Need help understanding the output of `DSolve`

Question

I give Mathematica 10 the following command:

DSolve[I D[P[x], {x, 2}] f[x] - (P'[x])^2 f[x] + 2 I P'[x] f'[x] == 0, P, x]

and I have the following output:

{{P -> Function[{x}, C[2] - I Log[C[1] - Integrate[-(I/f[K[1]]^2), {K[1], 1, x},
 Assumptions -> True]]]}}

What does Assumptions -> True mean? Which assumptions? I didn't make any assumption.

Answer 1

Assumptions -> True is the default for no assumptions. Setting this may seem unnecessary, but it overrides $Assumptions. One can see this in the following:

Integrate[Sin[k x], {x, 0, 2 Pi}]

Block[{$Assumptions = k ∈ Integers},
 Integrate[Sin[k x], {x, 0, 2 Pi}]
 ]

Block[{$Assumptions = k ∈ Integers},
 Integrate[Sin[k x], {x, 0, 2 Pi}, Assumptions -> True]
 ]
(*
  (2 Sin[k π]^2)/k
  0
  (2 Sin[k π]^2)/k
*)

So DSolve is protecting the solution it returned by using the option.

Ref: Usage of Assuming for Integration