eigen
own
Expresses possession of or belonging to a specific person or thing.
Das ist mein eigenes Fahrrad, nicht das von meinem Bruder.
That is my own bike, not my brother's.

own / peculiar
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own
Expresses possession of or belonging to a specific person or thing.
Das ist mein eigenes Fahrrad, nicht das von meinem Bruder.
That is my own bike, not my brother's.

The word 'eigen' (own) is very often used directly after a possessive pronoun (e.g. mein, dein, ihr). In this position, it strengthens the idea of personal ownership and emphasizes exclusivity.
Viele junge Leute träumen von einer eigenen Wohnung.
Many young people dream of having their own apartment.
'Eigen' (own) is the root for words related to possession and character. These include 'das Eigentum' (property), 'die Eigenschaft' (quality/characteristic), and the verb 'sich eignen für' (to be suitable for).
Geduld ist eine wichtige Eigenschaft für diesen Beruf.
Patience is an important quality for this profession.
In the meaning of 'besonders' (special) or 'charakteristisch' (characteristic), 'eigen' (distinctive/peculiar) is often combined with abstract nouns. Typical collocations are 'ein eigener Stil' (a distinctive style), 'ein eigener Kopf' (a mind of one's own), or 'eine eigene Welt' (a world of its own).
Der Regisseur ist für seinen ganz eigenen visuellen Stil bekannt.
The director is known for his very distinctive visual style.
To express that something is inherent in a thing, a formal structure is often used: the noun appears in the dative, followed by 'eigen' (inherent). This construction is typical of academic or technical texts.
Die dem Gerät eigenen Sicherheitsrisiken wurden unterschätzt.
The safety risks inherent in the device were underestimated.
The expression 'sich etwas zu eigen machen' (to adopt something as one's own) has a fixed grammatical structure. The person adopting something is in the dative (sich), while the thing being adopted is in the accusative.
Er hat sich die Denkweise seines Mentors völlig zu eigen gemacht.
He completely adopted his mentor's way of thinking as his own.
my
'mein' and 'eigen' can both have to do with possession or belonging. 'mein' directly says that something belongs to me or is connected to me. 'eigen' emphasizes more strongly that something belongs to a particular person themselves and not to someone else. It often appears together with a possessive article, for example 'mein eigenes Zimmer' (my own room). Learners may confuse the words because both express possession. In normal sentences, 'mein' is often enough. 'eigen' is especially suitable when you want to emphasize independence, distinction from others, or something personal.

Seit dem Umzug hat Lena endlich ihr eigenes Zimmer.
Since the move, Lena finally has her own room.
Mein Laptop ist vor dem Termin noch nicht geladen.
My laptop still isn't charged before the appointment.