Initial conditions for stability

In summary, the general solution for the given equations is x=4C1e^2t, y=C1e^2t-3C2e^-2t. To find a set of initial conditions (x0, y0) for which the solution is stable, the constants C1 and C2 need to be such that x and y do not go to infinity as t gets large. This can be solved by setting up the solution and finding the values of x(0) and y(0). The only unstable part of the general solution is e^-2t.
  • #1
hbomb
58
0
The general solution fo the following equations:

x'=2x-3y
y'=x-2y

Is, x=4C1e^2t, y=C1e^2t-3C2e^-2t

They ask for me to list a set of initial conditions (xo, yo) for which the solution is stable, i.e, (x, y)-->(0,0) for large t.

I don't understand this part of the problem.
 
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  • #2
Basically, for what constants C1 and C2 do x and y go to infinity as t gets large? From that, solve for what x(0) and y(0) can be
 
  • #3
Ok, I understand that for the general solution, the only unstable part of it is e^-2t. What does the setup of the solution look like?
 

1. What are initial conditions for stability?

Initial conditions for stability refer to the starting state of a system that determines whether it will remain in a stable state or if it will deviate from that state over time. These conditions can include factors such as the system's inputs, parameters, and external disturbances.

2. Why are initial conditions important for stability?

Initial conditions are important for stability because they can greatly influence the behavior of a system. A small change in the initial conditions can lead to a significantly different outcome, either maintaining stability or causing instability. Therefore, understanding and controlling initial conditions is crucial for ensuring the stability of a system.

3. How can initial conditions be controlled for stability?

There are various ways to control initial conditions for stability, depending on the type of system. One approach is to carefully design and adjust the system's parameters and inputs to ensure a stable starting state. Another approach is to use feedback control mechanisms to continuously monitor and adjust the system's behavior based on its initial conditions and external disturbances.

4. What happens if initial conditions are not stable?

If initial conditions are not stable, the system may experience undesired behavior, such as oscillations, divergent behavior, or failure. This can have serious consequences, especially in critical systems such as airplanes or nuclear reactors. Therefore, it is essential to carefully consider and control initial conditions to prevent instability.

5. Can initial conditions change over time?

Yes, initial conditions can change over time due to various factors, such as changes in the system's parameters or external disturbances. This can lead to a change in the system's stability, which may require adjusting the initial conditions or implementing feedback control to maintain stability.

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