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 yourworking when deriving the characteristic equation.Is it possible to solve for Bn analytically? If yes, solve it. If no, suggest anothermethod to use to solve for Bn.ii) |A – AI| = 0cosh(3,)sinh(8,) – sin(8,)=(cosh(B,) – 1)² – [sinh°(3„) – sin²(B,)] = 0> cosh (Bn) + X² – 21 cosh(3n) – sinh?(8n) + sin²(Bn) = 0Using, cosh (3,) – sinh?(Bn) = 1=1+X? – 2A cosh(B,)+ sin²(3n) = 0.sinh(B,) + sin(8,)|cosh(B„)= 0This is the characteristic equation. As the eigenvalue of the matrix A always satisfiesits characteristic equation, we must have1+ cos (Bn) + 2 cos(3n) cosh(B,) + sin (3m) = 0=2+2 cos(Bm) cosh(B,) = 0=1+ cos(Bn) cosh(Bn) = 0.(1)

Given: i) To evaluate the characteristic equation of matrix [A]. ii) To find that it is possible to…

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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Question

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)

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