Actually we will have two equations. One for each particle.

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:

Integrating:

Mr F edit in red. What follows might need a slight change.
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