Jeg kan godt lide violinens lyd.

Because at kunne lide is an idiomatic Danish expression meaning to like. You should learn it as a set phrase, not by translating each word separately.

• kan normally means can
• godt normally means well
• lide is related to to like / to be fond of

But in this expression, the whole phrase works together as the usual way to say that you like something.

In this phrase, godt usually does not need a direct English translation. It often just makes the expression sound natural and idiomatic.

Compare:

• Jeg kan lide violinens lyd — I like the sound of the violin
• Jeg kan godt lide violinens lyd — also I like the sound of the violin, often a little more natural in everyday Danish

Sometimes godt can give a slightly softer or warmer tone, but very often it is just part of the standard expression.

Because kan is part of the fixed expression kan lide / kan godt lide.

So although kan usually means can, here it does not express ability. You are not saying I am able to like in a literal English sense. It simply forms the Danish way of saying I like.

Violinens is the genitive form of violinen, meaning the violin’s or of the violin.

Here is the pattern:

• en violin — a violin
• violinen — the violin
• violinens — the violin’s

So violinens lyd literally means the violin’s sound, which English often expresses as the sound of the violin.

Because in Danish, when a noun is determined by a possessive or genitive, it usually does not take its own definite ending.

So:

• violinens lyd — correct
• violinens lyden — incorrect

This is the same idea as in:

• mandens bil — the man’s car
• min bog — my book

The noun is already made definite by what comes before it, so you do not add another definite marker.

Yes. Lyden af violinen is also possible and means almost the same thing.

Compare:

• violinens lyd — the violin’s sound / the sound of the violin
• lyden af violinen — the sound of the violin

The version with the genitive, violinens lyd, is a bit more compact. The version with af can feel a little more analytic or descriptive. Both are understandable and natural.

Because Danish main clauses normally put the finite verb in second position.

The structure here is:

• Jeg — subject
• kan — finite verb
• godt — adverb
• lide — infinitive
• violinens lyd — object

After a modal verb like kan, the next verb usually stays in the infinitive:

• Jeg kan lide det
• Hun vil købe den
• Vi skal høre musik

So lide appears after kan, and the thing being liked comes after lide.

Yes. Jeg kan lide violinens lyd is grammatical.

But jeg kan godt lide violinens lyd is often more natural in everyday Danish. Many speakers use godt very frequently in this expression. So omitting it is possible, but including it often sounds more idiomatic.

The dictionary form is at lide.

In this sentence, it appears as the infinitive lide because it follows the modal verb kan.

Compare:

• at lide — infinitive
• jeg kan lide — after a modal verb
• jeg lide — incorrect by itself

So if you look it up, you would normally look for lide or at kunne lide.

It is grammatically closer to the violin’s sound, because Danish uses the genitive -s.

But in natural English, the sound of the violin is often the better translation. So the Danish structure is possessive, while the most natural English version may use of instead.