Now sub ,
Now we have removed the coupling between the two equations. But I think solving this is still hard...
Then I will follow what Mr.F told you below:
So in our D.E:
As Mr.F pointed out, to eliminate C we need boundary conditions.
So continuing the integration:
Choosing the positive solution(its actually for one of the particles. For the other, it is negative, since they are moving towards each other)
The above integral can be solved with a trigonometric substitution: to get a very ugly answer
P.S: Thank you Mr.F. I did not see that trick coming