I have been trapped in solving the following ODE for a long time. I wonder if it has unique analytical solution \begin{equation} [b+c_B(\bar{\beta}^H-\bar{\beta}^L)]\frac{dF(x)}{dx}+c_BF(x)-c_BF(x+\bar{\beta}^H-\bar{\beta}^L-b/c_B)-c_B=0. \end{equation}

I could try to assume $F(x)$ is linear. But I was wondering if there is a way to prove or disprove the uniqueness of the solution of this ODE? Is there a systematic way to find all the solutions to this equation? Thanks for the comments. Now I know that this is a delay differential equation. In my problem, I have $x\geq b/c_B+\bar{\beta}^L-\bar{\beta}^H$. And in fact, $F(x)$ is a cumulative distribution function in my case.