Solving a specific differential equation

Question

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.

Answer 1

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 $\lambda=1/e$.

Answer 2

This is not an answer but a suggestion. There is a lot of "noise" in your equation. You can simplify it a bit. Set $$ A:= b+c_B(\bar{\beta}^H-\bar{\beta}^L),\;\;c:=c_B. $$ Then your equation reads $$ AF'(x)+cF(x)-cF(x+A/c)-c=0, $$ or $$ BF'(x)+F(x)-F(x+B)+1=0,\;\;B=A/c. $$ This is a linear autonomous delay equation. As some commented, try Fourier transform. Alternatively, have a look at Chapter VII in the book Ordinary and Delay Differential Equations, by R.D. Driver.