Why not separable (basic question)?

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In summary, the conversation is discussing Example 1.5 in Differential Equations Demystified, where the equation y' +2xy = x is given and the author claims it is not separable. The individual is trying to solve the equation but is unsure which approach is incorrect. They ask for help and express their need for assistance in understanding ODE. Another person questions the author's claim and the individual proves that the equation can be separated, leading the author to admit their mistake.
  • #1
jla2w
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In Example 1.5 of Differential Equations Demystified, the equation is y' +2xy = x, and the author claims this is not separable. Now, what am I missing. I try the following.

1) dy/dx = x - 2xy
2) dy/dx = x(1-2y)
3) dy/(1-2y) = xdx

I guess one of those is invalid but I just cannot identify which. Any help appreciated, thanks. This is part of an effort to brush up on ODE since it's been years since I took it and now I'm going for a Master's in Applied Math, need lots of help of this kind. Thanks
 
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  • #2
Hmm, are you sure that's the equation they gave? I don't see what's wrong with that.
 
  • #3
Yes, that's definitely the equation, and the author says
"This equation is plainly not separable (try and convince yourself that this is so)."
 
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  • #4
Maybe it isn't separable but I don't see how it's "plainly" not separable, you seem to agree
You just separated it, disproving the author's claim.
 
  • #5
Author wrote me back saying that it was a mistake, so that is in fact a separable equation.
 

Why not separable (basic question)?

Separability refers to the ability of a system to be broken down into individual components or parts. In science, this concept is often applied to physical, chemical, or biological systems. Here are the five most frequently asked questions about why a system may not be separable:

1. What factors determine if a system is separable or not?

The complexity of a system and the strength of the interactions between its components are the main factors that determine if a system is separable. A system with many interconnected components that have strong interactions will be less separable than a system with fewer and weaker interactions.

2. Can a non-separable system be simplified?

Yes, a non-separable system can be simplified by approximating it as separable under certain conditions. This is often done in mathematical models to make the system easier to analyze and understand. However, the simplification may not accurately represent the true behavior of the system.

3. Why is it important to know if a system is separable or not?

Understanding the separability of a system can help scientists predict its behavior and make accurate predictions about its future state. It can also provide insights into the underlying mechanisms and interactions within the system, which can be useful for designing experiments or interventions.

4. Are there any real-life examples of non-separable systems?

Yes, many real-life systems are non-separable, such as ecosystems, weather patterns, and human societies. These systems are complex and have many interconnected components with strong interactions that make them difficult to separate into individual parts.

5. Can a non-separable system ever become separable?

In some cases, a non-separable system may become separable under certain conditions. For example, if the interactions between components weaken or if new methods for separating the system are developed. However, in many cases, the non-separable nature of a system is inherent and cannot be changed.

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