For $x$ real I have a real function $F(x)$ (equal to a function I know) 
and a differential
equation $G$ with some complex functions for which $\Re(G)=0$
and the imaginary part doesn't necessarily vanish.

desolve (even with contrib_ode=True) fails to solve real(G),F(x),
though it solves  G,F(x) and when substituted the solution vanishes
(including the imaginary part). The solution has indefinite integrals
which I suppose are hopeless to solve.
Using assumptions, the solution has realpart and imagpart.

1. Is there some trick/other CAS to solve only for real()?
2. Can I numerically verify that the real part of the solution 
equals the known F(x) (this might not hold)?

Maple 13 claims to solve it with real(), but I don't get numerical
support for the solution (assuming have done it correctly).

Have numerical support that the differential equation holds for
the known function.

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