prove that there is only one a, such: $y'(x)=y(x)^3+\sin(x);y(0)=a$ and y(x) is a unique periodic solution

Question

prove that there is only one real a, such: $y'(x)=y(x)^3+\sin(x);y(0)=a$ which satisfies that y is a unique periodic solution. Find whether a is greater than 0 or smaller than 0. I believe that this equation is not solvable. I tried to get a contradiction of the existence of 2 solutions depending on a, but I found no such relationship as that.