The coefficient matrix for a system of linear differential equations of the form y^{1} = A_{y} has the given eigenvalues

tabita57i 2021-02-21 Answered

The coefficient matrix for a system of linear differential equations of the form y1=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system.
[λ1=1{[103]},λ2=3i{[2i1+i7i]},λ3=3i{[2+i1i7i]}]

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Expert Answer

AGRFTr
Answered 2021-02-22 Author has 95 answers

Step 1
The general solution is
y=c1y1+c2y2+c3y3
for y1=eλ1tu
y2=eat(sin(bt)Re(u)+cos(bt)Im(u))
y3=eat(cos(bt)Re(u)sin(bt)Im(u))
Step 2
We have
[λ1=1{[103]},λ2=3i{[2i1+i7i]},λ3=3i{[2+i1i7i]}]
Step 3
Then
y1=et([103])
y2=e0t((sin(3t)[212]+cos(3t)[111])
=((sin(3t)[212]+(cos(3t)[111])
y3=e0t(cos(3t)[210]sin(3t)[117])
=(cos(3t)[210]sin(3t)[117])
Step 4
Hence the general solution is
y=c1y1+c2y2+c3y3
=c1et([103])+c2((sin(3t)[212]+(cos(3t)[111])+c3(cos(3t)[210]sin(3t)[117])
Step 5
The individual functions are
y1=c1et(c2(2sin(3t)cos(3t))+c3(2cos(3t)sin(3t)))

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