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. 



Solving a specific differential equation
To continue Liviu Nicolaescu's simplification: put $F(x):=f(x/B)$ so the equation writes
$$f'(x)+f(x)-f(x+1)+1=0.$$
A particular solution of it is simply $f(x):=x^2$, so we are left with the homogeneous equation 
$$u'(x)+u(x)-u(x+1)=0.$$
If we put $u(x)=v(x)e^{-x}$ this becomes the (well-known)
$$v'(x)=\lambda v(x+1)$$
with $\lambda=1/e$.


