Prove that $y'=-y^3+\sin x $ has a unique solution satisfying $y(0)=y(2\pi)$

Asked Sep 08 '20 at 07:55

Active Sep 08 '20 at 07:55

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Prove that the ordinary differential equation $$y'=-y^3+\sin x $$ has a unique solution satisfying $y(0)=y(2\pi)$

It's really hard for me to figure out the existence and the uniqueness.

I don't have any clue about this problem. Maybe we can let $u(x)=y(x+2\pi)-y(x)$, then $u(x)$ should satisfy $u(0)=0$ and some differential equations.

asked Sep 08 '20 at 07:55

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This is, up to a change in the time direction, a duplicate of Solution to ODE $ x' = x^3+ \sin( t)$ and showing it has a periodic solution. After the change in the time direction, my answer there directly treats this problem. – Lutz Lehmann Sep 08 '20 at 09:02
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A much earlier question for this ODE asking Is this periodic solution unique? (ODE) – Lutz Lehmann Sep 08 '20 at 09:13

Prove that $y'=-y^3+\sin x $ has a unique solution satisfying $y(0)=y(2\pi)$

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