独自(dú zì) - alone, by oneself in Mandarin

独自 (dú zì) is an adverb, never an adjective. It describes how an action is done, so it sits directly in front of the verb: 独自旅行, 独自生活, 独自离开. You cannot say 我很独自.

It carries a faint literary weight and often a hint of solitude, which makes it a favourite in fiction and song lyrics. In plain conversation people usually say 一个人 instead, and 她一个人住 means exactly what 她独自住 means with none of the atmosphere.

The trap is 孤独, which looks close but is an adjective meaning lonely. 独自 says you were on your own; 孤独 says you felt it. A character can walk home 独自 and be perfectly content.