What is the sufficient condition for bounded solutions in this ODE system?

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SUMMARY

The discussion centers on finding sufficient conditions for bounded solutions in the given ordinary differential equation (ODE) system defined by x' = 2x + y - 7e^(-t) - 3 and y' = -x + 2y - 1. The eigenvalues of the associated matrix are 2 ± i, leading to a fundamental set of solutions for the homogeneous system. The user initially sought a simpler method to determine bounded solutions but ultimately resolved the problem independently.

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Homework Statement


Given this ode system:
x' = 2x+y-7e^(-t) -3
y'= -x+2y-1

Find all the bounded soloution in [a,infinity) when a is a real number...

I'm not really sure what is a sufficient condition for bounded soloution in this question...Maybe there's something we can do and then we will not even need to solve the system...


Help is Needed!

TNX a lot!

Homework Equations


The Attempt at a Solution


The eignvalues of the Matrix are: 2+-i...The eignvectors are: (1,i ) for 2+i & (1, -i ) for 2-i...
According to this we know that this is a fundamental set of soloutions for the homogenic system:
x1=e^2t[cost(1,0) -sint(0,1) ]
x2=e^2t[sint(1,0) +cost(0,1) ]
From here we can get to a private soloution of the whole system in several ways but they're all take very long time... I'm pretty sure there's an easier way to get to the bounded soloutions of the system...

HELP IS NEEDED ASAP!

TNX
 
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NVM...I've managed to solve it on my own...
TNX anyway
 

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