ODE: limit as parameters tend to 0

In summary, if you set a parameter to a specific eigenvalue of an ODE, you can get a solution with a qualitative change.
  • #1
Vey2000
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0
Maybe there's something I'm missing, but I just realized that if we take a certain general ODE such as y''+k^2 y=0 for example, and assume that in a specific case k=0 the solution is vastly different depending on whether k is set to 0 before or after solving the ODE.

The general solution to y''+k^2 y = 0 (with k>0) is y=A cos(kx)+B sin(kx).
If we then set k=0 the solution reduces to y=A.
Yet, if we set k=0 in the original ODE we get [1] y''=0, whose solution is [2] y=Bx+C.
Where A, B, and C are constants.

I get analogous results for y''-k^2 y = 0.

On the other hand, if I start with y''+2b y'+k^2 y = 0, then it seems that setting either b or k to 0, but not both, can be done either in the ODE itself or in the general solution.

Hrm, I just had another insight while writing this. The form of solution [2] is due to the characteristic equation having a repeated real root of r=0. Is this the reason why my initial attempt failed? Because I was setting the parameter to the same value as the eigenvalue of the ODE?
 
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  • #2
Basically, that's correct. In your first two examples, as k goes to 0, the characteristic equation goes from having two distinct solutions to one solution. For your third example, the characteristic equation is r2+ 2br+ k2= 0 which, in general has two distinct solutions. Taking either parameter 0 gives r2+ k2= 0 which still has 2 (imaginary) solutions or r2+ 2br= 0 which also still has two solutions (one r= 0).

In technical terms, for your first two examples, k= 0 is a "bifurcation point".

Generally, solutions will have distinct qualitative changes at a bifurcation point.
 
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  • #3
Thank you, now I have a starting point to look more into this. :smile:
 

What is an ODE?

An ODE, or ordinary differential equation, is a mathematical equation that describes how a variable changes over time based on its own value and the values of other variables.

What does it mean for parameters to tend to 0 in an ODE?

When parameters in an ODE tend to 0, it means that the values of those parameters are approaching 0 as the equation is solved. This can have a significant impact on the behavior and solutions of the equation.

Why is the limit as parameters tend to 0 important in ODEs?

The limit as parameters tend to 0 is important because it can help us understand the overall behavior and solutions of an ODE. It allows us to see how the equation behaves when certain variables are approaching 0, which can be a critical factor in finding solutions or making predictions.

How do scientists use the limit as parameters tend to 0 in their research?

Scientists may use the limit as parameters tend to 0 in various ways, depending on their specific research. Some may use it to analyze and understand the behavior of a system, while others may use it to optimize solutions or predict the outcomes of experiments.

Are there any real-world applications of ODEs with limits as parameters tend to 0?

Yes, there are many real-world applications of ODEs with limits as parameters tend to 0. For example, they can be used to model and understand physical systems such as weather patterns, population growth, and chemical reactions. They are also used in engineering, economics, and other fields to make predictions and optimize solutions.

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