In Gmail, sending emails via Fastmail's SMTP protocol alters the header From:

In the Gmail settings, under "Accounts and Import", I added a new email address under the "Send mail as" setting. This email address using my own domain, foobar@example.com, lets say. I've set up Gmail for sending via SMTP servers from Fastmail.

This domain has been configured on Fastmail. The MX records were defined and I added the alias, asking Fastmail to transfer all emails to my @ gmail.com address.

Receiving mail at foobar@example.com works as expected, the mail appears in my Gmail inbox. Sending from the Gmail web interface as foobar@example.com also works. However, the recipient receives the email as it comes from my email address @ fastmail.com. How can I solve this problem?

differential equations – The increase of MaxExtraPrecision arbitrarily alters the numerical result

I'm trying to confirm that a function $ f $ satisfies a particular differential equation of the type $ D f = 0 $, for a differential operator $ D $. I put $ Df as Diffeq in Mathematica and I tried to check if that gives zero for any numerical value of the variables $ f $ depend on. I wrote

NOT[Difeqx /. x -> 3/4 /. y -> 1/5 , 500]

I had the following:
"N :: meprec: internal precision limit $ MaxExtraPrecision = 50.` reached during the evaluation [the function]
Outside[1]= -1,214057 * 10 ^ -613 –
6.579195 * 10 ^ -614 I ".

I then used

Block[{$MaxExtraPrecision = 550}, N[Difeqx /. x -> 3/4 /. y -> 1/5, 500]]

and I came back
"N :: meprec: internal precision limit $ MaxExtraPrecision = 550.` reached during evaluation [the function]
Outside[2]= 4.131621 * 10 ^ -1089 +
2.022562 * 10 ^ -1089 I "

The numerical result changes (and becomes smaller for this particular case) as I increase MaxExtraPrecision.

The question:
From the different numerical results obtained, I can conclude with some uncertainty that indeed $ Df = $ 0 (or Difeq = 0) is satisfied. But, can I clarify? How can I be safer than it actually gives zero?

P.S. I do not think that Chop, which returns zero, makes it safer that the result is precisely zero.