Please show the work so I learn Determine the eigenvalues of the system: y1\'=8y

Question

   Please show the work so I learn  

Determine the eigenvalues of the system:
                 y1'=8y1-y2
                 y'2=y1+10y2
                 What is the stability of the critical point?

a.) lambda(base)1=2+i2,lambda= 2-i2, Stable
                 b.) lambda (base)1,lambda(base)2=9, Unstable
                 c.) lambda(base)1=-9+i2,lambda=-9-i2, Stable
                 d.) lambda(base)1=2+i2, lambda=2-i2, Stable and attractive
                 e.) None of the above; see Problem Work

Determine the eigenvalues of the system: y1'=8y1-y2 y'2=y1+10y2 What is the stability of the critical point? lambda(base)1=2+i2,lambda= 2-i2, Stable lambda (base)1,lambda(base)2=9, Unstable lambda(base)1=-9+i2,lambda=-9-i2, Stable lambda(base)1=2+i2, lambda=2-i2, Stable and attractive None of the above; see Problem Work

Explanation / Answer

Determine the eigenvalues of the system:
                 y1'=8y1-y2
                 y'2=y1+10y2
                 What is the stability of the critical point?        
writing in matrix form
y1' = 8 -1 y1
y2'    1   10 y2
finding eigen values of matrix as   determinent of |A-XI| = 0
8-x -1
1 10-x = 0
(8-x)*(10-x) +1 =0
80-10x-8x+x^2+1 = 0
x^2-18x+81 = 0
(x-9)^2= 0
x = 9,9

since roots are outside system is unstable.
b.) lambda (base)1,lambda(base)2=9, Unstable

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