# Periodic solutions of non autonomous differential equation $\dot{x}(t)=x(t)(1+\cos(t))-x(t)^3$

Find all $2\pi$ periodic solutions (either constant or non-constant) of the nonautonomous equation

$\dot{x}=x(1+\cos(t))-x^3$.

I know that the only equilibria is $x=0$ which is a source.

• But $x=\pm1$ are not equilibria, are they? – Did Nov 20 '16 at 19:24
• You're right, then is $x=0$ the only periodic orbit? – BronchoX Nov 20 '16 at 19:28
• Why should they? – Did Nov 20 '16 at 19:36
• I don't understand your question – BronchoX Nov 20 '16 at 19:40
• @Did Btw, me neither understand your approach. Since the question has already been answered, could you shed more light on your idea? – Evgeny Nov 25 '16 at 13:17

## 1 Answer

You can solve this as Bernoulli equation, set $u=r^{-2}$ then $$u'=-2r^{-3}r'=-2(1+\cos t)u+2$$ which now can be nicely solved as a first order linear ODE.

$A(t)=e^{2(t+\sin t)}$, then $(A(t)u(t))'=2A(t)$, $$A(2\pi)u(2\pi)=u(0)+\int_0^{2\pi}A(s)ds$$ So for $u(2\pi)=u(0)$ you need $$u(0)=\frac{\int_0^{2\pi}e^{2(s+\sin s)}ds}{e^{4\pi}-1}=\frac12e^{2\sin(\theta·2\pi)},\quad\theta\in(0,1)$$ which gives exactly one positive radius.

• Great answer, its so clear! – BronchoX Feb 7 '18 at 3:14