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

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In summary, the conversation discusses finding all bounded solutions in the interval [a, infinity) for a given ode system. The approach suggested is to find the eigenvalues and eigenvectors of the matrix and use them to obtain a fundamental set of solutions for the homogenous system. However, the conversation also mentions that there may be an easier way to find the bounded solutions. In the end, the person solves the problem on their own.
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TheForumLord
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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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  • #2
NVM...I've managed to solve it on my own...
TNX anyway
 

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

What is a bounded solution of an ODE?

A bounded solution of an ODE (ordinary differential equation) is a solution that remains within a finite range or limit as the independent variable (usually time) approaches infinity. In other words, the solution does not grow unbounded or approach infinity as time goes on.

How can you determine if a solution is bounded?

To determine if a solution is bounded, you can look at the behavior of the solution as the independent variable approaches infinity. If the solution remains within a finite range or approaches a constant value, then it is considered bounded. Additionally, you can also use techniques such as the comparison test or the Cauchy-Schwarz inequality to prove the boundedness of a solution.

What are the implications of a bounded solution?

A bounded solution has important implications in the stability of a system. In a physical system, a bounded solution means that the system will not grow out of control or exhibit chaotic behavior. In mathematical models, a bounded solution allows us to make predictions and analyze the behavior of the system over time.

Can all ODEs have bounded solutions?

No, not all ODEs have bounded solutions. Some ODEs may have solutions that grow unbounded or exhibit chaotic behavior over time. However, many physical systems and mathematical models are designed to have bounded solutions, as this ensures the stability and predictability of the system.

How is a bounded solution different from a stable solution?

A bounded solution is a type of stable solution, but not all stable solutions are necessarily bounded. A stable solution is one that remains close to its initial conditions over time. A bounded solution, on the other hand, has the additional requirement of remaining within a finite range as the independent variable approaches infinity.

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