Undergrad Find the only periodic solution of an ODE

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SUMMARY

The discussion focuses on finding the only periodic solution for the ordinary differential equation (ODE) defined as 𝑦′ + 𝑦 = 𝑏(π‘₯), where 𝑏 is a periodic function with a period of 2𝑇. The general solution is given by 𝑦(π‘₯) = exp(βˆ’π‘₯)⋅𝑐 + 1 or 𝑦(π‘₯) = exp(βˆ’π‘₯)⋅𝑐 βˆ’ 1. To ensure periodicity, the constant 𝑐 must be determined, taking into account the requirement for continuity in the solution. The discussion highlights the importance of continuity in defining the prefactors for the solution.

PREREQUISITES
  • Understanding of ordinary differential equations (ODEs)
  • Knowledge of periodic functions and their properties
  • Familiarity with continuity in mathematical functions
  • Basic skills in solving differential equations analytically
NEXT STEPS
  • Research methods for solving periodic boundary value problems in ODEs
  • Explore the role of continuity in determining solutions of differential equations
  • Study the implications of parameter selection in ODE solutions
  • Learn about the stability of solutions in the context of periodic functions
USEFUL FOR

Mathematicians, students studying differential equations, and researchers interested in periodic solutions of ODEs will benefit from this discussion.

michii15
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TL;DR
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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sorry forgot to mention the function should be continuous.
 
but for c = 0 you won't have a continuous solution
 
it jumps frome -1 to 1?
 
Sorry I misread your question. You are right. Deleting all my previous comments...
 
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?
 

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