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In summary: So, I assumed the initial condition to be y(0)=x(0)+1 and solved for x. I got x=-1.5 so the system is stable.

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Lancen said:

Did you try looking at the eigenvalues of the linearized system at the critical points?

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Did you notice it's separable? (Oh, and you really should be using parentheses where appropriate. 1+2/2+3 is 5, not 3/5)I can't find a integrating factor for this in terms or either x or y

But as LeBrad suggests, you should be able to figure out the answer without solving the equation. I suggest you do that first, and

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Recheck your equilibrium points! x= 1, y= -1/2 do NOT make dx/dt and dy/dt 0.

As Hurkyl and LeBrad suggested, you don't need to solve the system to determine if it is stable. In fact, that's the whole point! Most non-linear systems can't be solved that easily. The stability can be determined by looking at the "linearized" system at each equilibrium point. (That's what LeBrad was talking about: non-linear systems don't have eigen-values.)

As Hurkyl and LeBrad suggested, you don't need to solve the system to determine if it is stable. In fact, that's the whole point! Most non-linear systems can't be solved that easily. The stability can be determined by looking at the "linearized" system at each equilibrium point. (That's what LeBrad was talking about: non-linear systems don't have eigen-values.)

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Autonomous systems are systems that can operate independently without external control or influence. They are often used in fields such as robotics, artificial intelligence, and control theory.

Autonomous systems maintain stability through various control mechanisms, such as feedback control, predictive control, and adaptive control. These control methods allow the system to continuously monitor and adjust its behavior to achieve a desired outcome and avoid instability.

There are several benefits of using autonomous systems, including increased efficiency, improved accuracy, and reduced human error. Autonomous systems also have the potential to operate in hazardous or remote environments, reducing the risk to human operators.

One of the main challenges for autonomous systems in maintaining stability is uncertainty. This can be caused by external factors such as changing environments or unexpected events. Additionally, the complexity of autonomous systems can make it difficult to predict and control their behavior.

The stability of autonomous systems can be tested and evaluated through simulation, where different scenarios and inputs can be applied to assess the performance of the system. Additionally, real-world testing and validation can also be used to evaluate the stability of autonomous systems in practical applications.

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