In matrix form, the system can be written as-

$A = [\begin{matrix} 5 & - 9 \\ 2 & - 1 \end{matrix}]$

Step 2

Let $λ$ be the eigenvalues of the matrix A.

$\begin{array}{rcl} |A - λ I| = 0 \Rightarrow |\begin{matrix} 5 - λ & - 9 \\ 2 & - 1 - λ \end{matrix}| & = & 0 \\ \Rightarrow (5 - λ) (- 1 - λ) - 2 (- 9) & = & 0 \\ \Rightarrow λ^{2} - 4 λ - 5 + 18 & = & 0 \\ \Rightarrow λ^{2} - 4 λ + 13 & = & 0 \\ \Rightarrow λ & = & \frac{- (- 4) \pm \sqrt{{(- 4)}^{2} - 4 (1) (13)}}{2 (1)} \\ \Rightarrow λ & = & \frac{4 \pm \sqrt{- 36}}{2} \\ \Rightarrow λ & = & \frac{4 \pm 6 i}{2} \\ \Rightarrow λ & = & 2 \pm 3 i \end{array}$

So,

The eigenvalues are $2 + 3 i$ and $2 - 3 i$ .

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