Please help solve attached differential equations question below. Find the general solution of-1Discuss the type and stability of the critical point (0,0) as it depends on the parameter a.

Given xι= a1-1ax Let the given matrix be A=a1-1a Now we have to find eigen values first,…

Answered: Find the general solution of I' = -1…

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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Please help solve attached differential equations question below.

Find the general solution of
-1
Discuss the type and stability of the critical point (0,0) as it depends on the parameter a.

Expert Solution

Step 1

Given $x^{ι} = [\begin{matrix} a & 1 \\ - 1 & a \end{matrix}] x$

Let the given matrix be A= $[\begin{matrix} a & 1 \\ - 1 & a \end{matrix}]$

Now we have to find eigen values first,

$[\begin{matrix} a - λ & 1 \\ - 1 & a - λ \end{matrix}] = λ^{2} - 2 a λ + (a^{2} + 1) = (λ - (a - i)) \times (λ - (a + i))$ =0

Therefore $λ_{1} = (a - i)$ and $λ_{2} = (a + i)$

Step 2

For every λ we find its own vector

$λ_{1} = a - ⅈ$

$A - λ_{1} I = [\begin{matrix} i & 1 \\ - 1 & i \end{matrix}]$

So we have a homogeneous system of linear equations, we solve it by Gaussian Elimination:

$[\begin{matrix} i & 1 \\ - 1 & i \end{matrix}] P e r f o r m i n g R_{1} = - i \times R_{2} w e g e t [\begin{matrix} 1 & - i \\ - 1 & i \end{matrix}] (i \times i = - 1) P e r f o r m i n g R_{2} = R_{1} + R_{2}, w e g e t [\begin{matrix} 1 & - i \\ 0 & 0 \end{matrix}] t h e r e f o r e, x_{1} - i x_{2} = 0 x_{1} = i x_{2} x_{2} = x_{2} G e n e r a l s o l u t i o n i s X = (\frac{i . x_{2}}{x_{2}}) T h e s o l u t i o n s e t : \{x_{2} (\frac{i}{1})\} L e t x_{2} = 1, v_{1} = (\frac{i}{1})$

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