Final condition instead of initial condition

In summary, the conversation discusses a second order differential equation with initial and final conditions, and the conditions under which the solution for all initial and final conditions exists. The existence of the solution depends on the nature of the equation and is not always unique. Examples of such equations are requested.
  • #1
Petr Mugver
279
0
Let's consider a second order differential equation

[tex]f(x,\dot x,\ddot x,t)=0[/tex]

and let's suppose that f satisfies all the conditions of the Cauchy Theorem, i.e. f is such that the equation above with the initial conditions

[tex]x(t_0)=x_0\qquad\dot x(t_0)=v_0[/tex]

has an unique solution in a certain neighbourhood of t_0, for every t_0.

My question is, if instead of the two initial conditions above I have an initial and a final condition

[tex]x(t_0)=x_0\qquad x(t_1)=x_1[/tex]

under what further conditions on f the solution exists for all x_0 and x_1?
 
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  • #2
The existence depends crucially on the nature of the equation. The solution is, in general, not unique.
 
  • #3
Eynstone said:
The existence depends crucially on the nature of the equation. The solution is, in general, not unique.

Can you give me some examples? (of a f that satisfies the conditions of my first post but whose solution is not unique for some choice of initial and final conditions)
 

1. What is the concept of "final condition instead of initial condition"?

"Final condition instead of initial condition" is a scientific concept that describes a shift in perspective from focusing on the starting point of a system to focusing on the end result. This concept suggests that the final state of a system can be used to understand its behavior, rather than solely relying on the initial conditions.

2. Why is "final condition instead of initial condition" important in science?

This concept is important because it allows scientists to study complex systems with varying initial conditions and still make predictions about their final outcome. It also provides a more holistic view of how a system behaves and can lead to a deeper understanding of its underlying mechanisms.

3. How does "final condition instead of initial condition" apply to real-world situations?

This concept can be applied to a wide range of real-world situations, such as weather forecasting, economics, and ecology. By looking at the final outcome of a system, scientists can better predict future events and make informed decisions.

4. Are there any limitations to using "final condition instead of initial condition" in scientific research?

While this concept can be useful, it is not applicable in all situations. Some systems may be highly sensitive to initial conditions, making it difficult to rely solely on the final outcome. Additionally, it may be challenging to accurately determine the final state of a system in some cases.

5. How can the concept of "final condition instead of initial condition" be used in experimental design?

In experimental design, scientists can use this concept to choose the most relevant and informative final outcome to measure and analyze. This can help to streamline experiments and make data interpretation more straightforward. Additionally, it can provide a more comprehensive understanding of the system being studied.

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