"I have a pain in my ears."
Translation:У мене болять вуха.
That's a good question.
It just feels redundant to us, native speakers of Ukrainian. If you are to translate it literally from Ukrainian, it's more like "I have a pain in the ears", not just "in ears", because it's obvious from the context we refer back to our own ears. "У мене болять мої вуха" would sound oddly specific in Ukrainian, kind of like "I have a pain in my ears, which belong to me".
This is actually a general tendency when we refer to our body parts.
I wash my hands = Я мию руки (lit. I wash the hands).
He hurt his finger = Він ушкодив палець (lit. He hurt the finger)
Also true for other expressions:
I'm talking to my brother = Я розмовляю з братом (lit. I'm talking to the brother).
I love my parents = Я люблю батьків (lit. I love the parents)
In the last two examples it wouldn't be wrong to say "Я розмовляю зі своїм братом", Я люблю своїх батьків", it sounds natural with or without referring back to yourself with the word своїм-своїх.
Also, if you know a bit of French or Spanish, they do the same:
I have a pain in my back = Tengo dolor de espalda (spanish) = J'ai mal au dos (French), "my" is omitted in both languages.
If you need more examples please let me know.