daku, idaku 抱く

In Japanese, daku 抱く and idaku 抱く are two different words written with the same kanji that have multiple, different meanings.

The word daku だく means "to embrace [someone]," "to hug [someone]," "to have sex [with someone]," and "(for a bird) to sit [on eggs]."

The word idaku いだく means "to embrace," "to hug," and, unlike daku だく, it can be used toward thoughts: to embrace an idea, to embrace a feeling.

･･･昔から毎日ケンカをふっかけてたのはニャル子の気を引きたかったから･･･だから今ここでニャル子を･･･抱く･････････････････････ハ　ハグじゃなくて？セックスの方でぎっ

Manga: Haiyore! Nyaruko-san 這いよれ！ニャル子さん (Chapter 5, ぼくらのうちゅうＣＱＣ)

Usage

To Hug

Both words daku and idaku can mean "to hug" someone. This is a transitive verb, so whom you hug is marked by the wo を particle.

Hanako-san wo daku
花子さんを抱く
To hug Hanako-san.
Tarou-san ni dakareru
太郎さんに抱かれる
To be hugged by Tarou-san. (passive sentence.)

Tarou ga Hanako wo daite-iru
太郎は花子を抱いている
Tarou is hugging Hanako. (te-iru form.)

Often, this verb will be in the tai-form.

Hanako-san wo dakitai
花子さんを抱きたい
[I] want to hug Hanako-san.
Tarou-san ni dakaretai
太郎さんに抱かれたい
[I] want to be hugged by Tarou-san.

It's also often accompanied by the auxiliary verb shimeru 締める, "to tighten," forming the compound verb:

daki-shimeru
抱きしめる
To hug tightly.
Hanako-san wo dakishimetai
花子さんを抱きしめたい
[I] want to hug Hanako-san tightly.

Another compound word:

daki-makura
抱き枕
Hugging pillow. A pillow made for hugging. The infamous so-called "anime pillow."

Himeji Mizuki 姫路瑞希, hugging a dakimakura 抱き枕 featuring a crossdressing Yoshii Akihisa 吉井明久.

Manga: Baka to Test to Shoukanjuu バカとテストと召喚獣

To Have Sex With

Often, daku 抱く is used as an euphemism, so that "to embrace (someone)" means "to make love with (someone)."

Manga: Haiyore! Nyaruko-san 這いよれ！ニャル子さん (Chapter 5, ぼくらのうちゅうＣＱＣ)

Context: Cthuko クー子 wants to "hug" Nyaruko ニャル子, if you know what she means.
...mukashi kara mainichi kenka wo fukkaketeta no wa Nyaruko no ki wo hikitakatta kara
･･･昔から毎日ケンカをふっかけてたのはニャル子の気を引きたかったから
...[I] picked fights with Nyaruko every day since always because [I] wanted [your] affection.
- ki wo hiku
  気を引く
  To attract someone's affection.
- fukkaketa
  ふっかけてた
  (a contraction of fukkakete-ita ふかっけていた.)
...dakara ima koko de Nyaruko wo... daku.........
･･･だから今ここでニャル子を･･･抱く･････････
...that's why right here and now [I'll] embrace... Nyaruko...
- dakara
  だから
  Given that. Because of that. That's why.
............ha, hagu janakute?
････････････ハ　ハグじゃなくて？
[Y... you don't mean] hug?
sekkusu no hou de
セックスの方で
[I mean] sex.
- hou
  方
  Direction. Way.
- There are two "ways" a word can be interpreted, and, in this case, she means it's to be interpreted "with the way of sex," literally.
gi'
ぎっ
*shriek*

To Sit On Eggs

A rarer usage is for a bird to sit an egg. I guess sitting on eggs is how birds hug eggs in Japanese. The grammar is identical:

tori ga tamago wo daku
鳥が卵を抱く
The bird sits on the egg.

To Feel

The verb idaku 抱く doesn't have the same meanings as daku 抱く. It doesn't mean "to have sex with someone," nor does it mean "to sit on eggs."

The verb idaku 抱く can mean "to hug [someone]."

Hanako-san wo idaku
花子さんを抱く
To hug Hanako-san.

The verb idaku 抱く can also mean "to embrace [a feeling]," "to hold a feeling in your heart," in other words, "to feel" something. For example:

gimon wo idaku
疑問を抱く
To embrace a doubt.
To feel doubt.
kibou wo idaku
希望を抱く
To embrace hope.
To feel hope.

History

The word daku is supposed to be the same word as idaku. More specifically:

mudaku
むだく
(became...)
udaku
うだく
(which became...)
idaku
いだく
(which became...)
daku
だく
To hug.

Along with idaku, other verbs that started with i~ い～ have lost the i い in modern Japanese. For example:

idzu
出づ (いづ)
(became...)
deru
出る (でる)
To leave.

inu
寝ぬ (いぬ)
(became...)
neru
寝る (ねる)
To sleep.

The shimo-nidan 下二段 verbs idzu いづ and inu いぬ stopped being used completely as they were replaced by the shimo-ichidan 下一段 verbs deru でる and neru ねる.

However, the yodan 四段 verb idaku いだく wasn't completely replaced by daku だく. Both are used in modern Japanese as godan 五段 verbs, except that, as we've seen already, their usage differs somewhat.