back down vs forget it

back down

verb
  • To take a less aggressive position in a conflict than one previously has or has planned to. 

  • To withdraw from a commitment or position. 

forget it

verb
  • I don't want to continue with this conversation (usually used in frustration or anger). 

  • Don't let it bother you; put it out of your mind. 

  • It doesn't matter. 

  • A conventional reply to an expression of gratitude. 

  • Absolutely not; I forbid it or it is impossible. 

How often have the words back down and forget it occurred in a corpus of books? (source: Google Ngram Viewer )