Some Background:
I come from a computer science background and differential equations was not part of my course work. I'm still in contact with my professor that I got my Masters under, and he has both a PhD in math and CS. In the past, I've been able to use Mathematica to help me find answers to problems outside of my skill set. Based on this, I thought I might be able to help my professor with a problem he is working on, but I have not had much luck.
I used to own a copy of Mathematica, which I thought might be helpful in solving the following problem. But my license expired many moons ago, I also tried to find an online tool, such as http://www.wolframalpha.com/examples/DifferentialEquations.html but have not had luck finding something I could use to solve the problem. Failing to find the answer on my own, I've come here with the info to see if someone can help. My intent of this question is to try to find the answer to the problem, however if there is any open source program that could help solve problems similar to this problem, the information would be greatly appreciated as well.
The Original Presented Problem:
In the limiting case, the differential-difference equation then becomes
(2t + h(1-t)) * (2-h'(t)) = 4t
for 0 <= t <-1.
We can substitute variables to make the equation look more symmetric:
(x+1+f(-x))*(1-f'(x))=x+1
from -1 <= x <= 1.
The set of solutions, in our range of interest, is a 1-parameter family:
f(x) = (K+x)^((K-1)/2K)*(K-x)^((K+1)/2K)+x-1
where K > 1 is the parameter.
Follow Up Info Provided to me
The problem is to find the solution to
(x + 1 + f (−x)) · (1 − f ′ (x)) = x + 1
in the range -1 <= x <= 1. If it can find a closed form solution I will be very surprised.
The solution is actually a 1-parameter family, so you can't get a plot until you give a boundary condition. The optimal solution is where f(-1) is approximately -0.198, and there should be exactly two solutions with that boundary condition, and they should be very close. (If you were able to set f(-1) to the minimum possible value, which is sightly less, I would expect one solution, and there should be no solution for f(-1) = -0.2
If you are able to get a plot, I will also be surprised, since I don't know whether Mathematica can handle differential equations with reflection. On second thought, there could be extraneous solutions, whose plots should look very different.
You can transform a first order differential equation with reflection into an second order ordinary differential equation by differentiating again and substituting the occurences of $f(-x)$ and $f'(-x)$. If you have an initial condition you can get an extra one by the same process. Check https://arxiv.org/pdf/1707.01036.pdf for instance.