Can a multi-class Bard/Wizard write spells known from their Bard class, into their wizard spellbook?
e.g. they know the Comprehend Languages as a Bard spell, and want to write it directly into their wizard's spellbook?
And what is the cost? ... i.e. 1 hour and 10 gp because they know it, or 2 hours and 50 gp because they have to experiment with casting it as a wizard?
No, you can't write the spells you know as a Bard into your spellbook.
There are 2 ways (outside of levelling as a Wizard) that a Wizard can add spells to their spellbook. The first is by finding them:
When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a spell level you can prepare and if you can spare the time to decipher and copy it.
What does it mean to "find" a spell? Some examples are given immediately preceding this:
You might find other spells during your adventures. You could discover a spell recorded on a scroll in an evil wizard’s chest, for example, or in a dusty tome in an ancient library.
While this is not meant to be an exhaustive list, it's strongly suggestive that finding a spell means finding it written down. It certainly doesn't suggest that knowing it is the same as finding it.
The second method for Wizards to add spells to their spellbook is from their prepared spells.
If you lose your spellbook, you can use the same procedure to transcribe the spells that you have prepared into a new spellbook.
You don't prepare your Bard spells, so this method won't work either. Note that even if you did prepare Bard spells:
You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.
So even if you did prepare Bard spells, they wouldn't be considered to be prepared by the Wizard class and wouldn't be usable with Wizard class features, including scribing.
With all that said, you should be able to craft a spell scroll of Comprehend Languages as a Bard (Crafting a Magic Item, DMG pg. 128) and then copy it into your spellbook from there.
A wizard spell on a spell scroll can be copied just as spells in spellbooks can be copied.
However, both the rules on finding spells and the rules on spell scrolls agree that the spell must be a "wizard spell". Unfortunately, the rules never define what it means for a spell to be a wizard spell, or a bard spell, or any other class's spell. The best evidence I can find is this:
Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook for free.
The only interpretation I can come up with of this that makes any sense is that wizard spells just means spells on the wizard list. This would mean that spells common to both the can be scribed by a Bard and then copied by a Wizard.
This strategy is reliant on some ambiguous points, and requires explicit DM permission, since it's up to them whether you can craft items or not. There are also additional conditions to copying spell scrolls written in the DMG entry for spell scrolls.
At the time this question was initially asked and answered (August 2016), Xanathar's Guide to Everything had not been released (November 2017). This answer is the final paragraph from my recently revised answer to another question.
It's not without cost (time and money), but Xanathar's Guide to Everything provides rules for Scribing a Spell Scroll. Once the character has a scroll of a spell that exists on both lists, the character could use that scroll to add it to their wizard's spellbook. Just be sure to pay attention to the rules in the Dungeon Master's Guide for copying a scroll into a spellbook; there's a chance of wrecking the scroll. Of course, the character can just make a new one and try again.
