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I joined this amazing forum just today.I hope that my question will get answered soon.

So here it is.I am unable to understand a some steps in calculation. Please help me understand.

Here is a linear homogeneous first order differential equation

whose solution a research paper writes to be:

Now Integrating factor for solving the differential equation should have been:

And the final solution according to the standard procedure of solving linear homogeneous first order differential equation should have been:

But in the paper solution is not the same.Please guide me if I am wrong with my justifications.

Also in the final step delta function selects only s=t prime value.I am unable understand why theta function is coming.please elaborate on this step too.Any help will be deeply appreciated. I don't come from a maths background.