You can call the airline and ask for an emergency fare (at United, it's called a 'QRCARE' fare; an agent will know what that means). Two things:

Each airline has different rules as to what constitutes an "emergency"

While the terms of an emergency fare are flexible and allow changes and such, they are often not the cheapest. So, you might be able to find a cheaper, non-refundable fare either by calling or via a site such as Priceline or Hotwire.