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

Question

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.

Answer 1

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.