Домашня тварина вдома.

In Ukrainian, the present-tense form of to be is often omitted.

So Домашня тварина вдома literally looks like Pet at home, but it naturally means The pet is at home or A pet is at home, depending on context.

You can add є sometimes, but it usually sounds more emphatic or marked:

• Домашня тварина вдома. = neutral, normal
• Домашня тварина є вдома. = more emphatic, like The pet is at home

Because домашня has to agree with тварина.

The noun тварина is:

• feminine
• singular
• nominative

So the adjective also has to be:

• feminine
• singular
• nominative

That gives:

• dictionary form: домашній
• feminine nominative singular: домашня

This is basic adjective-noun agreement in Ukrainian.

It is in the nominative singular.

In this sentence, домашня тварина is the subject, so nominative is exactly what you expect.

You can recognize it from the ending too:

• тварина = nominative singular
• genitive would be тварини
• accusative for an animate feminine noun would also be тварину

In modern Ukrainian, вдома is best learned as an adverb meaning at home.

So in this sentence:

• домашня тварина = subject
• вдома = adverb of place

It does not change for case, number, or gender.

Even though historically it comes from a form related to home/house, for learners it is easiest to treat вдома as a fixed word meaning at home.

Yes. Literally, it is something like domestic/home animal.

In everyday Ukrainian, домашня тварина is a very common way to say pet.

Depending on context, though, it can also sound broader than English pet. If you want to be very precise:

• домашня тварина often = pet
• свійська тварина = domesticated animal, often more like farm/domestic livestock

So in many beginner contexts, translating домашня тварина as pet is perfectly natural.

Ukrainian has no articles.

So Домашня тварина can mean:

• a pet
• the pet

Context tells you which one is meant.

This is very normal in Ukrainian, and English speakers usually need some time to get used to it.

Yes. Вдома and удома both mean at home.

This is part of a very common Ukrainian pattern where в/у alternate for smoother pronunciation.

So both are standard:

• Домашня тварина вдома.
• Домашня тварина удома.

The meaning does not change.

No, Ukrainian word order is fairly flexible.

Домашня тварина вдома is a normal, straightforward order. But you may also hear other orders if the speaker wants to emphasize something.

For example:

• Домашня тварина вдома. = neutral
• Вдома домашня тварина. = puts more focus on at home

Even when the order changes, the endings and context usually make the meaning clear.

Sometimes, but not always, because they are not exactly the same.

• вдома / удома = at home
• у домі = in the house/building

So:

• Домашня тварина вдома. means the pet is at home in the general sense.
• Домашня тварина у домі. sounds more like the pet is inside the house/building.

English often uses at home, and Ukrainian usually matches that with вдома/удома.

The dictionary forms are:

• домашній = domestic, home, pet-related
• тварина = animal
• вдома = at home

Notice that adjectives are usually listed in the masculine singular form, so домашня appears in dictionaries as домашній.

You would change the noun and adjective to plural:

• Домашні тварини вдома. = The pets are at home / Pets are at home

Here:

• домашня → домашні
• тварина → тварини

The adverb вдома stays the same.