Find the only periodic solution of an ODE

In summary, there is an ODE with a periodic solution that has a period of 2T. The solution is 𝑦(π‘₯)=exp(βˆ’π‘₯)⋅𝑐+1 and 𝑦(π‘₯)=exp(βˆ’π‘₯)β‹…π‘βˆ’1, but the value of 𝑐 must be determined to make the solution continuous. This can be done by considering the given values for π‘₯(0,𝑇) and π‘₯(βˆ’π‘‡,0) and using continuity to determine the prefactors.
  • #1
michii15
4
0
TL;DR Summary
Find the only periodic solution for 𝑦′+𝑦=𝑏(π‘₯) with 𝑏:ℝ→ℝ has a period of 2𝑇 and is 1 for π‘₯(0,𝑇) and βˆ’1 for π‘₯(βˆ’π‘‡,0).
Find the only periodic solution for 𝑦′+𝑦=𝑏(π‘₯) with 𝑏:ℝ→ℝ has a period of 2𝑇 and is 1 for π‘₯(0,𝑇) and βˆ’1 for π‘₯(βˆ’π‘‡,0).

The ODE is easy to solve: 𝑦(π‘₯)=exp(βˆ’π‘₯)⋅𝑐+1 and 𝑦(π‘₯)=exp(βˆ’π‘₯)β‹…π‘βˆ’1. But how can I find the 𝑐 such that the solution is periodic with a period of 2𝑇?

The solution is: https://i.stack.imgur.com/S21Ze.png

Thanks for your help.
 
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  • #2
sorry forgot to mention the function should be continuous.
 
  • #3
but for c = 0 you won't have a continuous solution
 
  • #4
it jumps frome -1 to 1?
 
  • #5
Sorry I misread your question. You are right. Deleting all my previous comments...
 
  • #6
Well, you have two values given, that gives you ranges to play with, continuity will determine the prefactors.

I'm not sure if I understand your notation, however. Why does x depend on two parameters?
 

1. What is a periodic solution in an ODE?

A periodic solution in an ODE (ordinary differential equation) is a solution that repeats itself after a certain amount of time or distance. This means that the solution will have a recurring pattern or behavior.

2. How do you find the only periodic solution of an ODE?

To find the only periodic solution of an ODE, you can use the method of separation of variables or the method of undetermined coefficients. These methods involve manipulating the equation to isolate the dependent variable and then solving for its periodic behavior.

3. Can an ODE have more than one periodic solution?

Yes, an ODE can have multiple periodic solutions. This can occur when the equation has different initial conditions or when the equation has multiple solutions that satisfy the periodic behavior.

4. What are the applications of finding the only periodic solution of an ODE?

The applications of finding the only periodic solution of an ODE are numerous. This can be used to model and predict the behavior of physical systems such as oscillating springs, electrical circuits, and weather patterns. It can also be used in fields such as finance, biology, and chemistry.

5. Are there any limitations to finding the only periodic solution of an ODE?

Yes, there are limitations to finding the only periodic solution of an ODE. Some equations may not have a closed-form solution, meaning that an exact solution cannot be found. In these cases, numerical methods or approximations may be used to find an approximate periodic solution.

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