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The equation of motion which I am meant to get at the end point is: (with ' denoting derivative w.r.t time)

φ'' + 3Hφ' + dV/dφ = 0

Using the inflaton lagrangian:

with the euler-lagrange equation. However the best I have managed to achieve through this is:

φ'' + dV/dφ + (1/2)(φ')^2 + (H^3)φ'= 0

I effectively have the potential term and the φ'' term correct but the φ' terms are clearly wrong. I am unsure how I would ever get 3H as the prefix for the φ' term as I always get (a'/a)^3 which = H^3.

There is a lot of information online but it all seems to state that you just use the lagrangian with the euler-lagrange equations and get the equation of motion out but nothing has the full derivation!

Thank you for any help!