The poem S.E. Hinton quoted in The Outsiders was by Robert Frost, and set to music for the movie. Incidentally, Hinton was only 16 years old when she wrote the book, and her use of Frost's poem was a comment on youth in general. Frost's poem can be interpreted as meaning that all things wither and must accept the changes that happen. Hinton's use of the poem can be understood as a mantra for believing that you can retain the quality of youth that leaves one vital and pure in spirit, come what may.
Just a few thoughts accrued from teaching the novel to many students over the years.

The poem is entitled "Nothing Gold Can Stay", and goes a little something like this:

Nature's first green is gold
Her hardest hue to hold
Her early leaf's a flower
but only so an hour
then leaf subsides to leaf
so eden sank to grief
and dawn goes down to day
nothing gold can stay.