Dirac delta function under integral?

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chempinaki
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Hello all,
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
Equation: d/dt (δx(t) / δf(t')) = W'(x) δx(t) / δf(t') + δ(t − t'), a functional derivative relation written with partial-style notation.


whose solution a research paper writes to be:
Equation: δx(t)/δf(t') = ∫ from 0 to t of ds exp[∫ ds W(x(s'))] δ(s − t') = Θ(t − t') exp[∫ from t' to t of ds W(x(s))], with integrals and delta and theta functions as printed.


Now Integrating factor for solving the differential equation should have been:
Mathematical expression: exp[-∫ from s' to t of ds W(x(s'))], an exponential of the negative of an integral of W(x(s')) over s.


And the final solution according to the standard procedure of solving linear homogeneous first order differential equation should have been:
Functional derivative formula: delta x(t) over delta f(t') equals a ratio. Numerator is an integral from 0 to t of ds exp of minus the integral from s to t of ds' W(x(s')), times delta(s - t'). Denominator is exp of minus the integral from t' to t of ds' W(x(s')).


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.
 

Discussion

somebody please address the issue.thanks in advance.i am utterly confused.
 

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