The coefficient matrix for a system of linear differential equations of the form y_1=Ay has the given eigenvalues and eigenspace bases. Find the gener

cistG 2020-11-02 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=3+i{[2ii]},λ2=3i{[2ii]}
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Expert Answer

SoosteethicU
Answered 2020-11-03 Author has 102 answers

The general solution is
y=c1y1+c2y2
for y1=eat(cos(bt)Re(u)sin(bt)Im(u))
y2=eat(sin(bt)Re(u)+cos(bt)Im(u))
We have
λ1=3+i{[2ii]},λ2=3i{[2ii]}
Then
y1=e3t(cos(t)[00]sin(t)[21])
y2=et(sin(t)[00]+cos(t)[21])
Hence the general solution is
y=c1y1+c2y2
=c1e3t(cos(t)[00]sin(t)[21])+c2et(sin(t)[00]+cos(t)[21])
The individual functions are
y1=2c1e3t(sin(t)+cos(t))
y2=c2e3t(sin(t)+cos(t))

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