I Find the only periodic solution of an ODE

[michii15]

TL;DR Summary

Find the only periodic solution for 𝑦′+𝑦=𝑏(𝑥) with 𝑏:ℝ→ℝ has a period of 2𝑇 and is 1 for 𝑥(0,𝑇) and −1 for 𝑥(−𝑇,0).

Find the only periodic solution for 𝑦′+𝑦=𝑏(𝑥) with 𝑏:ℝ→ℝ has a period of 2𝑇 and is 1 for 𝑥(0,𝑇) and −1 for 𝑥(−𝑇,0).

The ODE is easy to solve: 𝑦(𝑥)=exp(−𝑥)⋅𝑐+1 and 𝑦(𝑥)=exp(−𝑥)⋅𝑐−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.

 

[michii15]

sorry forgot to mention the function should be continuous.

 

[michii15]

but for c = 0 you won't have a continuous solution

 

[michii15]

it jumps frome -1 to 1?

 

[member 587159]

Sorry I misread your question. You are right. Deleting all my previous comments...

 

[mfb]

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?