When and why did Bilbo write a song about Aragorn

Bilbo wrote this song for Aragorn:

All that is gold does not glitter,

Not all those who wander are lost;

The old that is strong does not wither,

Deep roots are not reached by the frost.

From the ashes a fire shall be woken,

A light from the shadows shall spring;

Renewed shall be Blade that was Broken,

The crownless again shall be king.

But based on the timeline of Aragorn, Aragorn was only 10 years old when they first met in the year 2941. Was this song written in 2941, when they first met, or it was written sometime later? Are there any records of Bilbo meeting Aragorn after the events of The Hobbit and before the events of The Lord of the Rings?

I am just curious about why anybody would write a song for a ten-year-old kid.