I solved this mathematics problem on my own. I solved this mathematics problem by myself.
我自己一個人解決這道數學難題。
on my own = by myself = alone, 表示自己單獨做，不假他人。
At night, she is afraid of going out on her own. At night, she is afraid of going out by herself.
晚上，她害怕她自己一個人出去。
on her own = by herself = alone, 表示她單獨一個人。
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