"On one's own" and "by oneself"

Hi teachers:

Could you please tell me if we can use either of the options given in the brackets for the sentences given below? "On one's own" and "by oneself" have the same meaning, don't they?

- As usual, she was playing in the garden (on her own / by herself).

- There was nobody to help me, so I had to get the furniture inside my room (on my own / by myself).

Thank you!