Consider the system dot(x)=x-y , dot(y)=x+y a) Write the system in matrix form and find the eigenvalues and eigenvectors

foass77W 2021-09-10 Answered
Consider the system x˙=xy,y˙=x+y
a) Write the system in matrix form and find the eigenvalues and eigenvectors (Note: they will be complex valued)
b) Write down the general solution for the system of differential equations using only real valued functions. [Hint : use the equality eiωt=cos(ωt)+isin(ωt) ]
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Expert Answer

hosentak
Answered 2021-09-11 Author has 100 answers

Step 1
Given:
dxdt=xy
dydt=x+y
To find: (a) = Write system in matrix form , find eigenvalues and eigenvectors.
(b). = General solution.
Step 2.
Solution: a)
dxdt=xy
dydt=x+y
x(t)=Ax(t) , where A is matrix
[xy]=[1111][xy]
Required matrix form of the system.
Eigenvalues of matrix A :
|AxI|=0
|1x111x|=0
(1x)2+1=0
x22x+2=0
x=2±482
=x=1±i
Hence the eigenvalues are in complex from which are 1+i , 1-i
Eigenvector corresponding to 1+i :
[A(1+i)I]v1=0
[11i1111i][xy]=0
[i11i][xy]=0
R2iR2+R1
[i100][xy]=0
  y is free variable , let  y=1
ix1=0
x=1i=i
Kv1=[i1]
Eigenvector corresponding to 1-i:
[A(1i)I]v2=0
[11+i1111+i][xy]=0
[i11i][xy]=0
R2iR2R1
[i100][xy]=0

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