Resolve has many meanings, some of which are: 1. they are unable to resolve their differences [they cannot work out their differences] 2. He resolved to cut down on fatty food. [made a decision to / made a resolution to] 3.I resolve a conflict; I resolve doubts about [successfully bring an end to; clear up] Solve: I solve a problem; I solve the crossword puzzle. [I find an answer to] The police solved the murder; Sherlock Holmes solved the mystery of 'The Engineer's Thumb'. [find the person who committed the murder; clear up the mystery of]

You can SOLVE a problem or a mystery. And you can RESOLVE an argument.

John and Mary resolved their differences and are now a much happier married couple. "Resolved" is often used to mean "solved" when talking of problems*, but the standard word to use for *differences of opinion or goals in a marriage or a business is "resolve", not "solve". When one does simple math problems, eg 1 + 1 = x, one "solves" the problems but does not "resolve" them. "copied". Source: http://www.englishforums.com/English/WhatsDifferenceBetweenSolve-Resolve/nhpng/post.htm