Sending a man to the Moon (and back).

Hilbert once remarked half-jokingly that catching a fly on the Moon would be the most important technological achievement. "Why? "Because the auxiliary technical problems which would have to be solved for such a result to be achieved imply the solution of almost all the material difficulties of mankind." (Quoted from [*Hilbert-Courant*][1] by Constance Reid, Springer, 1986, p. 92). 


The task obviously required solving plenty of scientific and technological problems. But the key breakthrough that made it all possible was [Richard Arenstorf's][2]  discovery of a stable 8-shaped orbit between the Earth and the Moon. This involved the development of a numerical algorithm for solving the restricted three-body problem which is just a special non-linear second order ODE  (see also my answer to the previous [MO question][3]).  

Another orbit, also mapped by Arenstorf, was later used in the dramatic rescue of the Apollo 13 crew. 


  [1]: http://books.google.com.ua/books?id=pPNirfJL86MC&printsec=frontcover&dq=reid+hilbert+courant&source=bl&ots=fY4R_ClWcq&sig=ZbYvvAb7G9jwM9tjv2-yXy3uEgw&hl=en&ei=Ra9mTdutJYvrOc7jkZwL&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBIQ6AEwAA#v=onepage&q&f=false
  [2]: http://en.wikipedia.org/wiki/Richard_Arenstorf
  [3]: https://mathoverflow.net/questions/52489/on-the-non-rigorous-calculations-of-the-trajectories-in-the-moon-landings/52490#52490