i) Derive the characteristic equation of matrix [A] in equation (2). Show al working when deriving the characteristic equation. ii) Is it possible to solve for Bn analytically? If yes, solve it. If no, suggest ar method to use to solve for Bn.
i) Derive the characteristic equation of matrix [A] in equation (2). Show al working when deriving the characteristic equation. ii) Is it possible to solve for Bn analytically? If yes, solve it. If no, suggest ar method to use to solve for Bn.
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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![1. Given that the eigenvalue for matrix [A] in equation (2) is A= -cos(ßn).
i)
Derive the characteristic equation of matrix [A] in equation (2). Show all your
working when deriving the characteristic equation.
Is it possible to solve for Bn analytically? If yes, solve it. If no, suggest another
method to use to solve for Bn.
ii)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57f33c8d-a162-41ea-8ce6-4d8ea1771ff0%2F43bc0c4c-ec01-45d2-9865-03392c7759f6%2Fn30xird_processed.png&w=3840&q=75)
Transcribed Image Text:1. Given that the eigenvalue for matrix [A] in equation (2) is A= -cos(ßn).
i)
Derive the characteristic equation of matrix [A] in equation (2). Show all your
working when deriving the characteristic equation.
Is it possible to solve for Bn analytically? If yes, solve it. If no, suggest another
method to use to solve for Bn.
ii)
![|A – AI| = 0
cosh(3,)
sinh(8,) – sin(8,)
=(cosh(B,) – 1)² – [sinh°(3„) – sin²(B,)] = 0
> cosh (Bn) + X² – 21 cosh(3n) – sinh?(8n) + sin²(Bn) = 0
Using, cosh (3,) – sinh?(Bn) = 1
=1+X? – 2A cosh(B,)+ sin²(3n) = 0.
sinh(B,) + sin(8,)|
cosh(B„)
= 0
This is the characteristic equation. As the eigenvalue of the matrix A always satisfies
its characteristic equation, we must have
1+ cos (Bn) + 2 cos(3n) cosh(B,) + sin (3m) = 0
=2+2 cos(Bm) cosh(B,) = 0
=1+ cos(Bn) cosh(Bn) = 0.
(1)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57f33c8d-a162-41ea-8ce6-4d8ea1771ff0%2F43bc0c4c-ec01-45d2-9865-03392c7759f6%2F7rh9bla_processed.png&w=3840&q=75)
Transcribed Image Text:|A – AI| = 0
cosh(3,)
sinh(8,) – sin(8,)
=(cosh(B,) – 1)² – [sinh°(3„) – sin²(B,)] = 0
> cosh (Bn) + X² – 21 cosh(3n) – sinh?(8n) + sin²(Bn) = 0
Using, cosh (3,) – sinh?(Bn) = 1
=1+X? – 2A cosh(B,)+ sin²(3n) = 0.
sinh(B,) + sin(8,)|
cosh(B„)
= 0
This is the characteristic equation. As the eigenvalue of the matrix A always satisfies
its characteristic equation, we must have
1+ cos (Bn) + 2 cos(3n) cosh(B,) + sin (3m) = 0
=2+2 cos(Bm) cosh(B,) = 0
=1+ cos(Bn) cosh(Bn) = 0.
(1)
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